Topological Free Entropy Dimension
in Unital C∗-algebras
Don Hadwin and Junhao Shen1
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824
Email: don@math.unh.edu and jog2@cisunix.unh.edu
Abstract: The notion of topological free entropy dimension of n-tuple of elements in a unital
C∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy
dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint
element in a unital C∗ algebra. We also calculate the values of topological free entropy dimensions
of any families of self-adjoint generators of some unital C∗ algebras, including irrational rotation
C∗ algebra, UHF algebra, and minimal tensor product of two reduced C∗ algebras of free groups.
Keywords: Topological free entropy dimension, C∗ algebra
2000 Mathematics Subject Classification: Primary 46L10, Secondary 46L54
1. Introduction
The theory of free probability and free entropy was developed by Voiculescu from 1990s. It
played a crucial role in the recent study of finite von Neumann algebras (see [1], [3], [4], [5], [6],
[7], [8], [11], [14], [15], [23], [24], [25]). The analogue of free entropy dimension in C∗ algebra
context, the notion of topological free entropy dimension of of n−tuple of elements in a unital
C∗ algebra, was also introduced by Voiculescu in [26].
After introducing the concept of topological free entropy dimension of n-tuple of elements
in a unital C∗ algebra, Voiculescu discussed some of its properties including subadditivity and
change of variables in [26]. In this paper, we will add one basic property into the list: topological
free entropy dimension of one variable. More specifically, suppose x is a self-adjoint element in a
unital C∗ algebra A and σ(x) is the spectrum of x in A. Then topological free entropy dimension
of x is equal to 1− 1
where n is the cardinality of the set σ(x) (see Theorem 4.1).
In [26], Voiculescu showed that (i) if x1, . . . , xn is a family of free semicircular elements
in a unital C∗ algebra with a tracial state, then δtop(x1, . . . , xn) = n, where δtop(x1, . . . , xn) is
the topological free entropy dimension of x1, . . . , xn; (ii) if x1, . . . , xn is the universal n-tuple
of self-adjoint contractions, then δtop(x1, . . . , xn) = n. Except in these two cases, very few has
been known on the values of topological free entropy dimensions in other C∗ algebras. Using the
inequality between topological free entropy dimension and Voiculescu’s free dimension capacity,
we are able to obtain an estimation of upper-bound of topological free entropy dimension for a
1The second author is supported by an NSF grant.
http://arxiv.org/abs/0704.0667v3
unital C∗ algebra with a unique tracial state (see Theorem 5.1). The lower-bound of topological
free entropy dimension is also obtained for infinite dimensional simple unital C∗ algebra with a
unique tracial state (see Theorem 5.2). As a corollary, we know that the topological free entropy
dimension of any family of self-adjoint generators of an irrational rotation C∗ algebra or UHF
algebra or C∗red(F2)⊗minC∗red(F2) is equal to 1 (see Theorem 5.3, 5.4, 5.5). For these C∗ algebras,
the value of the topological free entropy dimension is independent of the choice of generators.
The rest of the paper is devoted to study another invariant associated to n-tuple of elements
in C∗ algebras. This invariant, called topological free orbit dimension, is an analogue of free
orbit dimension in finite von Neumann algebras (see [11]). We show that the topological free
orbit dimension of a self-adjoint element in a unital C∗ algebra is equal to, according to some
measurement, the packing dimension of the spectrum of x (see Theorem 7.1).
The organization of the paper is as follows. In the section 2, we recall the definition of
topological free entropy dimension. Some technical lemmas are proved in section 3. In section
4, we compute the topological free entropy dimension of one self-adjoint element in a unital
C∗ algebra. In section 5, we study the relationship between topological free entropy dimension
and free capacity dimension of a unital C∗ algebra. Then we show that topological free entropy
dimension of of any family of generators of an infinite dimensional simple unital C∗ algebra with
a unique tracial state is always greater than or equal to 1. The concept of topological free orbit
dimension of n-tuple of elements in a C∗ algebra is introduced in section 6. Its value for one
variable is computed in section 7.
2. Definitions and preliminary
In this section, we are going to recall Voiculescu’s definition of topological free entropy
dimension of n-tuple of elements in a unital C∗ algebra.
2.1. A Covering of a set in a metric space. Suppose (X, d) is a metric space and K is
a subset of X . A family of balls in X is called a covering of K if the union of these balls covers
K and the centers of these balls lie in K.
2.2. Covering numbers in complex matrix algebra (Mk(C))n. Let Mk(C) be the
k × k full matrix algebra with entries in C, and τk be the normalized trace on Mk(C), i.e.,
Tr, where Tr is the usual trace on Mk(C). Let U(k) denote the group of all unitary
matrices in Mk(C). Let Mk(C)n denote the direct sum of n copies of Mk(C). Let Ms.ak (C) be
the subalgebra of Mk(C) consisting of all self-adjoint matrices of Mk(C). Let (Ms.ak (C))n be
the direct sum of n copies of Ms.ak (C). Let ‖ · ‖ be an operator norm on Mk(C)n defined by
‖(A1, . . . , An)‖ = max{‖A1‖, . . . , ‖An‖}
for all (A1, . . . , An) in Mk(C)n. Let ‖ · ‖2 denote the trace norm induced by τk on Mk(C)n, i.e.,
‖(A1, . . . , An)‖2 =
1A1) + . . .+ τk(A
for all (A1, . . . , An) in Mk(C)n.
For every ω > 0, we define the ω-‖ · ‖-ball Ball(B1, . . . , Bn;ω, ‖ · ‖) centered at (B1, . . . , Bn)
in Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that
‖(A1, . . . , An)− (B1, . . . , Bn)‖ < ω.
Definition 2.1. Suppose that Σ is a subset of Mk(C)n. We define the covering number ν∞(Σ, ω)
to be the minimal number of ω-‖ · ‖-balls that consist a covering of Σ in Mk(C)n.
For every ω > 0, we define the ω-‖·‖2-ball Ball(B1, . . . , Bn;ω, ‖·‖2) centered at (B1, . . . , Bn)
in Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that
‖(A1, . . . , An)− (B1, . . . , Bn)‖2 < ω.
Definition 2.2. Suppose that Σ is a subset of Mk(C)n. We define the covering number ν2(Σ, ω)
to be the minimal number of ω-‖ · ‖2-balls that consist a covering of Σ in Mk(C)n.
2.3. Noncommutative polynomials. In this article, we always assume that A is a unital
C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint elements inA. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉
be the unital noncommutative polynomials in the indeterminates X1, . . . , Xn, Y1, . . . , Ym. Let
{Pr}∞r=1 be the collection of all noncommutative polynomials in C〈X1, . . . , Xn, Y1, . . . , Ym〉 with
rational complex coefficients. (Here “rational complex coefficients” means that the real and
imaginary parts of all coefficients of Pr are rational numbers).
Remark 2.1. We alsways assume that 1 ∈ C〈X1, . . . , Xn, Y1, . . . , Ym〉.
2.4. Voiculescu’s Norm-microstates Space. For all integers r, k ≥ 1, real numbers
R, ǫ > 0 and noncommutative polynomials P1, . . . , Pr, we define
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr)
to be the subset of (Ms.ak (C))n+m consisting of all these
(A1, . . . , An, B1, . . . , Bm) ∈ (Ms.ak (C))n+m
satisfying
max{‖A1‖, . . . , ‖An‖, ‖B1‖, . . . , ‖Bm‖} ≤ R
|‖Pj(A1, . . . , An, B1, . . . , Bm)‖ − ‖Pj(x1, . . . , xn, y1, . . . , ym)‖| ≤ ǫ, ∀ 1 ≤ j ≤ r.
Remark 2.2. In the definition of norm-microstates space, we use the following assumption. If
Pj(x1, . . . , xn, y1, . . . , ym) = α0 · IA +
1≤i1,...,is≤n+m
αi1···iszi1 · · · zis
where z1, . . . , zn+m denotes x1, . . . , xn, y1, . . . , ym and α0, αi1···is are in C, then
Pj(A1, . . . , An, B1, . . . , Bm) = α0 · Ik +
1≤i1,...,is≤n+m
αi1···isZi1 · · ·Zis
where Z1, . . . , Zn+m denotes A1, . . . , An, B1, . . . , Bm and Ik is the identity matrix in Mk(C).
Remark 2.3. In the original definition of norm-microstates space in [26], the parameter R was
not introduced. Note the following observation: Let R > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖}.
When r is large enough so that
{X1, . . . , Xn, Y1, . . . , Ym} ⊂ {P1, . . . , Pr}
and 0 < ǫ < R−max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖}, we have
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) = Γtop(x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr)
for all k ≥ 1, where Γ(top)(x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) is the norm-microstates space
defined in [26]. Thus our definition agrees with the one in [26] for large R, r and small ǫ.
In the later sections, we need to construct the ultraproduct of some matrix algebras, it will
be convenient for us to include the parameter “R” in the definition of norm-microstate space.
Define the norm-microstates space of x1, . . . , xn in the presence of y1, . . . , ym, denoted by
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
as the projection of Γ
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) onto the space (Ms.ak (C))n via
the mapping
(A1, . . . , An, B1, . . . , Bm) → (A1, . . . , An).
2.5. Voiculescu’s topological free entropy dimension (see [26]). Define
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω)
to be the covering number of the set Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr) by ω-‖ · ‖-balls
in the metric space (Ms.ak (C))n equipped with operator norm.
Definition 2.3. Define
δtop(x1, . . . , xn : y1, . . . , ym;ω)
= sup
ǫ>0,r∈N
lim sup
log(ν∞(Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 log ω .
The topological entropy dimension of x1, . . . , xn in the presence of y1, . . . , ym is defined by
δtop(x1, . . . , xn : y1, . . . , ym) = lim sup
δtop(x1, . . . , xn : y1, . . . , ym;ω).
Remark 2.4. Let M > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖} be some positive number. By
Remark 2.3, we know
δtop(x1, . . . , xn : y1, . . . , ym)
= lim sup
ǫ>0,r∈N
lim sup
log(ν∞(Γ
(top)
M (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 logω .
2.6. C∗ algebra ultraproduct and von Neumann algebra ultraproduct. Suppose
{Mkm(C)}∞m=1 is a sequence of complex matrix algebras where km goes to infinity when m
approaches infinity. Let γ be a free ultrafilter in β(N)\N. We can introduce a unital C∗ algebra
m=1Mkm(C) as follows:
Mkm(C) = {(Ym)∞m=1 | ∀ m ≥ 1, Ym ∈ Mkm(C) and sup
‖Ym‖ <∞}.
We can also introduce the norm closed two sided ideals I∞ and I2 as follows.
I∞ = {(Ym)∞m=1 ∈
Mkm(C) | lim
‖Ym‖ = 0}
I2 = {(Ym)∞m=1 ∈
Mkm(C) | lim
‖Ym‖2 = 0}
Definition 2.4. The C∗ algebra ultraproduct of {Mkm(C)}∞m=1 along the ultrfilter γ, denoted
m=1Mkm(C), is defined to be the quotient algebra of
m=1Mkm(C) by the ideal I∞. The
image of (Ym)
m=1 ∈
m=1Mkm(C) in the quotient algebra is denoted by [(Ym)m].
Definition 2.5. The von Neumann algebra ultraproduct of {Mkm(C)}∞m=1 along the ultrfilter
γ, also denoted by
m=1Mkm(C) if no confusion arises, is defined to be the quotient algebra of
m=1Mkm(C) by the ideal I2. The image of (Ym)∞m=1 ∈
m=1Mkm(C) in the quotient algebra
is denoted by [(Ym)m].
Remark 2.5. The von Neumann algebra ultraproduct
m=1Mkm(C) is a finite factor (see
[16]).
2.7. Topological free entropy dimension of elements in a non-unital C∗ algebra.
Topological free entropy dimension can also be defined for n-tuple of elements in a non-unital
C∗ algebra. Suppose that A is a non-unital C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint
elements in A. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉 ⊖ C be the noncommutative polynomials in the
indeterminates X1, . . . , Xn, Y1, . . . , Ym without constant terms. Let {Pr}∞r=1 be the collection
of all noncommutative polynomials in C〈X1, . . . , Xn, Y1, . . . , Ym〉 ⊖ C with rational complex
coefficients. Then norm-mocrostate space
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
can be defined similarly as in section 2.4. So topological free entropy dimension
δtop(x1, . . . , xn : y1, . . . , ym)
can also be defined similarly as in section 2.5.
In the paper, we will focus on the case when A is a unital C∗ algebra.
3. Some technical lemmas
3.1. Suppose x is a self-adjoint element in a unital C∗ algebra A. Let σ(x) be the spectrum
of x in A.
Theorem 3.1. Let R > ‖x‖. For any ω > 0, we have the following.
(1) There are some integer n ≥ 1 and distinct real numbers λ1, λ2, · · · , λn in σ(x) satisfying
(i) |λi−λj | ≥ ω for all 1 ≤ i 6= j ≤ n; and (ii) for any λ in σ(x), there is some λj with
1 ≤ j ≤ n such that |λ− λj| ≤ ω.
(2) There are some r0 > 0 and ǫ0 > 0 such that the following holds: when r > r0, ǫ < ǫ0,
for any A in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), there are positive integers 1 ≤ k1, . . . , kn ≤ k with
k1 + k2 + · · ·+ kn = k and some unitary matrix U in Mk(C) satisfying
‖U∗AU −
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λnIkn
‖ ≤ 2ω,
where Ikj is the kj × kj identity matrix in Mkj (C) for 1 ≤ j ≤ n.
Proof. The proof of part (1) is trivial. We will only prove part (2). Assume that the result
in (2) does not hold. Then there is some ω > 0 so that the following holds: for all m ≥ 1, there
are km ≥ 1 and some Am in Γ(top)R (x; km, 1m , P1, . . . , Pm) such that
‖U∗AmU −
λ1Is1 0 · · · 0
0 λ2Is2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λnIsn
‖ > 2ω, (∗)
for every 1 ≤ s1, . . . , sn ≤ km with s1 + · · ·+ sn = km and every unitary matrix U in Mkm(C).
Let γ be a free ultrafilter in β(N)\N. Let B =
m=1 Mkm(C) be the C∗ algebra ultraproduct
of {Mkm(C)}∞m=1 along the ultrafilter α, i.e.
m=1 Mkm(C) is the quotient algebra of the
C∗ algebra
m=1Mkm(C) by I∞, the 0-ideal of the norm ‖ · ‖, where I∞ = {(Am)∞m=1 ∈
Mkm(C) | limm→γ ‖Am‖ = 0}. Let a = [(U∗AmU)∞m=1] be a self-adjoint element in B. By
mapping x to a, there is a unital ∗-isomorphism from the C∗ subalgebra generated by {IA, x}
in A onto the C∗ subalgebra generated by {IB, a} in B. Thus σ(x) = σ(a). It is not hard to see
that Hausdorff-dist(σ(U∗AmU), σ(a)) → 0 as m goes to γ, which contradicts with the results in
part (1) and (∗).
3.2. In this subsection, we will use the following notation.
(i) Let n,m be some positive integers with n ≥ m.
(ii) Let δ, θ be some positive numbers.
(iii) Let {λ1, λ2, . . . , λm} ∪ {λm+1, . . . , λn} be a family of real numbers such that
|λi − λj | ≥ θ for all 1 ≤ i < j ≤ m.
(iv) Let k be a positive integer such that k − (n−m) is divided by m. We let
k − n +m
(v) We let
B = diag(λm+1, . . . , λn)
be a diagonal matrix in Mn−m(C) and
A = diag(λ1It, λ2It, . . . , λmIt, B)
be a block-diagonal matrix in Mk(C), where It is the identity matrix in Mt(C).
(vi) We let A be defined as above and
Ω(A) = {U∗AU | U is in U(k)}.
(vii) Assume that {eij}ki,j=1 is a canonical basis of Mk(C). We let
V1 = span{eij | |λ[ i
]+1 − λ[ j
]+1| ≥ θ, with 1 ≤ i, j < mt}; and
V2 = Mk(C)⊖ V1,
where [ i
], or [ j
], denotes the largest integer ≤ [ i
], or [ j
] respectively.
Lemma 3.1. We follow the notation as above. Suppose ‖U1AU∗1 − U2AU∗2‖2 ≤ δ for some
unitary matrices U1 and U2 in U(k). Then the following hold.
(1) There exists some S ∈ V2 such that ‖S‖2 ≤ 1 and
‖U1 − U2S‖2 ≤
(2) If n = m, then there is a unitary matrix W in V2 such that
‖U1 − U2W‖2 ≤
Proof. Assume that
U∗2U1 =
U11 U12 · · · U1,m+1
U21 U22 · · · U2,m+1
· · · · · · · · · · · ·
Um+1,1 Um+1,2 · · · Um+1,m+1
where Ui,j is a t × t matrix, Ui,m+1 a t × (n − m) matrix, Um+1,j a (n − m) × t matrix for
1 ≤ i, j ≤ m and Um+1,m+1 is a (n−m)× (n−m) matrix.
(1) Let
U11 0 · · · 0 U1,m+1
0 U22 · · · 0 U2,m+1
· · · · · · . . . · · · · · ·
0 0 · · · Um,m Um,m+1
Um+1,1 Um+1,2 · · · Um+1,m Um+1,m+1
It is easy to see that S is in V2, ‖S‖2 ≤ 1 and
δ2 ≥ ‖U1AU∗1 − U2AU∗2‖22 =
Tr((U∗2U1A− AU∗2U1)(U∗2U1A− AU∗2U1)∗)
1≤i 6=j≤m
Tr(|λi − λj|2UijU∗ij)
1≤i 6=j≤m
Tr(UijU
Hence
‖U1 − U2S‖22 = ‖U∗2U1 − S‖22 =
1≤i 6=j≤m
Tr(UijU
ij) ≤
It follows that
‖U1 − U2S‖2 ≤
(2) If n = m, then
V2 = Mt(C)⊕Mt(C)⊕ · · · ⊕Mt(C).
By the construction of S, we can assume S =WH is a polar decomposition of S in V2 for some
unitary matrix W and positive matrix H in V2. Again by the construction of S, we know that
‖S‖ ≤ 1, whence ‖H‖ ≤ 1. From the proven fact that ‖U∗2U1 − S‖2 ≤ δθ , we know that
‖H2 − I‖2 = ‖S∗S − I‖2 ≤
‖H − I‖2 ≤ ‖H2 − I‖2 ≤
It follows that
‖U1 − U2W‖2 ≤ ‖U1 − U2WH‖2 + ‖U2WH − U2W‖2 = ‖U1 − U2S‖2 + ‖H − I‖2 ≤
Lemma 3.2. We have the following results.
(1) For every U ∈ U(k), let
Σ(U) = {W ∈ U(k) | ∃ S ∈ V2 such that ‖S‖2 ≤ 1 and ‖W − US‖2 ≤
Then the volume of Σ(U) is bounded above by
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)2mt2+4m(n−m)t+2(n−m)2
where µ is the normalized Haar measure on the unitary group U(k) and C,C1 are some
constants independent of k, δ, θ.
(2) When n = m, for every U ∈ U(k), let
Σ̃(U) = {W ∈ U(k) | ∃ a unitary matrix W1 in V2 such that ‖W − UW1‖2 ≤
µ(Σ̃(U)) ≤ (C1 · 8δ/θ)k
Proof. (1) By computing the covering number of the set {S | S ∈ V2, such that ‖S‖2 ≤ 1}
by δ/θ-‖ · ‖2-balls in Mk(C), we know
ν2({S | S ∈ V2, ‖S‖2 ≤ 1},
)real dimension of of V2
)2mt2+4m(n−m)t+2(n−m)2
where C is a universal constant. Thus the covering number of the set Σ(U) by the 4δ/θ-‖·‖2-balls
in Mk(C) is bounded by
ν2(Σ(U),
) ≤ ν2({S | S ∈ V2, ‖S‖2 ≤ 1},
)2mt2+4m(n−m)t+2(n−m)2
But the ball of radius 4δ/θ in U(k) has the volume bounded by
µ(ball of radius 4δ/θ) ≤ (C1 · 4δ/θ)k
where C1 is a universal constant. Thus
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)2mt2+4m(n−m)t+2(n−m)2
(2) A slight adaption of the proof of part (1) gives us the proof of part (2). �
Lemma 3.3. Let Ω(A) be defined as in (vi) at the beginning of this subsection.
(1) The covering number of Ω(A) by the 1
δ-‖ · ‖2-balls in Mk(C) is bounded below by
ν2(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
(2) If n = m, then
ν2(Ω(A),
δ) ≥ (C1 · 8δ/θ)−k
)−mt2
Proof. (1) For every U ∈ U(k), define
Σ(U) = {W ∈ U(k) | ∃ S = S∗ ∈ V1, such that ‖S‖2 ≤ 1, ‖W − US‖2 ≤
By preceding lemma, we have
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)mt2+4m(n−m)t+2(n−m)2
A “parking” (or exhausting) argument will show the existence of a family of unitary elements
{Ui}Ni=1 ⊂ U(k) such that
N ≥ (C1 · 4δ/θ)−k
)−(mt2+4m(n−m)t+2(n−m)2)
Ui is not contained in ∪i−1j=1 Σ(Uj), ∀ i = 1, . . . , N.
From the definition of each Σ(Uj), it follows that
‖Ui − UjS‖2 >
, ∀ S ∈ V2, with ‖S‖2 ≤ 1, ∀1 ≤ j < i ≤ N.
By Lemma 3.1, we know that
‖UiAU∗i − UjAU∗j ‖2 > δ, ∀1 ≤ j < i ≤ N,
which implies that
ν2(Ω(A),
δ) ≥ N ≥ (C1 · 4δ/θ)−k
)−(mt2+4m(n−m)t+2(n−m)2)
(2) is similar as (1). �
3.3. We have following theorem.
Theorem 3.2. Let n ≥ m, δ, θ > 0 and {λ1, λ2, . . . , λm} ∪ {λm+1, . . . , λn} be a family of real
numbers such that
|λi − λj| ≥ θ
for all 1 ≤ i < j ≤ m. Let k be a positive integer such that k − (n−m) is divided by m and
k − n +m
B = diag(λm+1, . . . , λn)
be a diagonal matrix in Mn−m(C) and
A = diag(λ1It, λ2It, . . . , λmIt, B)
be a block-diagonal matrix in Mk(C), where It is the identity matrix in Mt(C). We let
Ω(A) = {U∗AU | U is in U(k)}.
Then the covering number of Ω(A) by the 1
δ-‖ · ‖-balls in Mk(C) is bounded below by
ν∞(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
where C,C1 are some universal constants.
When n = m, we have
ν∞(Ω(A),
δ) ≥ (C1 · 8δ/θ)−k
)−mt2
Proof. Note that
ν∞(Ω(A),
) ≥ ν2(Ω(A),
), ∀ δ > 0.
The result follows directly from preceding lemma. �
3.4. The following proposition, whose proof is skipped, is an easy extension of Lemma 3.3.
Proposition 3.1. Let m, k be some positive integers and θ, δ be some positive numbers. Let
T1, T2, . . . , Tm+1 is a partition of the set {1, 2, . . . , k}, i.e. ∪m+1i=1 Ti = {1, 2, . . . , k} and Ti∩Tj = ∅
for 1 ≤ i 6= j ≤ m+ 1. Let λ1, . . . , λk be some real numbers such that, if 1 ≤ j1 6= j2 ≤ m then
|λi1 − λi2 | > θ, ∀ i1 ∈ Tj1, i2 ∈ Tj2 .
Let A = diag(λ1, λ2, . . . , λk) be a self-adjoint matrix in Mk(C) and
Ω(A) = {U∗AU | U ∈ U(k)}
be a subset of Mk(C).
Let sj be the cardinality of the set Tj for 1 ≤ j ≤ m+ 1. Then the covering number of Ω(A)
by the 1
δ-‖ · ‖2-balls in Mk(C) is bounded below by
ν2(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−2s2
−···−2s2m+1−4(s1+···+sm)sm+1
where C,C1 are some universal constants.
4. Topological free entropy dimension of one variable
Suppose x is a self-adjoint element of a unital C∗ algebra A. In this section, we are going to
compute the topological entropy dimension of x.
4.1. Upperbound.
Proposition 4.1. Suppose x in A is a self-adjoint element with the spectrum σ(x). Then
δtop(x) ≤ 1−
where n is the cardinality of σ(x). Here we assume that 1
Proof. By [26], we know that the inequality always holds when n is infinity. We need only
to show that
δtop(x) ≤ 1−
when n <∞.
Assume that λ1, . . . , λn are in the spectrum of x in A.
Let R > ‖x‖. By Theorem 3.1, for every ω > 0, there are r0 > 0 and ǫ0 > 0 such that, for
all r > r0, ǫ < ǫ0,
A ∈ Γ(top)R (x; k, ǫ, P1, . . . , Pr),
there are some 1 ≤ k1, . . . , kn ≤ k, with k1 + · · · + kn = k and a unitary matrix U in Mk(C)
satisfying
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
0 0 · · · λnIkn
≤ 2ω. (∗∗)
Ω(k1, . . . , kn) =
λ1Ik1 0 · · · 0 0
0 λ2Ik2 · · · 0 0
0 0 · · · λn−1Ikn−1 0
0 0 · · · 0 λnIkn
U∗ | U is in Uk
By Corollary 12 in [21] or Theorem 3 in [2], the covering number of Ω(k1, . . . , kn−1, kn) by
ω-‖ · ‖-balls in Mk(C) is upperbounded by
ν∞(Ω(k1, . . . , kn−1, kn), ω) ≤
where C2 is a constant which does not depend on k, k1, . . . , kn (may depend on n and ‖x‖).
Let I be the set consisting of all these (k1, . . . , kn) in Zn such that 1 ≤ k1, . . . , kn ≤ k and
k1 + · · ·+ kn = k. Then the cardinality of the set I is equal to
(k − 1)!
(n− 1)!(k − n)! .
Note that
k2i ≥ k2/n
for all 1 ≤ k1, . . . , kn ≤ k with k1 + · · ·+ kn = k; and by (∗∗)
(top)
R (x; k, ǫ, P1, . . . , Pr)
is contained in 2ω-neighborhood of the set
(k1,...,kn)∈I
Ω(k1, . . . , kn).
It follows that the covering number of the set
(top)
R (x; k, ǫ, P1, . . . , Pr)
by 3ω-‖ · ‖-balls in Mk(C) is upperbounded by
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω) ≤
(k − 1)!
(n− 1)!(k − n)! ·
)k2−k2/n
δtop(x) ≤ lim sup
lim sup
(k−1)!
(n−1)!(k−n)!
)k2−k2/n
−k2 log(3ω) = 1−
4.2. Lower-bound. We follow the notation from last subsection.
Proposition 4.2. Suppose that x is a self-adjoint element with the finite spectrum σ(x) in A.
δtop(x) ≥ 1−
where n is the cardinality of the set σ(x).
Proof. Suppose that λ1, . . . , λn are distinct spectrum of x. There is some positive number
θ such that
|λi − λj| > θ, ∀ 1 ≤ i 6= j ≤ n.
Assume k = nt for some positive integer t. Let
Ak = diag(λ1It, . . . , λnIt)
be a diagonal matrix in Mk(C) where It is the t× t identity matrix. It is easy to see that, for
all R > ‖x‖, r ≥ 1 and ǫ > 0, we have
Ak ∈ Γ(top)R (x; k, ǫ, P1, . . . , Pr).
For any ω > 0, applying Theorem 3.2 for n = m and δ = 1
ω, we have
(top)
R (x; k, ǫ, P1, . . . , Pr), ω) ≥ (C1 · 8δ/θ)−k
)−mt2
= (16C1ω/θ)
−k2 ·
)−mt2
Note that k = nt = mt and θ is some fixed number. A quick computation shows that
δtop(x) ≥ 1−
Proposition 4.3. Suppose that x is a self-adjoint element in A with infinite spectrum. Then
δtop(x) ≥ 1.
Proof. For any 0 < θ < 1, there are λ1, . . . , λm in the spectrum of x, σ(x), satisfying (i)
|λi − λj | ≥ θ;
and (ii) for any λ in σ(x), there is some λj with |λ − λj| ≤ θ. By functional calculus, for any
R > ‖x‖, r ≥ 1 and ǫ > 0, there are some positive integer n ≥ m and real numbers λm+1, . . . , λn
in σ(x) satisfying: for every t ≥ 1 the matrix
A = diag(λ1It, λ2It, . . . , λmIt, λm+1, . . . , λn)
is in
(top)
R (x; k, ǫ, P1, . . . , Pr),
where we assume that k = mt + n−m. For any ω > 0, let δ = 1
ω. By Theorem 3.2, we know
(top)
R (x; k, ǫ, P1, . . . , Pr), ω) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
lim sup
log(ν∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), ω))
−k2 logω ≥
log(4C1
)− log θ
+ 1 +
log(2C) + log θ
log ω
Then,
δtop(x) ≥ 1−
When θ goes to 0, m goes to infinity as σ(x) has infinitely many elements. Therefore,
δtop(x) ≥ 1.
4.3. Topological free entropy dimension in one variable case. By Proposition 4.1,
Proposition 4.2 and Proposition 4.3, we have the following result.
Theorem 4.1. Suppose x is a self-adjoint element in a unital C∗ algebra A. Then
δtop(x) = 1−
where n is the cardinality of the set σ(x) and σ(x) is the set of spectrum of x in A. Here we
assume that 1
5. Topological free entropy dimension of n-tuple in unital C∗ algebras
5.1. An equivalent definition of topological free entropy dimension. Suppose that
A is a unital C∗ algebra and x1, . . . , xn, y1, . . . , ym are self-adjoint elements in A. For every
R, ǫ > 0 and positive integers r, k, let
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
be Voiculescu’s norm-microstate space defined in section 2.4.
Define
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω)
to be the covering number of the set Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr) by ω-‖·‖2-balls
in the metric space (Ms.ak (C))n equipped with trace norm (see Definition 2.2).
Definition 5.1. Define
δ̃top(x1, . . . , xn : y1, . . . , ym;ω)
= sup
ǫ>0,r∈N
lim sup
log(ν2(Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 logω
δ̃top(x1, . . . , xn : y1, . . . , ym) = lim sup
δ̃top(x1, . . . , xn : y1, . . . , ym;ω)
The following proposition was pointed out by Voiculescu in [26]. For the sake of complete-
ness, we also include a proof here.
Proposition 5.1. Suppose that A is a unital C∗ algebra and x1, . . . , xn, y1, . . . , ym are self-
adjoint elements in A. Then
δ̃top(x1, . . . , xn : y1, . . . , ym) = δtop(x1, . . . , xn : y1, . . . , ym),
where δtop(x1, . . . , xn : y1, . . . , ym) is the topological free entropy dimension of x1, . . . , xn in
presence of y1, . . . , ym.
Proof. This is an easy consequence of Lemma 1 in [21]. Let λ be the Lebesgue measure
on (Ms.ak (C))n. Let, for every ω > 0,
B∞(ω) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)‖ ≤ ω}
B2(ω) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)‖2 ≤ ω}
It follows from the results in [21] or Theorem 8 in [2] that, for some M1,M2 independent of k, ω
such that
λ(B∞(1)) ≤ λ(B∞(ω/4))
λ(B2(2
nω)) ≤ λ(B2(1)). (5.1.1)
For every ω > 0 and any subset set K of (Ms.ak (C))n , let
K(ω, ‖ · ‖) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)− (D1, . . . , Dn)‖ ≤ ω
for some (D1, . . . , Dn) ∈ K}
K(ω, ‖ · ‖2) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)− (D1, . . . , Dn)‖2 ≤ ω
for some (D1, . . . , Dn) ∈ K}
Note the following fact:
‖(A1, . . . , An)‖2 ≤
n‖(A1, . . . , An)‖, ∀ (A1, . . . , An) ∈ (Ms.ak (C))n.
It follows from Lemma 1 in [21] that
ν∞(K,ω) ≤
λ(K(ω, ‖ · ‖))
λ(B∞(ω/4))
nω, ‖ · ‖2))
λ(B2(2
≤ ν2(K(
nω, ‖ · ‖2), 2
Combining with the equalities (5.1.1), we get
ν∞(K,ω) ≤
λ(K(ω, ‖ · ‖))
λ(B∞(ω/4))
λ(K(ω, ‖ · ‖))
λ(B∞(1))
nω, ‖ · ‖2))
λ(B∞(1))
nω, ‖ · ‖2))
λ(B2(1))
≤ λ(K(
nω, ‖ · ‖2))
λ(B2(2
≤ ν2(K(
nω, ‖·‖2), 2
nω) ≤ ν2(K,
Therefore, we have
ν2(K,
nω) ≤ ν∞(K,ω) ≤
λ(B2(1))
λ(B∞(1))
ν2(K,
It is a well-known fact (for example see Theorem 8 in [2]) that
λ(B2(1))
λ(B∞(1))
≤ Cnk23
for some universal constant C3 > 0. Hence
ν2(K,
nω) ≤ ν∞(K,ω) ≤
nM1C3
ν2(K,
Let K be Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr). By the definitions of δ̃top and δtop, we
δ̃top(x1, . . . , xn : y1, . . . , ym) = δtop(x1, . . . , xn : y1, . . . , ym).
5.2. Upper-bound of topological free entropy dimension in a unital C∗ algebra.
Let us recall Voiculescu’s definition of free dimension capacity in [26].
Definition 5.2. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) is the set consisting of all tracial states of A. If TS(A) 6= ∅,
define Voiculescu’s free dimension capacity κδ(x1, . . . , xn) of x1, . . . , xn as follows,
κδ(x1, . . . , xn) = sup
τ∈TS(A)
δ0(x1, . . . , xn : τ),
where δ0(x1, . . . , xn : τ) is Voiculescu’s (von Neumann algebra) free entropy dimension of
x1, . . . , xn in 〈A, τ〉.
The relationship between topological free entropy dimension of a unital C∗ algebra with a
unique tracial state and its free dimension capacity is indicated by the following result.
Theorem 5.1. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) is the set consisting of all tracial states of A. If TS(A) is a set
with a single element, then
δtop(x1, . . . , xn) ≤ κδ(x1, . . . , xn).
To prove the preceding theorem, we need the following lemma.
Sublemma 5.2.1. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) 6= ∅ is the set consisting of all tracial states of A. Let R >
max{‖x1‖, . . . , ‖xn‖} be some positive number. Then for any m ≥ 1, there is some rm ∈ N such
(top)
R (x1, . . . , xn; k,
, P1, . . . , Prm) ⊆ ∪τ∈TS(A)ΓR(x1, . . . , xn; k,m,
; τ), ∀ k ≥ 1
where ΓR(x1, . . . , xn; k,m,
; τ) is microstate space of x1, . . . , xn with respect to τ (see [23]).
Proof of Sublemma 5.2.1: We will prove the result by contradiction. Suppose, to the
contrary, there is some m0 ≥ 1 so that following holds: for any r ∈ N, there are some kr ≥ 1
and some
1 , A
2 , . . . , A
n ) ∈ Γ
(top)
R (x1, . . . , xn; kr,
, P1, . . . , Pr)
satisfying
1 , A
2 , . . . , A
n ) /∈ ∪τ∈TS(A)ΓR(x1, . . . , xn; kr, m,
; τ). (5.2.1)
Let α be a free ultrafilter in β(N)\N. Let N =
r=1Mkr(C) be the von Neumann algebra ultra-
product of {Mkr(C)}∞r=1 along the ultrafilter α, i.e.
r=1Mkr(C) is the quotient algebra of the
C∗ algebra
r=1Mkr(C) by I2, the 0-ideal of the trace τα, where τα((Ar)∞r=1) = limr→α
Tr(Ar)
Let, for each 1 ≤ j ≤ n, aj = [(A(j)r )∞r=1] be a self-adjoint element in N . By mapping xj to aj ,
there is a unital ∗-homomorphism ψ from the C∗ algebra A onto the C∗ subalgebra generated
by {a1, . . . , an} in N .
Let τ0 be the tracial state on A which is induced by τα on ψ(A), i.e.
τ0(x) = τα(ψ(x)), ∀ x ∈ A.
It follows that when r is large enough,
1 , A
2 , . . . , A
n ) ∈ ΓR(x1, . . . , xn; kr, m,
; τ0),
which contradicts with the inequality (5.2.1). This complete the proof. �
Proof of Theorem 5.1: Let R > max{‖x1‖, . . . , ‖xn‖}. Let τ be the unique trace of
A. By Sublemma 5.2.1, for any m ≥ 1, there is r ∈ N such that
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr) ⊆ ΓR(x1, . . . , xn; k,m,
; τ), ∀ k ≥ 1.
Therefore, for any 1 > ω > 0, we have
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr), ω) ≤ ν2(ΓR(x1, . . . , xn; k,m,
; τ), ω), ∀ k ≥ 1.
Now it is easy to check that
δ̃top(x1, . . . , xn) ≤ δ(x1, . . . , xn; τ) = κδ(x1, . . . , xn).
By Proposition 5.1, we know that
δtop(x1, . . . , xn) ≤ κδ(x1, . . . , xn).
Remark 5.1. Combining Theorem 5.1 with the results in [11] or [14], we will be able to compute
the upper-bound of topological free entropy dimension for a large class of unital C∗ algebras. For
example, δtop(x1, . . . , xn) ≤ 1 if x1, . . . , xn is a family of self-adjoint operators that generates an
irrational rotation algebra A.
5.3. Lower-bound of topological free entropy dimension in a unital C∗ algebra.
In this subsection, we assume that A is a finitely generated, infinite dimensional, unital simple
C∗ algebra with a unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint
generators of A. Let H be the Hilbert space L2(A, τ). Without loss of generality, we might
assume that A is faithfully represented on the Hilbert space H . Let M be the von Neumann
algebra generated by A on H . It is not hard to see that M is a diffuse von Neumann algebra
with a tracial state τ .
For each positive integer m, there is a family of mutually orthogonal projections p1, . . . , pm
in M such that τ(pj) = 1/m for 1 ≤ j ≤ m. Let
ym = 1 · p1 + 2 · p2 + · · ·+m · pm =
j · pj ∈ M.
Let {Pr(x1, . . . , xn)}∞r=1 be defined as in section 2.3. Thus {Pr(x1, . . . , xn)}∞r=1 is dense in M
with respect to the strong operator topology. Hence, for each m ≥ 1, there is some self-adjoint
element Prm(x1, . . . , xn) in A such that
‖ym − Prm(x1, . . . , xn)‖2 ≤
where ‖a‖2 =
τ(a∗a) for all a ∈ M.
Lemma 5.1. Let A be finitely generated, infinite dimensional, unital simple C∗ algebra with a
unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint generators of A. Let
H, M be defined as above. For each m ≥ 1, let ym and Prm(x1, . . . , xn) be chosen as above.
δtop(x1, . . . , xn) ≥ δtop(Prm(x1, . . . , xn) : x1, . . . , xn).
Proof. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}. There exists a positive constant
D > 1 such that
‖Prm(A1, . . . , An)− Prm(B1, . . . , Bm)‖ ≤ D‖(A1, . . . , An)− (B1, . . . , Bm)‖
for all A1, . . . , An, B1, . . . , Bn in Mk(C) satisfying 0 ≤ ‖A1‖, . . . , ‖An‖, ‖B1‖, . . . , ‖Bn‖ ≤ R.
Then it is not hard to verify that, for ω > 0,
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω)
≤ ν∞(Γ(top)R (x1, . . . , xn; k, ǫ, P1, . . . , Pr),
for each r ≥ rm and ǫ < ω4 . By definition of δtop and Remark 2.3, we have
δtop(Prm(x1, . . . , xn) : x1, . . . , xn) ≤ δtop(x1, . . . , xn).
Definition 5.3. Suppose A is a unital C∗ algebra and x1, . . . , xn is a family of self-adjoint ele-
ments of A that generates A as a C∗ algebra. If for any R > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖},
r > 0, ǫ > 0, there is a sequence of positive integers k1 < k2 < · · · such that
(top)
R (x1, . . . , xn : y1, . . . , ym; ks, ǫ, P1, . . . , Pr) 6= ∅, ∀ s ≥ 1
then A is called having approximation property.
Lemma 5.2. Let A be a finitely generated, infinite dimensional, unital simple C∗ algebra with a
unique tracial state τ . Assume that A has approximation property. Assume that x1, . . . , xn is a
family of self-adjoint generators of A. Let H, M be defined as above. Let m be a positive integer.
Let ym and Prm(x1, . . . , xn) be chosen as above. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}.
Then there is some positive integer r > rm so that the following hold: ∀ k ≥ 1, if
(B,A1, . . . , An) ∈ Γ(top)R (Prm(x1, . . . , xn), x1, . . . , xn; k,
, P1, . . . , Pr),
then there are some 1 ≤ k1, . . . , km ≤ k with 1m −
for each 1 ≤ j ≤ m and
k1 + · · ·+ km = k, and a unitary matrix U in U(k) satisfying
‖B − U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
Proof. We will prove the result by contradiction. Assume, to the contrary, for all r ≥ rm
there are some kr ≥ 1 and some
(B(r), A
1 , . . . , A
n ) ∈ Γ
(top)
R (Prm(x1, . . . , xn), x1, . . . , xn; kr,
, P1, . . . , Pr),
satisfying
‖B(r) − U
1 · Is1 0 · · · 0
0 2 · Is2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ism
U∗‖2 >
, (5.3.1)
for all 1 ≤ s1, . . . , sm ≤ kr with 1m −
for each 1 ≤ j ≤ n and s1+ · · ·+ sm = kr,
and all unitary matrix U in U(k).
Let α be a free ultrafilter in β(N) \N. Let N =
r=1Mkr(C) be the von Neumann algebra
ultraproduct of {Mkr(C)}∞r=1 along the ultrafilter α, i.e.
r=1Mkr(C) is the quotient of the
C∗ algebra
r=1Mkr(C) by I2, the 0-ideal of the trace τα, where τα((Ar)∞r=1) = limr→α
Tr(Ar)
Let, for each 1 ≤ j ≤ n, aj = [(A(j)r )∞r=1] be a self-adjoint element in N . By mapping xj to aj ,
there is a unital ∗-homomorphism ψ from the C∗ algebra A onto the C∗ subalgebra generated
by {a1, . . . , an} in N . Since A is a simple C∗ algebra and ψ(IA) = IN , ψ actually is a ∗-
isomorphism. Since A has a unique trace τ , ψ induces a ∗-isomorphism (still denoted by ψ)
from M onto the von Neumann subalgebra generated by a1, . . . , an in N . Therefore,
‖ym − Prm(x1, . . . , xn)‖2 = ‖ψ(ym)− Prm(a1, . . . , an)‖2,τα ≤
This contradicts with the definition of ym and inequality (5.3.1). �
The following lemma is well-known (for example, see Lemma 4.1 in [23]).
Lemma 5.3. Suppose A, or B, is a self-adjoint matrix in Ms.a.k (C) with a list of eigenvalues
λ1 ≤ λ2 ≤ · · · ≤ λk, or µ1 ≤ µ2 ≤ · · · ≤ µk respectively. Then
|λj − µj |2 ≤ Tr((A− UBU∗)2),
where U is any unitary matrix in U(k).
Lemma 5.4. Let r,m be some positive integer with 4 < m < r. Suppose k1, . . . , km is a family
of positive integers such that 1
for all 1 ≤ j ≤ k and k1 + · · ·+ km = k. If A
is a self-adjoint matrix in Mk(C) such that, for some unitary matrix U in U(k),
‖A− U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
then, for any ω > 0 we have
ν2(Ω(A), ω) ≥ (8C1ω)−k
for some constants C1, C > 1 independent of k, ω, where
Ω(A) = {W ∗AW | W ∈ U(k)}.
Proof. Suppose that λ1 ≤ λ2 ≤ . . . ≤ λk are the eigenvalues of A. For each 1 ≤ j ≤ m, let
Tj = {i ∈ N | (
kt) + 1 ≤ i ≤
kt and |λi − j| ≤
T̂j = {(
kt) + 1, (
kt) + 2, · · · ,
kt} \ Tj ,
here we assume that k0 = 0. Let B = diag(1 · Ik1, · · · , m · Ikm) be a diagonal matrix in Mk(C).
By Lemma 5.3, we have
≥ Tr((A− UBU∗)2) ≥
i ∈T̂j
|λi − j|2 ≥
card(T̂j),
where card(T̂j) is the cardinality of the set T̂j . Thus
card(T̂j) ≤
, for 1 ≤ j ≤ m.
Let sj = card(Tj) for 1 ≤ j ≤ m, whence
≥ kj ≥ sj = kj − card(T̃j) ≥ kj −
, ∀ 1 ≤ j ≤ m.
Tm+1 = {1, 2, . . . , k} \ (∪nj=1Tj)
and sm+1 be the cardinality of the set Tm+1. Thus
sm+1 = k − s1 − · · · − sm =
card(T̂j) ≤
It is not hard to see that T1, . . . , Tm+1 is a partition of the set {1, 2, . . . , k}. Moreover, if
1 ≤ j1 6= j2 ≤ m then for any
i1 ∈ Tj1 , and i2 ∈ Tj2
we have
|λi1 − λi2| ≥ |j2 − j1| − |λi2 − j2| − |λi1 − j1| ≥ 1−
Applying Proposition 3.1 for such T1, . . . , Tm, Tm+1, θ = 1/2 and ω = δ/2, we have
ν2(Ω(A), ω) ≥ (8C1ω)−k
)−2s2
−···−2s2m−2s
−4(s1+···+sm)sm+1
≥ (8C1ω)−k
)−2(k2
+···+k2m+(
)2+2k· 4k
≥ (8C1ω)−k
)−2(( k
)2+···+( k
)2+ 16k
≥ (8C1ω)−k
for some constants C,C1 > 1 independent of k, ω.
Lemma 5.5. Let A be a finitely generated, infinite dimensional, simple unital C∗ algebra with a
unique tracial state τ . Assume that A has approximation property. Assume that x1, . . . , xn is a
family of self-adjoint generators of A. Let H, M be defined as above. Let m be a positive integer.
Let ym and Prm(x1, . . . , xn) be chosen as above. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}.
When r is large enough and ǫ is small enough, for any ω > 0, we have
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω) ≥ (8C1ω)−k
Proof. By Lemma 5.2, when r is large enough and ǫ is small enough, the following hold:
∀ k ≥ 1, if
(B,A1, . . . , An) ∈ Γ(top)R (Prm(x1, . . . , xn), x1, . . . , xn; k, ǫ, P1, . . . , Pr),
then there are some 1 ≤ k1, . . . , km ≤ k with 1m −
for each 1 ≤ j ≤ m and
k1 + · · ·+ km = k, and a unitary matrix U in U(k) satisfying
‖B − U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
Combining with Lemma 5.4, we know that if
B ∈ Γ(top)R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr)
then, for any ω > 0,
ν2(Ω(B), ω) ≥ (8C1ω)−k
where
Ω(B) = {W ∗BW | W ∈ U(k)}.
Note that Ω(B) ⊂ Γ(top)R (Prm(x1, . . . , xn) : x1, . . . , xn; k, r, ǫ). It follows that, for any ω > 0,
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, r, ǫ), ω) ≥ (8C1ω)−k
−56k2
Now we have the following result.
Theorem 5.2. Let A be a finitely generated, infinite dimensional, simple unital C∗ algebra with
a unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint generators of A. If
A has approximation property, then δtop(x1, . . . , xn) ≥ 1.
Proof. Let H be the Hilbert space L2(A, τ). Without loss of generality, we might assume
that A is faithfully represented on the Hilbert space H . Let M be the von Neumann algebra
generated by A on H . It is not hard to see that M is a diffuse von Neumann algebra with a
tracial state τ . For each positive integer m, there is a family of mutually orthogonal projections
p1, . . . , pm in M such that τ(pj) = 1/m for 1 ≤ j ≤ m. Let
ym = 1 · p1 + 2 · p2 + · · ·+m · pm =
j · pj.
Let {Pr(x1, . . . , xn)}∞r=1 be defined as in section 2.3. Thus {Pr(x1, . . . , xn)}∞r=1 is dense in M
with respect to the strong operator topology. Hence, for each m ≥ 1, there is some self-adjoint
element Prm(x1, . . . , xn) in A such that
‖ym − Prm(x1, . . . , xn)‖2 ≤
By Lemma 5.5, for any ω > 0, when r is large enough and ǫ is small enough, we have for
some constants C1, C > 1 independent of k, ω
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω) ≥ (8C1ω)−k
Therefore,
δ̃top(Prm(x1, . . . , xn) : x1, . . . , xn) ≥ 1−
By Proposition 5.1, we get
δtop(Prm(x1, . . . , xn) : x1, . . . , xn) ≥ 1−
By Lemma 5.1,
δtop(x1, . . . , xn) ≥ 1−
Since m is an arbitrary positive integer, we obtain
δtop(x1, . . . , xn) ≥ 1.
5.4. Values of topological free entropy dimensions in some unital C∗ algebras. In
this subsection, we are going to compute the values of topological free entropy dimensions in
some unital C∗ algebras by using the results from preceding subsection.
Theorem 5.3. Let Aθ be an irrational rotation C∗ algebra. Then
δtop(x1, . . . , xn) = 1
where x1, . . . , xn is a family of self-adjoint operators that generates Aθ.
Proof. Note that Aθ is an infinite dimensional, unital simple C∗ algebra with a unique
tracial state τ . By [24] or [11] and Theorem 5.1, we know that
δtop(x1, . . . , xn) ≤ 1.
It follows from [18] that Aθ has approximation property. Therefore
δtop(x1, . . . , xn) ≥ 1.
Hence
δtop(x1, . . . , xn) = 1.
Theorem 5.4. Let A be a UHF algebra (uniformly hyperfinite C∗ algebra). Then
δtop(x1, . . . , xn) = 1
where x1, . . . , xn is a family of self-adjoint operators that generates A.
Proof. By [17], we know that A is generated by two self-adjoint elements. It is not hard
to see that A is an infinite dimensional, unital simple C∗ algebra with a unique tracial state τ .
By [24] or [11] and Theorem 5.1, we know that
δtop(x1, . . . , xn) ≤ 1.
It is easy to check that A has approximation property. Therefore
δtop(x1, . . . , xn) ≥ 1.
Hence
δtop(x1, . . . , xn) = 1.
Recall that for any sequence (Am)∞m=1 of C∗ algebras,we can introduce two C∗ algebras
Am = {(am)∞m=1 | am ∈ Am, sup
‖am‖ <∞}
Am = {(am)∞m=1 | am ∈ Am, lim
‖am‖ = 0}
The norm in the quotient C∗ algebra
m Am/
mAm is given by
‖ρ((am)∞m=1)‖ = lim sup
‖xm‖,
where ρ is the quotient map from
mAm onto
If A is an exact C∗ algebra, then the sequence
0 → A⊗min
Mm(C) → A⊗min
Mm(C) → A⊗min (
Mm(C)/
Mm(C)) → 0
is exact. Therefore, we have the following natural identification
A⊗min (
Mm(C)/
Mm(C)) = (A⊗min
Mm(C))/(A⊗min
Mm(C)).
On the other hand, we have the following natural embedding
A⊗min
Mm(C) ⊆
Mm(A)
and the identification
A⊗min
Mm(C) =
Mm(A)
Thus we have for any exact C∗ algebra A a natural embedding
ψ : A⊗min (
Mm(C)/
Mm(C) ⊆
Mm(A)/
Mm(A).
Lemma 5.6. Suppose that A and B are unital C∗ algebras and ρ is an unital embedding
ρ : A →
Mm(B)/
Mm(B).
Suppose that x1, . . . , xn is a family of elements in A. Suppose r is a positive integer and
{Pj(x1, . . . , xn)}rj=1 is a family of noncommutative polynomials of x1, . . . , xn. Then there are
some k ∈ N and a(k)1 , . . . , a
n in Mk(B) so that
|‖Pj(a(k)1 , . . . , a(k)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Proof. We might assume that
ρ(xi) = [(x
i )m] ∈
Mm(B)/
Mm(B), ∀ 1 ≤ i ≤ n.
By the definition of
mMm(B)/
mMm(B), there are some positive integers m1 ≤ m2 such
|( sup
m1≤l≤m2
‖Pj(x(l)1 , . . . , x(l)n )‖)− ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Let k =
j and
i = ⊕m2l=m1x
i ∈ Mk(B), ∀ 1 ≤ i ≤ n.
Then, it is not hard to check that
|‖Pj(a(k)1 , . . . , a(k)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Theorem 5.5. Let p ≥ 2 be a positive integer and Fp be the free group on p generators. Let
C∗red(Fp)⊗minC∗red(Fp) be a minimal tensor product of two reduced C∗ algebras of free groups Fp.
δtop(x1, . . . , xn) = 1,
where x1, . . . , xn is any family of self-adjoint generators of C
red(Fp)⊗min C∗red(Fp).
Proof. Note that C∗red(Fp)⊗minC∗red(Fp) is an infinite dimensional, unital simple C∗ algebra
with a unique tracial state. By the result from [5] or [11] and Theorem 5.1, Theorem 5.2, to
show δtop(x1, . . . , xn) = 1, we need only to show that C
red(Fp)⊗min C∗red(Fp) has approximation
property. Therefore, it suffices to show the following: Let R > max{‖x1‖, . . . , ‖xn‖}. For any
r ≥ 1, there is some k ∈ N so that
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr) 6= ∅.
By the result from [9], we know there is a unital embedding
φ1 : C
red(Fp) →
Mm(C)/
Mm(C),
which induce a unital embedding
φ2 : C
red(Fp)⊗min C∗red(Fp) → C∗red(Fp)⊗min (
Mm(C)/
Mm(C))
Note that C∗red(Fp) is an exact C
∗ algebra. From the explanation preceding the theorem it
follows that there is a unital embedding
φ3 : C
red(Fp)⊗min C∗red(Fp) →
Mm(C∗red(Fp))/
Mm(C∗red(Fp)).
By Lemma 5.6, for a family of elements x1, . . . , xn in C
red(Fp) ⊗min C∗red(Fp) and r ≥ 1, there
are some m ∈ N and some a(m)1 , . . . , a
n in Mm(C∗red(Fp)) so that max{‖a1‖, . . . , ‖an‖} < R
|‖Pj(a(m)1 , . . . , a(m)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 0 ≤ j ≤ r.
On the other hand, by the existence of embedding
φ1 : C
red(Fp) →
Mm′(C)/
Mm′(C),
it follows that there is a unital embedding
φ4 : Mm(C∗red(Fp)) = Mm(C)⊗min C∗red(Fp) → Mm(C)⊗min (
Mm′(C)/
Mm′(C))
Mm(C)⊗min (
Mm′(C)/
Mm′(C)) =
Mm′m(C)/
Mm′m(C).
Hence for such a
1 , . . . , a
n in Mm(C∗red(Fp)) and r ≥ 1, by Lemma 5.6, there are some k ∈ N
and A1, . . . , An in Mk(C) so that max{‖A1‖, . . . , ‖An‖} < R and
|‖Pj(a(m)1 , . . . , a(m)n )‖ − ‖Pj(A1, . . . , An)‖| ≤
, ∀ 0 ≤ j ≤ r.
Altogether, we have
|‖Pj(x1, . . . , xn)‖ − ‖Pj(A1, . . . , An)‖| ≤
, ∀ 0 ≤ j ≤ r,
which implies that C∗red(Fp)⊗min C∗red(Fp) has approximation property.
Hence
δtop(x1, . . . , xn) = 1,
for any family of self-adjoint elements x1, . . . , xn that generates C
red(Fp)⊗min C∗red(Fp). �
Theorem 5.6. Suppose that K be the C∗ algebra consisting of all compact operators on a sepa-
rable Hilbert space H. Suppose A = C
K is the unitization of K. If x1, . . . , xn is a family of
self-adjoint elements that generate A as a C∗ algebra, then
δtop(x1, . . . , xn) = 0.
Proof. By [17], we know that unital C∗ algebra A is generated by two self-adjoint elements
in A. Note that A has a unique trace τ , which is defined by
τ((λ, x)) = λ, ∀ (λ, x) ∈ A.
By Theorem 5.1, it is not hard to see that
δtop(x1, . . . , xn) = 0,
where x1, . . . , xn is a family of self-adjoint generators of A.
6. Topological free orbit dimension of C∗ algebras
Assume that A is a unital C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint elements
in A. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉 be the noncommutative polynomials in the indeterminates
X1, . . . , Xn, Y1, . . . , Ym. Let {Pr}∞r=1 be the collection of all noncommutative polynomials in
C〈X1, . . . , Xn, Y1, . . . , Ym〉 with rational coefficients.
6.1. Unitary orbits of balls in Mk(C)n. We let Mk(C) be the k× k full matrix algebra
with entries in C, and U(k) be the group of all unitary matrices in Mk(C). Let Mk(C)n denote
the direct sum of n copies of Mk(C). Let Ms.ak (C) be the subalgebra of Mk(C) consisting of
all self-adjoint matrices of Mk(C). Let (Ms.ak (C))n be the direct sum of n copies of Ms.ak (C).
For every ω > 0, we define the ω-orbit-‖ · ‖-ball U(B1, . . . , Bn;ω) centered at (B1, . . . , Bn) in
Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that there
exists some unitary matrix W in U(k) satisfying
‖(A1, . . . , An)− (WB1W ∗, . . . ,WBnW ∗)‖ < ω.
6.2. Norm-microstate space. For all integers r, k ≥ 1, real numbers R, ǫ > 0 and non-
commutative polynomials P1, . . . , Pr, we let
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
be as defined as in section 2.4.
6.3. Topological free orbit dimension.
Definition 6.1. For ω > 0, we define the covering number
(top)
R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr), ω)
to be the minimal number of ω-orbit–‖·‖-balls that cover Γ(top)R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr)
with the centers of these ω-orbit-‖ · ‖-balls in Γ(top)R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr)
For each function f : N× N× R+ → R, we define,
kf(x1, . . . , xn : y1, . . . , yp;ω,R)
= inf
r∈N,ǫ>0
lim sup
f(o∞(Γ
(top)
R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr), ω), k, ω)
kf(x1, . . . , xn : y1, . . . , yp;ω) = sup
kf(x1, . . . , xn : y1, . . . , yp;ω,R)
kf(x1, . . . , xn : y1, . . . , yp) = lim sup
kf(x1, . . . , xn : y1, . . . , yp;ω),
where kf(x1, , . . . , xn : y1, . . . , yp) is called the topological f(·)-free-orbit-dimension of x1, . . . , xn
in the presence of y1, . . . , yp.
6.4. Topological free entropy dimension and topological free orbit dimension.
The following result follows directly from the definitions of topological free entropy dimension
and topological free orbit dimension of n-tuple of self-adjoint elements in a C∗ algebra.
Theorem 6.1. Suppose that A is a unital C∗ algebra and x1, . . . , xn is a family of self-adjoint
elements of A. Let f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
−k2 logω
for s, k ∈ N, ω > 0. Then
δtop(x1, . . . , xn) ≤ kf(x1, . . . , xn) + 1.
7. Topological free orbit dimension of one variable
We recall the packing number of a set in a metric space as follows.
Definition 7.1. Suppose that X is a metric space with a metric distance d. (i) The packing
number of a set K by ω-balls in X, denoted by P (K,ω), is the maximal cardinality of the subsets
F in K satisfying for all a, b in F either a = b or d(a, b) ≥ ω. (ii) The packing dimension of
the set K in X, denoted by d(K), is defined by
d(K) = lim sup
log(P (K,ω))
− log ω .
7.1. Upper-bound of the topological free orbit dimension of one variable. Suppose
that x = x∗ is a self-adjoint element in a unital C∗ algebra A and σ(x) is the spectrum of x in
For any ω > 0, let m = P (K,ω) be the packing number of σ(x) in R. Thus there exists a
family of elements λ1, . . . , λm in σ(x) such that (i) |λi − λj | ≥ ω for all 1 ≤ i 6= j ≤ m; and (ii)
for any λ in σ(x), there is some λj with 1 ≤ j ≤ m satisfying |λ− λj | ≤ ω.
Lemma 7.1. For any given R > ‖x‖, when r is large enough and ǫ is small enough, we have
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω)
log k
Proof. By Theorem 3.1, there exist some r0 ≥ 1 and ǫ0 > 0 such that the following
holds: when r > r0, ǫ < ǫ0, for any A in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), there are positive integers
1 ≤ k1, . . . , km ≤ k with k1 + · · ·+ km = k and some unitary matrix U in Mk(C) satisfying
‖U∗AU −
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λmIkm
‖ ≤ 2ω,
where Ikj is the kj × kj identity matrix for 1 ≤ j ≤ m.
Ω(k1, . . . , km) =
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λmIkm
U | U is in Uk
Let J be the set consisting of all these (k1, . . . , km) ∈ Nm with k1 + · · ·+ km = k. Then the
cardinality of the set J is equal to
(k − 1)!
(m− 1)!(k −m)! .
(top)
R (x; k, ǫ, P1, . . . , Pr)
is contained in 2ω-neighborhood of the set
(k1,...,km)∈J
Ω(k1, . . . , km).
It follows that
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω) ≤ o∞(
(k1,...,km)∈J
Ω(k1, . . . , km), ω) ≤ |J | =
(k − 1)!
(m− 1)!(k −m)! .
Therefore,
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω)
log k
= lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), ω)
log k
≤ lim sup
(k−1)!
(m−1)!(k−m)!
log k
= m− 1.
7.2. Lower-bound. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A
and σ(x) is the spectrum of x in A.
Lemma 7.2. We have
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr),
log k
≥ m− 1.
Proof. For any ω > 0, let m = P (K,ω) be the packing number of σ(x) in R. Thus there
exists a family of elements λ1, . . . , λm in σ(x) such that (i) |λi − λj | ≥ ω for all 1 ≤ i 6= j ≤ m;
and (ii) for any λ in σ(x), there is some λj with 1 ≤ j ≤ m satisfying |λ− λj| ≤ ω.
For any R > ‖x‖, r ≥ 1 and ǫ > 0, by functional calculus, there are λm+1, . . . , λn in σ(x)
such that for every 1 ≤ t1, . . . , tm ≤ k − n with 2nt1 + . . .+ 2ntm = k − n, the matrix
A = diag(λ1I2nt1 ,λ2I2nt2 , . . . , λmI2ntm , λ1, . . . , λm, . . . , λn)
is in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), (7.2.1)
where we assume that 2n|(k − n).
Let J be the set consisting of all these (t1, . . . , tm) ∈ Nm with 2nt1 + . . . + 2ntm = k − n.
Then the cardinality of the set J is equal to
!(m− 1)!
By Weyl’s theorem in [27] on the distance of unitary orbits of two self-adjoint matrices, for
any two distinct elements
(s1, . . . , sm) and (t1, . . . , tm)
in J and any W in U(k), we have
‖A1 −WA2W ∗‖ ≥ ω,
where
A1 = diag(λ1I2nt1 , λ2I2nt2 , . . . , λmI2ntm , λ1, . . . , λm, . . . , λn)
A2 = diag(λ1I2ns1, λ2I2ns2 , . . . , λmI2nsm, λ1, . . . , λm, . . . , λn)
are two diagonal self-adjoint matrices in Mk(C). Combining with (7.2.1), we have
(top)
R (x; k, ǫ, P1, . . . , Pr),
) ≥ |J | ≥
!(m− 1)!
Hence
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr),
log k
≥ lim sup
( k−n2n −1)!
( k−n2n −m)!(m−1)!
log k
= m− 1.
7.3. Topological free orbit dimension of one self-adjoint element.
Theorem 7.1. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A and σ(x)
is the spectrum of x in A. Let d(σ(x)) be the packing dimension of the set σ(x) in R. Let
f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
log k
− log ω
for s, k ∈ N, ω > 0. Then
kf(x) = d(σ(x)).
Proof. The result follows directly from Lemma 7.1, Lemma 7.2 and Definition 7.1. �
Theorem 7.2. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A. Let
f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
−k2 logω
for s, k ∈ N, ω > 0. Then
kf(x) = 0.
Proof. The result follows directly from Lemma 7.1 and Definition 7.1. �
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http://arxiv.org/abs/math/0510576
http://arxiv.org/abs/math/0603669
	1. Introduction
	2. Definitions and preliminary
	3. Some technical lemmas
	4. Topological free entropy dimension of one variable
	5. Topological free entropy dimension of n-tuple in unital C* algebras 
	6. Topological free orbit dimension of C* algebras
	7. Topological free orbit dimension of one variable
	References

  The notion of topological free entropy dimension of $n-$tuples of elements in
a unital C$^*$ algebra was introduced by Voiculescu. In the paper, we compute
topological free entropy dimension of one self-adjoint element and topological
orbit dimension of one self-adjoint element in a unital C$^*$ algebra.
Moreover, we calculate the values of topological free entropy dimensions of
families of generators of some unital C$^*$ algebras (for example: irrational
rotation C$^*$ algebras or minimal tensor product of two reduced C$^*$ algebras
of free groups).

Introduction
The theory of free probability and free entropy was developed by Voiculescu from 1990s. It
played a crucial role in the recent study of finite von Neumann algebras (see [1], [3], [4], [5], [6],
[7], [8], [11], [14], [15], [23], [24], [25]). The analogue of free entropy dimension in C∗ algebra
context, the notion of topological free entropy dimension of of n−tuple of elements in a unital
C∗ algebra, was also introduced by Voiculescu in [26].
After introducing the concept of topological free entropy dimension of n-tuple of elements
in a unital C∗ algebra, Voiculescu discussed some of its properties including subadditivity and
change of variables in [26]. In this paper, we will add one basic property into the list: topological
free entropy dimension of one variable. More specifically, suppose x is a self-adjoint element in a
unital C∗ algebra A and σ(x) is the spectrum of x in A. Then topological free entropy dimension
of x is equal to 1− 1
where n is the cardinality of the set σ(x) (see Theorem 4.1).
In [26], Voiculescu showed that (i) if x1, . . . , xn is a family of free semicircular elements
in a unital C∗ algebra with a tracial state, then δtop(x1, . . . , xn) = n, where δtop(x1, . . . , xn) is
the topological free entropy dimension of x1, . . . , xn; (ii) if x1, . . . , xn is the universal n-tuple
of self-adjoint contractions, then δtop(x1, . . . , xn) = n. Except in these two cases, very few has
been known on the values of topological free entropy dimensions in other C∗ algebras. Using the
inequality between topological free entropy dimension and Voiculescu’s free dimension capacity,
we are able to obtain an estimation of upper-bound of topological free entropy dimension for a
1The second author is supported by an NSF grant.
http://arxiv.org/abs/0704.0667v3
unital C∗ algebra with a unique tracial state (see Theorem 5.1). The lower-bound of topological
free entropy dimension is also obtained for infinite dimensional simple unital C∗ algebra with a
unique tracial state (see Theorem 5.2). As a corollary, we know that the topological free entropy
dimension of any family of self-adjoint generators of an irrational rotation C∗ algebra or UHF
algebra or C∗red(F2)⊗minC∗red(F2) is equal to 1 (see Theorem 5.3, 5.4, 5.5). For these C∗ algebras,
the value of the topological free entropy dimension is independent of the choice of generators.
The rest of the paper is devoted to study another invariant associated to n-tuple of elements
in C∗ algebras. This invariant, called topological free orbit dimension, is an analogue of free
orbit dimension in finite von Neumann algebras (see [11]). We show that the topological free
orbit dimension of a self-adjoint element in a unital C∗ algebra is equal to, according to some
measurement, the packing dimension of the spectrum of x (see Theorem 7.1).
The organization of the paper is as follows. In the section 2, we recall the definition of
topological free entropy dimension. Some technical lemmas are proved in section 3. In section
4, we compute the topological free entropy dimension of one self-adjoint element in a unital
C∗ algebra. In section 5, we study the relationship between topological free entropy dimension
and free capacity dimension of a unital C∗ algebra. Then we show that topological free entropy
dimension of of any family of generators of an infinite dimensional simple unital C∗ algebra with
a unique tracial state is always greater than or equal to 1. The concept of topological free orbit
dimension of n-tuple of elements in a C∗ algebra is introduced in section 6. Its value for one
variable is computed in section 7.
2. Definitions and preliminary
In this section, we are going to recall Voiculescu’s definition of topological free entropy
dimension of n-tuple of elements in a unital C∗ algebra.
2.1. A Covering of a set in a metric space. Suppose (X, d) is a metric space and K is
a subset of X . A family of balls in X is called a covering of K if the union of these balls covers
K and the centers of these balls lie in K.
2.2. Covering numbers in complex matrix algebra (Mk(C))n. Let Mk(C) be the
k × k full matrix algebra with entries in C, and τk be the normalized trace on Mk(C), i.e.,
Tr, where Tr is the usual trace on Mk(C). Let U(k) denote the group of all unitary
matrices in Mk(C). Let Mk(C)n denote the direct sum of n copies of Mk(C). Let Ms.ak (C) be
the subalgebra of Mk(C) consisting of all self-adjoint matrices of Mk(C). Let (Ms.ak (C))n be
the direct sum of n copies of Ms.ak (C). Let ‖ · ‖ be an operator norm on Mk(C)n defined by
‖(A1, . . . , An)‖ = max{‖A1‖, . . . , ‖An‖}
for all (A1, . . . , An) in Mk(C)n. Let ‖ · ‖2 denote the trace norm induced by τk on Mk(C)n, i.e.,
‖(A1, . . . , An)‖2 =
1A1) + . . .+ τk(A
for all (A1, . . . , An) in Mk(C)n.
For every ω > 0, we define the ω-‖ · ‖-ball Ball(B1, . . . , Bn;ω, ‖ · ‖) centered at (B1, . . . , Bn)
in Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that
‖(A1, . . . , An)− (B1, . . . , Bn)‖ < ω.
Definition 2.1. Suppose that Σ is a subset of Mk(C)n. We define the covering number ν∞(Σ, ω)
to be the minimal number of ω-‖ · ‖-balls that consist a covering of Σ in Mk(C)n.
For every ω > 0, we define the ω-‖·‖2-ball Ball(B1, . . . , Bn;ω, ‖·‖2) centered at (B1, . . . , Bn)
in Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that
‖(A1, . . . , An)− (B1, . . . , Bn)‖2 < ω.
Definition 2.2. Suppose that Σ is a subset of Mk(C)n. We define the covering number ν2(Σ, ω)
to be the minimal number of ω-‖ · ‖2-balls that consist a covering of Σ in Mk(C)n.
2.3. Noncommutative polynomials. In this article, we always assume that A is a unital
C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint elements inA. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉
be the unital noncommutative polynomials in the indeterminates X1, . . . , Xn, Y1, . . . , Ym. Let
{Pr}∞r=1 be the collection of all noncommutative polynomials in C〈X1, . . . , Xn, Y1, . . . , Ym〉 with
rational complex coefficients. (Here “rational complex coefficients” means that the real and
imaginary parts of all coefficients of Pr are rational numbers).
Remark 2.1. We alsways assume that 1 ∈ C〈X1, . . . , Xn, Y1, . . . , Ym〉.
2.4. Voiculescu’s Norm-microstates Space. For all integers r, k ≥ 1, real numbers
R, ǫ > 0 and noncommutative polynomials P1, . . . , Pr, we define
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr)
to be the subset of (Ms.ak (C))n+m consisting of all these
(A1, . . . , An, B1, . . . , Bm) ∈ (Ms.ak (C))n+m
satisfying
max{‖A1‖, . . . , ‖An‖, ‖B1‖, . . . , ‖Bm‖} ≤ R
|‖Pj(A1, . . . , An, B1, . . . , Bm)‖ − ‖Pj(x1, . . . , xn, y1, . . . , ym)‖| ≤ ǫ, ∀ 1 ≤ j ≤ r.
Remark 2.2. In the definition of norm-microstates space, we use the following assumption. If
Pj(x1, . . . , xn, y1, . . . , ym) = α0 · IA +
1≤i1,...,is≤n+m
αi1···iszi1 · · · zis
where z1, . . . , zn+m denotes x1, . . . , xn, y1, . . . , ym and α0, αi1···is are in C, then
Pj(A1, . . . , An, B1, . . . , Bm) = α0 · Ik +
1≤i1,...,is≤n+m
αi1···isZi1 · · ·Zis
where Z1, . . . , Zn+m denotes A1, . . . , An, B1, . . . , Bm and Ik is the identity matrix in Mk(C).
Remark 2.3. In the original definition of norm-microstates space in [26], the parameter R was
not introduced. Note the following observation: Let R > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖}.
When r is large enough so that
{X1, . . . , Xn, Y1, . . . , Ym} ⊂ {P1, . . . , Pr}
and 0 < ǫ < R−max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖}, we have
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) = Γtop(x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr)
for all k ≥ 1, where Γ(top)(x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) is the norm-microstates space
defined in [26]. Thus our definition agrees with the one in [26] for large R, r and small ǫ.
In the later sections, we need to construct the ultraproduct of some matrix algebras, it will
be convenient for us to include the parameter “R” in the definition of norm-microstate space.
Define the norm-microstates space of x1, . . . , xn in the presence of y1, . . . , ym, denoted by
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
as the projection of Γ
(top)
R (x1, . . . , xn, y1, . . . , ym; k, ǫ, P1, . . . , Pr) onto the space (Ms.ak (C))n via
the mapping
(A1, . . . , An, B1, . . . , Bm) → (A1, . . . , An).
2.5. Voiculescu’s topological free entropy dimension (see [26]). Define
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω)
to be the covering number of the set Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr) by ω-‖ · ‖-balls
in the metric space (Ms.ak (C))n equipped with operator norm.
Definition 2.3. Define
δtop(x1, . . . , xn : y1, . . . , ym;ω)
= sup
ǫ>0,r∈N
lim sup
log(ν∞(Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 log ω .
The topological entropy dimension of x1, . . . , xn in the presence of y1, . . . , ym is defined by
δtop(x1, . . . , xn : y1, . . . , ym) = lim sup
δtop(x1, . . . , xn : y1, . . . , ym;ω).
Remark 2.4. Let M > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖} be some positive number. By
Remark 2.3, we know
δtop(x1, . . . , xn : y1, . . . , ym)
= lim sup
ǫ>0,r∈N
lim sup
log(ν∞(Γ
(top)
M (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 logω .
2.6. C∗ algebra ultraproduct and von Neumann algebra ultraproduct. Suppose
{Mkm(C)}∞m=1 is a sequence of complex matrix algebras where km goes to infinity when m
approaches infinity. Let γ be a free ultrafilter in β(N)\N. We can introduce a unital C∗ algebra
m=1Mkm(C) as follows:
Mkm(C) = {(Ym)∞m=1 | ∀ m ≥ 1, Ym ∈ Mkm(C) and sup
‖Ym‖ <∞}.
We can also introduce the norm closed two sided ideals I∞ and I2 as follows.
I∞ = {(Ym)∞m=1 ∈
Mkm(C) | lim
‖Ym‖ = 0}
I2 = {(Ym)∞m=1 ∈
Mkm(C) | lim
‖Ym‖2 = 0}
Definition 2.4. The C∗ algebra ultraproduct of {Mkm(C)}∞m=1 along the ultrfilter γ, denoted
m=1Mkm(C), is defined to be the quotient algebra of
m=1Mkm(C) by the ideal I∞. The
image of (Ym)
m=1 ∈
m=1Mkm(C) in the quotient algebra is denoted by [(Ym)m].
Definition 2.5. The von Neumann algebra ultraproduct of {Mkm(C)}∞m=1 along the ultrfilter
γ, also denoted by
m=1Mkm(C) if no confusion arises, is defined to be the quotient algebra of
m=1Mkm(C) by the ideal I2. The image of (Ym)∞m=1 ∈
m=1Mkm(C) in the quotient algebra
is denoted by [(Ym)m].
Remark 2.5. The von Neumann algebra ultraproduct
m=1Mkm(C) is a finite factor (see
[16]).
2.7. Topological free entropy dimension of elements in a non-unital C∗ algebra.
Topological free entropy dimension can also be defined for n-tuple of elements in a non-unital
C∗ algebra. Suppose that A is a non-unital C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint
elements in A. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉 ⊖ C be the noncommutative polynomials in the
indeterminates X1, . . . , Xn, Y1, . . . , Ym without constant terms. Let {Pr}∞r=1 be the collection
of all noncommutative polynomials in C〈X1, . . . , Xn, Y1, . . . , Ym〉 ⊖ C with rational complex
coefficients. Then norm-mocrostate space
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
can be defined similarly as in section 2.4. So topological free entropy dimension
δtop(x1, . . . , xn : y1, . . . , ym)
can also be defined similarly as in section 2.5.
In the paper, we will focus on the case when A is a unital C∗ algebra.
3. Some technical lemmas
3.1. Suppose x is a self-adjoint element in a unital C∗ algebra A. Let σ(x) be the spectrum
of x in A.
Theorem 3.1. Let R > ‖x‖. For any ω > 0, we have the following.
(1) There are some integer n ≥ 1 and distinct real numbers λ1, λ2, · · · , λn in σ(x) satisfying
(i) |λi−λj | ≥ ω for all 1 ≤ i 6= j ≤ n; and (ii) for any λ in σ(x), there is some λj with
1 ≤ j ≤ n such that |λ− λj| ≤ ω.
(2) There are some r0 > 0 and ǫ0 > 0 such that the following holds: when r > r0, ǫ < ǫ0,
for any A in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), there are positive integers 1 ≤ k1, . . . , kn ≤ k with
k1 + k2 + · · ·+ kn = k and some unitary matrix U in Mk(C) satisfying
‖U∗AU −
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λnIkn
‖ ≤ 2ω,
where Ikj is the kj × kj identity matrix in Mkj (C) for 1 ≤ j ≤ n.
Proof. The proof of part (1) is trivial. We will only prove part (2). Assume that the result
in (2) does not hold. Then there is some ω > 0 so that the following holds: for all m ≥ 1, there
are km ≥ 1 and some Am in Γ(top)R (x; km, 1m , P1, . . . , Pm) such that
‖U∗AmU −
λ1Is1 0 · · · 0
0 λ2Is2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λnIsn
‖ > 2ω, (∗)
for every 1 ≤ s1, . . . , sn ≤ km with s1 + · · ·+ sn = km and every unitary matrix U in Mkm(C).
Let γ be a free ultrafilter in β(N)\N. Let B =
m=1 Mkm(C) be the C∗ algebra ultraproduct
of {Mkm(C)}∞m=1 along the ultrafilter α, i.e.
m=1 Mkm(C) is the quotient algebra of the
C∗ algebra
m=1Mkm(C) by I∞, the 0-ideal of the norm ‖ · ‖, where I∞ = {(Am)∞m=1 ∈
Mkm(C) | limm→γ ‖Am‖ = 0}. Let a = [(U∗AmU)∞m=1] be a self-adjoint element in B. By
mapping x to a, there is a unital ∗-isomorphism from the C∗ subalgebra generated by {IA, x}
in A onto the C∗ subalgebra generated by {IB, a} in B. Thus σ(x) = σ(a). It is not hard to see
that Hausdorff-dist(σ(U∗AmU), σ(a)) → 0 as m goes to γ, which contradicts with the results in
part (1) and (∗).
3.2. In this subsection, we will use the following notation.
(i) Let n,m be some positive integers with n ≥ m.
(ii) Let δ, θ be some positive numbers.
(iii) Let {λ1, λ2, . . . , λm} ∪ {λm+1, . . . , λn} be a family of real numbers such that
|λi − λj | ≥ θ for all 1 ≤ i < j ≤ m.
(iv) Let k be a positive integer such that k − (n−m) is divided by m. We let
k − n +m
(v) We let
B = diag(λm+1, . . . , λn)
be a diagonal matrix in Mn−m(C) and
A = diag(λ1It, λ2It, . . . , λmIt, B)
be a block-diagonal matrix in Mk(C), where It is the identity matrix in Mt(C).
(vi) We let A be defined as above and
Ω(A) = {U∗AU | U is in U(k)}.
(vii) Assume that {eij}ki,j=1 is a canonical basis of Mk(C). We let
V1 = span{eij | |λ[ i
]+1 − λ[ j
]+1| ≥ θ, with 1 ≤ i, j < mt}; and
V2 = Mk(C)⊖ V1,
where [ i
], or [ j
], denotes the largest integer ≤ [ i
], or [ j
] respectively.
Lemma 3.1. We follow the notation as above. Suppose ‖U1AU∗1 − U2AU∗2‖2 ≤ δ for some
unitary matrices U1 and U2 in U(k). Then the following hold.
(1) There exists some S ∈ V2 such that ‖S‖2 ≤ 1 and
‖U1 − U2S‖2 ≤
(2) If n = m, then there is a unitary matrix W in V2 such that
‖U1 − U2W‖2 ≤
Proof. Assume that
U∗2U1 =
U11 U12 · · · U1,m+1
U21 U22 · · · U2,m+1
· · · · · · · · · · · ·
Um+1,1 Um+1,2 · · · Um+1,m+1
where Ui,j is a t × t matrix, Ui,m+1 a t × (n − m) matrix, Um+1,j a (n − m) × t matrix for
1 ≤ i, j ≤ m and Um+1,m+1 is a (n−m)× (n−m) matrix.
(1) Let
U11 0 · · · 0 U1,m+1
0 U22 · · · 0 U2,m+1
· · · · · · . . . · · · · · ·
0 0 · · · Um,m Um,m+1
Um+1,1 Um+1,2 · · · Um+1,m Um+1,m+1
It is easy to see that S is in V2, ‖S‖2 ≤ 1 and
δ2 ≥ ‖U1AU∗1 − U2AU∗2‖22 =
Tr((U∗2U1A− AU∗2U1)(U∗2U1A− AU∗2U1)∗)
1≤i 6=j≤m
Tr(|λi − λj|2UijU∗ij)
1≤i 6=j≤m
Tr(UijU
Hence
‖U1 − U2S‖22 = ‖U∗2U1 − S‖22 =
1≤i 6=j≤m
Tr(UijU
ij) ≤
It follows that
‖U1 − U2S‖2 ≤
(2) If n = m, then
V2 = Mt(C)⊕Mt(C)⊕ · · · ⊕Mt(C).
By the construction of S, we can assume S =WH is a polar decomposition of S in V2 for some
unitary matrix W and positive matrix H in V2. Again by the construction of S, we know that
‖S‖ ≤ 1, whence ‖H‖ ≤ 1. From the proven fact that ‖U∗2U1 − S‖2 ≤ δθ , we know that
‖H2 − I‖2 = ‖S∗S − I‖2 ≤
‖H − I‖2 ≤ ‖H2 − I‖2 ≤
It follows that
‖U1 − U2W‖2 ≤ ‖U1 − U2WH‖2 + ‖U2WH − U2W‖2 = ‖U1 − U2S‖2 + ‖H − I‖2 ≤
Lemma 3.2. We have the following results.
(1) For every U ∈ U(k), let
Σ(U) = {W ∈ U(k) | ∃ S ∈ V2 such that ‖S‖2 ≤ 1 and ‖W − US‖2 ≤
Then the volume of Σ(U) is bounded above by
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)2mt2+4m(n−m)t+2(n−m)2
where µ is the normalized Haar measure on the unitary group U(k) and C,C1 are some
constants independent of k, δ, θ.
(2) When n = m, for every U ∈ U(k), let
Σ̃(U) = {W ∈ U(k) | ∃ a unitary matrix W1 in V2 such that ‖W − UW1‖2 ≤
µ(Σ̃(U)) ≤ (C1 · 8δ/θ)k
Proof. (1) By computing the covering number of the set {S | S ∈ V2, such that ‖S‖2 ≤ 1}
by δ/θ-‖ · ‖2-balls in Mk(C), we know
ν2({S | S ∈ V2, ‖S‖2 ≤ 1},
)real dimension of of V2
)2mt2+4m(n−m)t+2(n−m)2
where C is a universal constant. Thus the covering number of the set Σ(U) by the 4δ/θ-‖·‖2-balls
in Mk(C) is bounded by
ν2(Σ(U),
) ≤ ν2({S | S ∈ V2, ‖S‖2 ≤ 1},
)2mt2+4m(n−m)t+2(n−m)2
But the ball of radius 4δ/θ in U(k) has the volume bounded by
µ(ball of radius 4δ/θ) ≤ (C1 · 4δ/θ)k
where C1 is a universal constant. Thus
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)2mt2+4m(n−m)t+2(n−m)2
(2) A slight adaption of the proof of part (1) gives us the proof of part (2). �
Lemma 3.3. Let Ω(A) be defined as in (vi) at the beginning of this subsection.
(1) The covering number of Ω(A) by the 1
δ-‖ · ‖2-balls in Mk(C) is bounded below by
ν2(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
(2) If n = m, then
ν2(Ω(A),
δ) ≥ (C1 · 8δ/θ)−k
)−mt2
Proof. (1) For every U ∈ U(k), define
Σ(U) = {W ∈ U(k) | ∃ S = S∗ ∈ V1, such that ‖S‖2 ≤ 1, ‖W − US‖2 ≤
By preceding lemma, we have
µ(Σ(U)) ≤ (C1 · 4δ/θ)k
)mt2+4m(n−m)t+2(n−m)2
A “parking” (or exhausting) argument will show the existence of a family of unitary elements
{Ui}Ni=1 ⊂ U(k) such that
N ≥ (C1 · 4δ/θ)−k
)−(mt2+4m(n−m)t+2(n−m)2)
Ui is not contained in ∪i−1j=1 Σ(Uj), ∀ i = 1, . . . , N.
From the definition of each Σ(Uj), it follows that
‖Ui − UjS‖2 >
, ∀ S ∈ V2, with ‖S‖2 ≤ 1, ∀1 ≤ j < i ≤ N.
By Lemma 3.1, we know that
‖UiAU∗i − UjAU∗j ‖2 > δ, ∀1 ≤ j < i ≤ N,
which implies that
ν2(Ω(A),
δ) ≥ N ≥ (C1 · 4δ/θ)−k
)−(mt2+4m(n−m)t+2(n−m)2)
(2) is similar as (1). �
3.3. We have following theorem.
Theorem 3.2. Let n ≥ m, δ, θ > 0 and {λ1, λ2, . . . , λm} ∪ {λm+1, . . . , λn} be a family of real
numbers such that
|λi − λj| ≥ θ
for all 1 ≤ i < j ≤ m. Let k be a positive integer such that k − (n−m) is divided by m and
k − n +m
B = diag(λm+1, . . . , λn)
be a diagonal matrix in Mn−m(C) and
A = diag(λ1It, λ2It, . . . , λmIt, B)
be a block-diagonal matrix in Mk(C), where It is the identity matrix in Mt(C). We let
Ω(A) = {U∗AU | U is in U(k)}.
Then the covering number of Ω(A) by the 1
δ-‖ · ‖-balls in Mk(C) is bounded below by
ν∞(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
where C,C1 are some universal constants.
When n = m, we have
ν∞(Ω(A),
δ) ≥ (C1 · 8δ/θ)−k
)−mt2
Proof. Note that
ν∞(Ω(A),
) ≥ ν2(Ω(A),
), ∀ δ > 0.
The result follows directly from preceding lemma. �
3.4. The following proposition, whose proof is skipped, is an easy extension of Lemma 3.3.
Proposition 3.1. Let m, k be some positive integers and θ, δ be some positive numbers. Let
T1, T2, . . . , Tm+1 is a partition of the set {1, 2, . . . , k}, i.e. ∪m+1i=1 Ti = {1, 2, . . . , k} and Ti∩Tj = ∅
for 1 ≤ i 6= j ≤ m+ 1. Let λ1, . . . , λk be some real numbers such that, if 1 ≤ j1 6= j2 ≤ m then
|λi1 − λi2 | > θ, ∀ i1 ∈ Tj1, i2 ∈ Tj2 .
Let A = diag(λ1, λ2, . . . , λk) be a self-adjoint matrix in Mk(C) and
Ω(A) = {U∗AU | U ∈ U(k)}
be a subset of Mk(C).
Let sj be the cardinality of the set Tj for 1 ≤ j ≤ m+ 1. Then the covering number of Ω(A)
by the 1
δ-‖ · ‖2-balls in Mk(C) is bounded below by
ν2(Ω(A),
δ) ≥ (C1 · 4δ/θ)−k
)−2s2
−···−2s2m+1−4(s1+···+sm)sm+1
where C,C1 are some universal constants.
4. Topological free entropy dimension of one variable
Suppose x is a self-adjoint element of a unital C∗ algebra A. In this section, we are going to
compute the topological entropy dimension of x.
4.1. Upperbound.
Proposition 4.1. Suppose x in A is a self-adjoint element with the spectrum σ(x). Then
δtop(x) ≤ 1−
where n is the cardinality of σ(x). Here we assume that 1
Proof. By [26], we know that the inequality always holds when n is infinity. We need only
to show that
δtop(x) ≤ 1−
when n <∞.
Assume that λ1, . . . , λn are in the spectrum of x in A.
Let R > ‖x‖. By Theorem 3.1, for every ω > 0, there are r0 > 0 and ǫ0 > 0 such that, for
all r > r0, ǫ < ǫ0,
A ∈ Γ(top)R (x; k, ǫ, P1, . . . , Pr),
there are some 1 ≤ k1, . . . , kn ≤ k, with k1 + · · · + kn = k and a unitary matrix U in Mk(C)
satisfying
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
0 0 · · · λnIkn
≤ 2ω. (∗∗)
Ω(k1, . . . , kn) =
λ1Ik1 0 · · · 0 0
0 λ2Ik2 · · · 0 0
0 0 · · · λn−1Ikn−1 0
0 0 · · · 0 λnIkn
U∗ | U is in Uk
By Corollary 12 in [21] or Theorem 3 in [2], the covering number of Ω(k1, . . . , kn−1, kn) by
ω-‖ · ‖-balls in Mk(C) is upperbounded by
ν∞(Ω(k1, . . . , kn−1, kn), ω) ≤
where C2 is a constant which does not depend on k, k1, . . . , kn (may depend on n and ‖x‖).
Let I be the set consisting of all these (k1, . . . , kn) in Zn such that 1 ≤ k1, . . . , kn ≤ k and
k1 + · · ·+ kn = k. Then the cardinality of the set I is equal to
(k − 1)!
(n− 1)!(k − n)! .
Note that
k2i ≥ k2/n
for all 1 ≤ k1, . . . , kn ≤ k with k1 + · · ·+ kn = k; and by (∗∗)
(top)
R (x; k, ǫ, P1, . . . , Pr)
is contained in 2ω-neighborhood of the set
(k1,...,kn)∈I
Ω(k1, . . . , kn).
It follows that the covering number of the set
(top)
R (x; k, ǫ, P1, . . . , Pr)
by 3ω-‖ · ‖-balls in Mk(C) is upperbounded by
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω) ≤
(k − 1)!
(n− 1)!(k − n)! ·
)k2−k2/n
δtop(x) ≤ lim sup
lim sup
(k−1)!
(n−1)!(k−n)!
)k2−k2/n
−k2 log(3ω) = 1−
4.2. Lower-bound. We follow the notation from last subsection.
Proposition 4.2. Suppose that x is a self-adjoint element with the finite spectrum σ(x) in A.
δtop(x) ≥ 1−
where n is the cardinality of the set σ(x).
Proof. Suppose that λ1, . . . , λn are distinct spectrum of x. There is some positive number
θ such that
|λi − λj| > θ, ∀ 1 ≤ i 6= j ≤ n.
Assume k = nt for some positive integer t. Let
Ak = diag(λ1It, . . . , λnIt)
be a diagonal matrix in Mk(C) where It is the t× t identity matrix. It is easy to see that, for
all R > ‖x‖, r ≥ 1 and ǫ > 0, we have
Ak ∈ Γ(top)R (x; k, ǫ, P1, . . . , Pr).
For any ω > 0, applying Theorem 3.2 for n = m and δ = 1
ω, we have
(top)
R (x; k, ǫ, P1, . . . , Pr), ω) ≥ (C1 · 8δ/θ)−k
)−mt2
= (16C1ω/θ)
−k2 ·
)−mt2
Note that k = nt = mt and θ is some fixed number. A quick computation shows that
δtop(x) ≥ 1−
Proposition 4.3. Suppose that x is a self-adjoint element in A with infinite spectrum. Then
δtop(x) ≥ 1.
Proof. For any 0 < θ < 1, there are λ1, . . . , λm in the spectrum of x, σ(x), satisfying (i)
|λi − λj | ≥ θ;
and (ii) for any λ in σ(x), there is some λj with |λ − λj| ≤ θ. By functional calculus, for any
R > ‖x‖, r ≥ 1 and ǫ > 0, there are some positive integer n ≥ m and real numbers λm+1, . . . , λn
in σ(x) satisfying: for every t ≥ 1 the matrix
A = diag(λ1It, λ2It, . . . , λmIt, λm+1, . . . , λn)
is in
(top)
R (x; k, ǫ, P1, . . . , Pr),
where we assume that k = mt + n−m. For any ω > 0, let δ = 1
ω. By Theorem 3.2, we know
(top)
R (x; k, ǫ, P1, . . . , Pr), ω) ≥ (C1 · 4δ/θ)−k
)−(2mt2+4m(n−m)t+2(n−m)2)
lim sup
log(ν∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), ω))
−k2 logω ≥
log(4C1
)− log θ
+ 1 +
log(2C) + log θ
log ω
Then,
δtop(x) ≥ 1−
When θ goes to 0, m goes to infinity as σ(x) has infinitely many elements. Therefore,
δtop(x) ≥ 1.
4.3. Topological free entropy dimension in one variable case. By Proposition 4.1,
Proposition 4.2 and Proposition 4.3, we have the following result.
Theorem 4.1. Suppose x is a self-adjoint element in a unital C∗ algebra A. Then
δtop(x) = 1−
where n is the cardinality of the set σ(x) and σ(x) is the set of spectrum of x in A. Here we
assume that 1
5. Topological free entropy dimension of n-tuple in unital C∗ algebras
5.1. An equivalent definition of topological free entropy dimension. Suppose that
A is a unital C∗ algebra and x1, . . . , xn, y1, . . . , ym are self-adjoint elements in A. For every
R, ǫ > 0 and positive integers r, k, let
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
be Voiculescu’s norm-microstate space defined in section 2.4.
Define
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω)
to be the covering number of the set Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr) by ω-‖·‖2-balls
in the metric space (Ms.ak (C))n equipped with trace norm (see Definition 2.2).
Definition 5.1. Define
δ̃top(x1, . . . , xn : y1, . . . , ym;ω)
= sup
ǫ>0,r∈N
lim sup
log(ν2(Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr), ω))
−k2 logω
δ̃top(x1, . . . , xn : y1, . . . , ym) = lim sup
δ̃top(x1, . . . , xn : y1, . . . , ym;ω)
The following proposition was pointed out by Voiculescu in [26]. For the sake of complete-
ness, we also include a proof here.
Proposition 5.1. Suppose that A is a unital C∗ algebra and x1, . . . , xn, y1, . . . , ym are self-
adjoint elements in A. Then
δ̃top(x1, . . . , xn : y1, . . . , ym) = δtop(x1, . . . , xn : y1, . . . , ym),
where δtop(x1, . . . , xn : y1, . . . , ym) is the topological free entropy dimension of x1, . . . , xn in
presence of y1, . . . , ym.
Proof. This is an easy consequence of Lemma 1 in [21]. Let λ be the Lebesgue measure
on (Ms.ak (C))n. Let, for every ω > 0,
B∞(ω) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)‖ ≤ ω}
B2(ω) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)‖2 ≤ ω}
It follows from the results in [21] or Theorem 8 in [2] that, for some M1,M2 independent of k, ω
such that
λ(B∞(1)) ≤ λ(B∞(ω/4))
λ(B2(2
nω)) ≤ λ(B2(1)). (5.1.1)
For every ω > 0 and any subset set K of (Ms.ak (C))n , let
K(ω, ‖ · ‖) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)− (D1, . . . , Dn)‖ ≤ ω
for some (D1, . . . , Dn) ∈ K}
K(ω, ‖ · ‖2) = {(A1, . . . , An) ∈ (Ms.ak (C))n | ‖(A1, . . . , An)− (D1, . . . , Dn)‖2 ≤ ω
for some (D1, . . . , Dn) ∈ K}
Note the following fact:
‖(A1, . . . , An)‖2 ≤
n‖(A1, . . . , An)‖, ∀ (A1, . . . , An) ∈ (Ms.ak (C))n.
It follows from Lemma 1 in [21] that
ν∞(K,ω) ≤
λ(K(ω, ‖ · ‖))
λ(B∞(ω/4))
nω, ‖ · ‖2))
λ(B2(2
≤ ν2(K(
nω, ‖ · ‖2), 2
Combining with the equalities (5.1.1), we get
ν∞(K,ω) ≤
λ(K(ω, ‖ · ‖))
λ(B∞(ω/4))
λ(K(ω, ‖ · ‖))
λ(B∞(1))
nω, ‖ · ‖2))
λ(B∞(1))
nω, ‖ · ‖2))
λ(B2(1))
≤ λ(K(
nω, ‖ · ‖2))
λ(B2(2
≤ ν2(K(
nω, ‖·‖2), 2
nω) ≤ ν2(K,
Therefore, we have
ν2(K,
nω) ≤ ν∞(K,ω) ≤
λ(B2(1))
λ(B∞(1))
ν2(K,
It is a well-known fact (for example see Theorem 8 in [2]) that
λ(B2(1))
λ(B∞(1))
≤ Cnk23
for some universal constant C3 > 0. Hence
ν2(K,
nω) ≤ ν∞(K,ω) ≤
nM1C3
ν2(K,
Let K be Γ
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr). By the definitions of δ̃top and δtop, we
δ̃top(x1, . . . , xn : y1, . . . , ym) = δtop(x1, . . . , xn : y1, . . . , ym).
5.2. Upper-bound of topological free entropy dimension in a unital C∗ algebra.
Let us recall Voiculescu’s definition of free dimension capacity in [26].
Definition 5.2. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) is the set consisting of all tracial states of A. If TS(A) 6= ∅,
define Voiculescu’s free dimension capacity κδ(x1, . . . , xn) of x1, . . . , xn as follows,
κδ(x1, . . . , xn) = sup
τ∈TS(A)
δ0(x1, . . . , xn : τ),
where δ0(x1, . . . , xn : τ) is Voiculescu’s (von Neumann algebra) free entropy dimension of
x1, . . . , xn in 〈A, τ〉.
The relationship between topological free entropy dimension of a unital C∗ algebra with a
unique tracial state and its free dimension capacity is indicated by the following result.
Theorem 5.1. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) is the set consisting of all tracial states of A. If TS(A) is a set
with a single element, then
δtop(x1, . . . , xn) ≤ κδ(x1, . . . , xn).
To prove the preceding theorem, we need the following lemma.
Sublemma 5.2.1. Suppose that A is a unital C∗ algebra with a family of self-adjoint generators
x1, . . . , xn. Suppose that TS(A) 6= ∅ is the set consisting of all tracial states of A. Let R >
max{‖x1‖, . . . , ‖xn‖} be some positive number. Then for any m ≥ 1, there is some rm ∈ N such
(top)
R (x1, . . . , xn; k,
, P1, . . . , Prm) ⊆ ∪τ∈TS(A)ΓR(x1, . . . , xn; k,m,
; τ), ∀ k ≥ 1
where ΓR(x1, . . . , xn; k,m,
; τ) is microstate space of x1, . . . , xn with respect to τ (see [23]).
Proof of Sublemma 5.2.1: We will prove the result by contradiction. Suppose, to the
contrary, there is some m0 ≥ 1 so that following holds: for any r ∈ N, there are some kr ≥ 1
and some
1 , A
2 , . . . , A
n ) ∈ Γ
(top)
R (x1, . . . , xn; kr,
, P1, . . . , Pr)
satisfying
1 , A
2 , . . . , A
n ) /∈ ∪τ∈TS(A)ΓR(x1, . . . , xn; kr, m,
; τ). (5.2.1)
Let α be a free ultrafilter in β(N)\N. Let N =
r=1Mkr(C) be the von Neumann algebra ultra-
product of {Mkr(C)}∞r=1 along the ultrafilter α, i.e.
r=1Mkr(C) is the quotient algebra of the
C∗ algebra
r=1Mkr(C) by I2, the 0-ideal of the trace τα, where τα((Ar)∞r=1) = limr→α
Tr(Ar)
Let, for each 1 ≤ j ≤ n, aj = [(A(j)r )∞r=1] be a self-adjoint element in N . By mapping xj to aj ,
there is a unital ∗-homomorphism ψ from the C∗ algebra A onto the C∗ subalgebra generated
by {a1, . . . , an} in N .
Let τ0 be the tracial state on A which is induced by τα on ψ(A), i.e.
τ0(x) = τα(ψ(x)), ∀ x ∈ A.
It follows that when r is large enough,
1 , A
2 , . . . , A
n ) ∈ ΓR(x1, . . . , xn; kr, m,
; τ0),
which contradicts with the inequality (5.2.1). This complete the proof. �
Proof of Theorem 5.1: Let R > max{‖x1‖, . . . , ‖xn‖}. Let τ be the unique trace of
A. By Sublemma 5.2.1, for any m ≥ 1, there is r ∈ N such that
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr) ⊆ ΓR(x1, . . . , xn; k,m,
; τ), ∀ k ≥ 1.
Therefore, for any 1 > ω > 0, we have
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr), ω) ≤ ν2(ΓR(x1, . . . , xn; k,m,
; τ), ω), ∀ k ≥ 1.
Now it is easy to check that
δ̃top(x1, . . . , xn) ≤ δ(x1, . . . , xn; τ) = κδ(x1, . . . , xn).
By Proposition 5.1, we know that
δtop(x1, . . . , xn) ≤ κδ(x1, . . . , xn).
Remark 5.1. Combining Theorem 5.1 with the results in [11] or [14], we will be able to compute
the upper-bound of topological free entropy dimension for a large class of unital C∗ algebras. For
example, δtop(x1, . . . , xn) ≤ 1 if x1, . . . , xn is a family of self-adjoint operators that generates an
irrational rotation algebra A.
5.3. Lower-bound of topological free entropy dimension in a unital C∗ algebra.
In this subsection, we assume that A is a finitely generated, infinite dimensional, unital simple
C∗ algebra with a unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint
generators of A. Let H be the Hilbert space L2(A, τ). Without loss of generality, we might
assume that A is faithfully represented on the Hilbert space H . Let M be the von Neumann
algebra generated by A on H . It is not hard to see that M is a diffuse von Neumann algebra
with a tracial state τ .
For each positive integer m, there is a family of mutually orthogonal projections p1, . . . , pm
in M such that τ(pj) = 1/m for 1 ≤ j ≤ m. Let
ym = 1 · p1 + 2 · p2 + · · ·+m · pm =
j · pj ∈ M.
Let {Pr(x1, . . . , xn)}∞r=1 be defined as in section 2.3. Thus {Pr(x1, . . . , xn)}∞r=1 is dense in M
with respect to the strong operator topology. Hence, for each m ≥ 1, there is some self-adjoint
element Prm(x1, . . . , xn) in A such that
‖ym − Prm(x1, . . . , xn)‖2 ≤
where ‖a‖2 =
τ(a∗a) for all a ∈ M.
Lemma 5.1. Let A be finitely generated, infinite dimensional, unital simple C∗ algebra with a
unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint generators of A. Let
H, M be defined as above. For each m ≥ 1, let ym and Prm(x1, . . . , xn) be chosen as above.
δtop(x1, . . . , xn) ≥ δtop(Prm(x1, . . . , xn) : x1, . . . , xn).
Proof. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}. There exists a positive constant
D > 1 such that
‖Prm(A1, . . . , An)− Prm(B1, . . . , Bm)‖ ≤ D‖(A1, . . . , An)− (B1, . . . , Bm)‖
for all A1, . . . , An, B1, . . . , Bn in Mk(C) satisfying 0 ≤ ‖A1‖, . . . , ‖An‖, ‖B1‖, . . . , ‖Bn‖ ≤ R.
Then it is not hard to verify that, for ω > 0,
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω)
≤ ν∞(Γ(top)R (x1, . . . , xn; k, ǫ, P1, . . . , Pr),
for each r ≥ rm and ǫ < ω4 . By definition of δtop and Remark 2.3, we have
δtop(Prm(x1, . . . , xn) : x1, . . . , xn) ≤ δtop(x1, . . . , xn).
Definition 5.3. Suppose A is a unital C∗ algebra and x1, . . . , xn is a family of self-adjoint ele-
ments of A that generates A as a C∗ algebra. If for any R > max{‖x1‖, . . . , ‖xn‖, ‖y1‖, . . . , ‖ym‖},
r > 0, ǫ > 0, there is a sequence of positive integers k1 < k2 < · · · such that
(top)
R (x1, . . . , xn : y1, . . . , ym; ks, ǫ, P1, . . . , Pr) 6= ∅, ∀ s ≥ 1
then A is called having approximation property.
Lemma 5.2. Let A be a finitely generated, infinite dimensional, unital simple C∗ algebra with a
unique tracial state τ . Assume that A has approximation property. Assume that x1, . . . , xn is a
family of self-adjoint generators of A. Let H, M be defined as above. Let m be a positive integer.
Let ym and Prm(x1, . . . , xn) be chosen as above. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}.
Then there is some positive integer r > rm so that the following hold: ∀ k ≥ 1, if
(B,A1, . . . , An) ∈ Γ(top)R (Prm(x1, . . . , xn), x1, . . . , xn; k,
, P1, . . . , Pr),
then there are some 1 ≤ k1, . . . , km ≤ k with 1m −
for each 1 ≤ j ≤ m and
k1 + · · ·+ km = k, and a unitary matrix U in U(k) satisfying
‖B − U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
Proof. We will prove the result by contradiction. Assume, to the contrary, for all r ≥ rm
there are some kr ≥ 1 and some
(B(r), A
1 , . . . , A
n ) ∈ Γ
(top)
R (Prm(x1, . . . , xn), x1, . . . , xn; kr,
, P1, . . . , Pr),
satisfying
‖B(r) − U
1 · Is1 0 · · · 0
0 2 · Is2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ism
U∗‖2 >
, (5.3.1)
for all 1 ≤ s1, . . . , sm ≤ kr with 1m −
for each 1 ≤ j ≤ n and s1+ · · ·+ sm = kr,
and all unitary matrix U in U(k).
Let α be a free ultrafilter in β(N) \N. Let N =
r=1Mkr(C) be the von Neumann algebra
ultraproduct of {Mkr(C)}∞r=1 along the ultrafilter α, i.e.
r=1Mkr(C) is the quotient of the
C∗ algebra
r=1Mkr(C) by I2, the 0-ideal of the trace τα, where τα((Ar)∞r=1) = limr→α
Tr(Ar)
Let, for each 1 ≤ j ≤ n, aj = [(A(j)r )∞r=1] be a self-adjoint element in N . By mapping xj to aj ,
there is a unital ∗-homomorphism ψ from the C∗ algebra A onto the C∗ subalgebra generated
by {a1, . . . , an} in N . Since A is a simple C∗ algebra and ψ(IA) = IN , ψ actually is a ∗-
isomorphism. Since A has a unique trace τ , ψ induces a ∗-isomorphism (still denoted by ψ)
from M onto the von Neumann subalgebra generated by a1, . . . , an in N . Therefore,
‖ym − Prm(x1, . . . , xn)‖2 = ‖ψ(ym)− Prm(a1, . . . , an)‖2,τα ≤
This contradicts with the definition of ym and inequality (5.3.1). �
The following lemma is well-known (for example, see Lemma 4.1 in [23]).
Lemma 5.3. Suppose A, or B, is a self-adjoint matrix in Ms.a.k (C) with a list of eigenvalues
λ1 ≤ λ2 ≤ · · · ≤ λk, or µ1 ≤ µ2 ≤ · · · ≤ µk respectively. Then
|λj − µj |2 ≤ Tr((A− UBU∗)2),
where U is any unitary matrix in U(k).
Lemma 5.4. Let r,m be some positive integer with 4 < m < r. Suppose k1, . . . , km is a family
of positive integers such that 1
for all 1 ≤ j ≤ k and k1 + · · ·+ km = k. If A
is a self-adjoint matrix in Mk(C) such that, for some unitary matrix U in U(k),
‖A− U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
then, for any ω > 0 we have
ν2(Ω(A), ω) ≥ (8C1ω)−k
for some constants C1, C > 1 independent of k, ω, where
Ω(A) = {W ∗AW | W ∈ U(k)}.
Proof. Suppose that λ1 ≤ λ2 ≤ . . . ≤ λk are the eigenvalues of A. For each 1 ≤ j ≤ m, let
Tj = {i ∈ N | (
kt) + 1 ≤ i ≤
kt and |λi − j| ≤
T̂j = {(
kt) + 1, (
kt) + 2, · · · ,
kt} \ Tj ,
here we assume that k0 = 0. Let B = diag(1 · Ik1, · · · , m · Ikm) be a diagonal matrix in Mk(C).
By Lemma 5.3, we have
≥ Tr((A− UBU∗)2) ≥
i ∈T̂j
|λi − j|2 ≥
card(T̂j),
where card(T̂j) is the cardinality of the set T̂j . Thus
card(T̂j) ≤
, for 1 ≤ j ≤ m.
Let sj = card(Tj) for 1 ≤ j ≤ m, whence
≥ kj ≥ sj = kj − card(T̃j) ≥ kj −
, ∀ 1 ≤ j ≤ m.
Tm+1 = {1, 2, . . . , k} \ (∪nj=1Tj)
and sm+1 be the cardinality of the set Tm+1. Thus
sm+1 = k − s1 − · · · − sm =
card(T̂j) ≤
It is not hard to see that T1, . . . , Tm+1 is a partition of the set {1, 2, . . . , k}. Moreover, if
1 ≤ j1 6= j2 ≤ m then for any
i1 ∈ Tj1 , and i2 ∈ Tj2
we have
|λi1 − λi2| ≥ |j2 − j1| − |λi2 − j2| − |λi1 − j1| ≥ 1−
Applying Proposition 3.1 for such T1, . . . , Tm, Tm+1, θ = 1/2 and ω = δ/2, we have
ν2(Ω(A), ω) ≥ (8C1ω)−k
)−2s2
−···−2s2m−2s
−4(s1+···+sm)sm+1
≥ (8C1ω)−k
)−2(k2
+···+k2m+(
)2+2k· 4k
≥ (8C1ω)−k
)−2(( k
)2+···+( k
)2+ 16k
≥ (8C1ω)−k
for some constants C,C1 > 1 independent of k, ω.
Lemma 5.5. Let A be a finitely generated, infinite dimensional, simple unital C∗ algebra with a
unique tracial state τ . Assume that A has approximation property. Assume that x1, . . . , xn is a
family of self-adjoint generators of A. Let H, M be defined as above. Let m be a positive integer.
Let ym and Prm(x1, . . . , xn) be chosen as above. Let R > max{‖Prm(x1, . . . , xn)‖, ‖x1‖, . . . , ‖xn‖}.
When r is large enough and ǫ is small enough, for any ω > 0, we have
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω) ≥ (8C1ω)−k
Proof. By Lemma 5.2, when r is large enough and ǫ is small enough, the following hold:
∀ k ≥ 1, if
(B,A1, . . . , An) ∈ Γ(top)R (Prm(x1, . . . , xn), x1, . . . , xn; k, ǫ, P1, . . . , Pr),
then there are some 1 ≤ k1, . . . , km ≤ k with 1m −
for each 1 ≤ j ≤ m and
k1 + · · ·+ km = k, and a unitary matrix U in U(k) satisfying
‖B − U
1 · Ik1 0 · · · 0
0 2 · Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · m · Ikm
U∗‖2 ≤
Combining with Lemma 5.4, we know that if
B ∈ Γ(top)R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr)
then, for any ω > 0,
ν2(Ω(B), ω) ≥ (8C1ω)−k
where
Ω(B) = {W ∗BW | W ∈ U(k)}.
Note that Ω(B) ⊂ Γ(top)R (Prm(x1, . . . , xn) : x1, . . . , xn; k, r, ǫ). It follows that, for any ω > 0,
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, r, ǫ), ω) ≥ (8C1ω)−k
−56k2
Now we have the following result.
Theorem 5.2. Let A be a finitely generated, infinite dimensional, simple unital C∗ algebra with
a unique tracial state τ . Assume that x1, . . . , xn is a family of self-adjoint generators of A. If
A has approximation property, then δtop(x1, . . . , xn) ≥ 1.
Proof. Let H be the Hilbert space L2(A, τ). Without loss of generality, we might assume
that A is faithfully represented on the Hilbert space H . Let M be the von Neumann algebra
generated by A on H . It is not hard to see that M is a diffuse von Neumann algebra with a
tracial state τ . For each positive integer m, there is a family of mutually orthogonal projections
p1, . . . , pm in M such that τ(pj) = 1/m for 1 ≤ j ≤ m. Let
ym = 1 · p1 + 2 · p2 + · · ·+m · pm =
j · pj.
Let {Pr(x1, . . . , xn)}∞r=1 be defined as in section 2.3. Thus {Pr(x1, . . . , xn)}∞r=1 is dense in M
with respect to the strong operator topology. Hence, for each m ≥ 1, there is some self-adjoint
element Prm(x1, . . . , xn) in A such that
‖ym − Prm(x1, . . . , xn)‖2 ≤
By Lemma 5.5, for any ω > 0, when r is large enough and ǫ is small enough, we have for
some constants C1, C > 1 independent of k, ω
(top)
R (Prm(x1, . . . , xn) : x1, . . . , xn; k, ǫ, P1, . . . , Pr), ω) ≥ (8C1ω)−k
Therefore,
δ̃top(Prm(x1, . . . , xn) : x1, . . . , xn) ≥ 1−
By Proposition 5.1, we get
δtop(Prm(x1, . . . , xn) : x1, . . . , xn) ≥ 1−
By Lemma 5.1,
δtop(x1, . . . , xn) ≥ 1−
Since m is an arbitrary positive integer, we obtain
δtop(x1, . . . , xn) ≥ 1.
5.4. Values of topological free entropy dimensions in some unital C∗ algebras. In
this subsection, we are going to compute the values of topological free entropy dimensions in
some unital C∗ algebras by using the results from preceding subsection.
Theorem 5.3. Let Aθ be an irrational rotation C∗ algebra. Then
δtop(x1, . . . , xn) = 1
where x1, . . . , xn is a family of self-adjoint operators that generates Aθ.
Proof. Note that Aθ is an infinite dimensional, unital simple C∗ algebra with a unique
tracial state τ . By [24] or [11] and Theorem 5.1, we know that
δtop(x1, . . . , xn) ≤ 1.
It follows from [18] that Aθ has approximation property. Therefore
δtop(x1, . . . , xn) ≥ 1.
Hence
δtop(x1, . . . , xn) = 1.
Theorem 5.4. Let A be a UHF algebra (uniformly hyperfinite C∗ algebra). Then
δtop(x1, . . . , xn) = 1
where x1, . . . , xn is a family of self-adjoint operators that generates A.
Proof. By [17], we know that A is generated by two self-adjoint elements. It is not hard
to see that A is an infinite dimensional, unital simple C∗ algebra with a unique tracial state τ .
By [24] or [11] and Theorem 5.1, we know that
δtop(x1, . . . , xn) ≤ 1.
It is easy to check that A has approximation property. Therefore
δtop(x1, . . . , xn) ≥ 1.
Hence
δtop(x1, . . . , xn) = 1.
Recall that for any sequence (Am)∞m=1 of C∗ algebras,we can introduce two C∗ algebras
Am = {(am)∞m=1 | am ∈ Am, sup
‖am‖ <∞}
Am = {(am)∞m=1 | am ∈ Am, lim
‖am‖ = 0}
The norm in the quotient C∗ algebra
m Am/
mAm is given by
‖ρ((am)∞m=1)‖ = lim sup
‖xm‖,
where ρ is the quotient map from
mAm onto
If A is an exact C∗ algebra, then the sequence
0 → A⊗min
Mm(C) → A⊗min
Mm(C) → A⊗min (
Mm(C)/
Mm(C)) → 0
is exact. Therefore, we have the following natural identification
A⊗min (
Mm(C)/
Mm(C)) = (A⊗min
Mm(C))/(A⊗min
Mm(C)).
On the other hand, we have the following natural embedding
A⊗min
Mm(C) ⊆
Mm(A)
and the identification
A⊗min
Mm(C) =
Mm(A)
Thus we have for any exact C∗ algebra A a natural embedding
ψ : A⊗min (
Mm(C)/
Mm(C) ⊆
Mm(A)/
Mm(A).
Lemma 5.6. Suppose that A and B are unital C∗ algebras and ρ is an unital embedding
ρ : A →
Mm(B)/
Mm(B).
Suppose that x1, . . . , xn is a family of elements in A. Suppose r is a positive integer and
{Pj(x1, . . . , xn)}rj=1 is a family of noncommutative polynomials of x1, . . . , xn. Then there are
some k ∈ N and a(k)1 , . . . , a
n in Mk(B) so that
|‖Pj(a(k)1 , . . . , a(k)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Proof. We might assume that
ρ(xi) = [(x
i )m] ∈
Mm(B)/
Mm(B), ∀ 1 ≤ i ≤ n.
By the definition of
mMm(B)/
mMm(B), there are some positive integers m1 ≤ m2 such
|( sup
m1≤l≤m2
‖Pj(x(l)1 , . . . , x(l)n )‖)− ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Let k =
j and
i = ⊕m2l=m1x
i ∈ Mk(B), ∀ 1 ≤ i ≤ n.
Then, it is not hard to check that
|‖Pj(a(k)1 , . . . , a(k)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 1 ≤ j ≤ r.
Theorem 5.5. Let p ≥ 2 be a positive integer and Fp be the free group on p generators. Let
C∗red(Fp)⊗minC∗red(Fp) be a minimal tensor product of two reduced C∗ algebras of free groups Fp.
δtop(x1, . . . , xn) = 1,
where x1, . . . , xn is any family of self-adjoint generators of C
red(Fp)⊗min C∗red(Fp).
Proof. Note that C∗red(Fp)⊗minC∗red(Fp) is an infinite dimensional, unital simple C∗ algebra
with a unique tracial state. By the result from [5] or [11] and Theorem 5.1, Theorem 5.2, to
show δtop(x1, . . . , xn) = 1, we need only to show that C
red(Fp)⊗min C∗red(Fp) has approximation
property. Therefore, it suffices to show the following: Let R > max{‖x1‖, . . . , ‖xn‖}. For any
r ≥ 1, there is some k ∈ N so that
(top)
R (x1, . . . , xn; k,
, P1, . . . , Pr) 6= ∅.
By the result from [9], we know there is a unital embedding
φ1 : C
red(Fp) →
Mm(C)/
Mm(C),
which induce a unital embedding
φ2 : C
red(Fp)⊗min C∗red(Fp) → C∗red(Fp)⊗min (
Mm(C)/
Mm(C))
Note that C∗red(Fp) is an exact C
∗ algebra. From the explanation preceding the theorem it
follows that there is a unital embedding
φ3 : C
red(Fp)⊗min C∗red(Fp) →
Mm(C∗red(Fp))/
Mm(C∗red(Fp)).
By Lemma 5.6, for a family of elements x1, . . . , xn in C
red(Fp) ⊗min C∗red(Fp) and r ≥ 1, there
are some m ∈ N and some a(m)1 , . . . , a
n in Mm(C∗red(Fp)) so that max{‖a1‖, . . . , ‖an‖} < R
|‖Pj(a(m)1 , . . . , a(m)n )‖ − ‖Pj(x1, . . . , xn)‖| ≤
, ∀ 0 ≤ j ≤ r.
On the other hand, by the existence of embedding
φ1 : C
red(Fp) →
Mm′(C)/
Mm′(C),
it follows that there is a unital embedding
φ4 : Mm(C∗red(Fp)) = Mm(C)⊗min C∗red(Fp) → Mm(C)⊗min (
Mm′(C)/
Mm′(C))
Mm(C)⊗min (
Mm′(C)/
Mm′(C)) =
Mm′m(C)/
Mm′m(C).
Hence for such a
1 , . . . , a
n in Mm(C∗red(Fp)) and r ≥ 1, by Lemma 5.6, there are some k ∈ N
and A1, . . . , An in Mk(C) so that max{‖A1‖, . . . , ‖An‖} < R and
|‖Pj(a(m)1 , . . . , a(m)n )‖ − ‖Pj(A1, . . . , An)‖| ≤
, ∀ 0 ≤ j ≤ r.
Altogether, we have
|‖Pj(x1, . . . , xn)‖ − ‖Pj(A1, . . . , An)‖| ≤
, ∀ 0 ≤ j ≤ r,
which implies that C∗red(Fp)⊗min C∗red(Fp) has approximation property.
Hence
δtop(x1, . . . , xn) = 1,
for any family of self-adjoint elements x1, . . . , xn that generates C
red(Fp)⊗min C∗red(Fp). �
Theorem 5.6. Suppose that K be the C∗ algebra consisting of all compact operators on a sepa-
rable Hilbert space H. Suppose A = C
K is the unitization of K. If x1, . . . , xn is a family of
self-adjoint elements that generate A as a C∗ algebra, then
δtop(x1, . . . , xn) = 0.
Proof. By [17], we know that unital C∗ algebra A is generated by two self-adjoint elements
in A. Note that A has a unique trace τ , which is defined by
τ((λ, x)) = λ, ∀ (λ, x) ∈ A.
By Theorem 5.1, it is not hard to see that
δtop(x1, . . . , xn) = 0,
where x1, . . . , xn is a family of self-adjoint generators of A.
6. Topological free orbit dimension of C∗ algebras
Assume that A is a unital C∗-algebra. Let x1, . . . , xn, y1, . . . , ym be self-adjoint elements
in A. Let C〈X1, . . . , Xn, Y1, . . . , Ym〉 be the noncommutative polynomials in the indeterminates
X1, . . . , Xn, Y1, . . . , Ym. Let {Pr}∞r=1 be the collection of all noncommutative polynomials in
C〈X1, . . . , Xn, Y1, . . . , Ym〉 with rational coefficients.
6.1. Unitary orbits of balls in Mk(C)n. We let Mk(C) be the k× k full matrix algebra
with entries in C, and U(k) be the group of all unitary matrices in Mk(C). Let Mk(C)n denote
the direct sum of n copies of Mk(C). Let Ms.ak (C) be the subalgebra of Mk(C) consisting of
all self-adjoint matrices of Mk(C). Let (Ms.ak (C))n be the direct sum of n copies of Ms.ak (C).
For every ω > 0, we define the ω-orbit-‖ · ‖-ball U(B1, . . . , Bn;ω) centered at (B1, . . . , Bn) in
Mk(C)n to be the subset of Mk(C)n consisting of all (A1, . . . , An) in Mk(C)n such that there
exists some unitary matrix W in U(k) satisfying
‖(A1, . . . , An)− (WB1W ∗, . . . ,WBnW ∗)‖ < ω.
6.2. Norm-microstate space. For all integers r, k ≥ 1, real numbers R, ǫ > 0 and non-
commutative polynomials P1, . . . , Pr, we let
(top)
R (x1, . . . , xn : y1, . . . , ym; k, ǫ, P1, . . . , Pr)
be as defined as in section 2.4.
6.3. Topological free orbit dimension.
Definition 6.1. For ω > 0, we define the covering number
(top)
R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr), ω)
to be the minimal number of ω-orbit–‖·‖-balls that cover Γ(top)R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr)
with the centers of these ω-orbit-‖ · ‖-balls in Γ(top)R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr)
For each function f : N× N× R+ → R, we define,
kf(x1, . . . , xn : y1, . . . , yp;ω,R)
= inf
r∈N,ǫ>0
lim sup
f(o∞(Γ
(top)
R (x1, . . . , xn : y1, . . . , yp; k, ǫ, P1, . . . , Pr), ω), k, ω)
kf(x1, . . . , xn : y1, . . . , yp;ω) = sup
kf(x1, . . . , xn : y1, . . . , yp;ω,R)
kf(x1, . . . , xn : y1, . . . , yp) = lim sup
kf(x1, . . . , xn : y1, . . . , yp;ω),
where kf(x1, , . . . , xn : y1, . . . , yp) is called the topological f(·)-free-orbit-dimension of x1, . . . , xn
in the presence of y1, . . . , yp.
6.4. Topological free entropy dimension and topological free orbit dimension.
The following result follows directly from the definitions of topological free entropy dimension
and topological free orbit dimension of n-tuple of self-adjoint elements in a C∗ algebra.
Theorem 6.1. Suppose that A is a unital C∗ algebra and x1, . . . , xn is a family of self-adjoint
elements of A. Let f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
−k2 logω
for s, k ∈ N, ω > 0. Then
δtop(x1, . . . , xn) ≤ kf(x1, . . . , xn) + 1.
7. Topological free orbit dimension of one variable
We recall the packing number of a set in a metric space as follows.
Definition 7.1. Suppose that X is a metric space with a metric distance d. (i) The packing
number of a set K by ω-balls in X, denoted by P (K,ω), is the maximal cardinality of the subsets
F in K satisfying for all a, b in F either a = b or d(a, b) ≥ ω. (ii) The packing dimension of
the set K in X, denoted by d(K), is defined by
d(K) = lim sup
log(P (K,ω))
− log ω .
7.1. Upper-bound of the topological free orbit dimension of one variable. Suppose
that x = x∗ is a self-adjoint element in a unital C∗ algebra A and σ(x) is the spectrum of x in
For any ω > 0, let m = P (K,ω) be the packing number of σ(x) in R. Thus there exists a
family of elements λ1, . . . , λm in σ(x) such that (i) |λi − λj | ≥ ω for all 1 ≤ i 6= j ≤ m; and (ii)
for any λ in σ(x), there is some λj with 1 ≤ j ≤ m satisfying |λ− λj | ≤ ω.
Lemma 7.1. For any given R > ‖x‖, when r is large enough and ǫ is small enough, we have
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω)
log k
Proof. By Theorem 3.1, there exist some r0 ≥ 1 and ǫ0 > 0 such that the following
holds: when r > r0, ǫ < ǫ0, for any A in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), there are positive integers
1 ≤ k1, . . . , km ≤ k with k1 + · · ·+ km = k and some unitary matrix U in Mk(C) satisfying
‖U∗AU −
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λmIkm
‖ ≤ 2ω,
where Ikj is the kj × kj identity matrix for 1 ≤ j ≤ m.
Ω(k1, . . . , km) =
λ1Ik1 0 · · · 0
0 λ2Ik2 · · · 0
· · · · · · . . . · · ·
0 0 · · · λmIkm
U | U is in Uk
Let J be the set consisting of all these (k1, . . . , km) ∈ Nm with k1 + · · ·+ km = k. Then the
cardinality of the set J is equal to
(k − 1)!
(m− 1)!(k −m)! .
(top)
R (x; k, ǫ, P1, . . . , Pr)
is contained in 2ω-neighborhood of the set
(k1,...,km)∈J
Ω(k1, . . . , km).
It follows that
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω) ≤ o∞(
(k1,...,km)∈J
Ω(k1, . . . , km), ω) ≤ |J | =
(k − 1)!
(m− 1)!(k −m)! .
Therefore,
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), 3ω)
log k
= lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), ω)
log k
≤ lim sup
(k−1)!
(m−1)!(k−m)!
log k
= m− 1.
7.2. Lower-bound. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A
and σ(x) is the spectrum of x in A.
Lemma 7.2. We have
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr),
log k
≥ m− 1.
Proof. For any ω > 0, let m = P (K,ω) be the packing number of σ(x) in R. Thus there
exists a family of elements λ1, . . . , λm in σ(x) such that (i) |λi − λj | ≥ ω for all 1 ≤ i 6= j ≤ m;
and (ii) for any λ in σ(x), there is some λj with 1 ≤ j ≤ m satisfying |λ− λj| ≤ ω.
For any R > ‖x‖, r ≥ 1 and ǫ > 0, by functional calculus, there are λm+1, . . . , λn in σ(x)
such that for every 1 ≤ t1, . . . , tm ≤ k − n with 2nt1 + . . .+ 2ntm = k − n, the matrix
A = diag(λ1I2nt1 ,λ2I2nt2 , . . . , λmI2ntm , λ1, . . . , λm, . . . , λn)
is in Γ
(top)
R (x; k, ǫ, P1, . . . , Pr), (7.2.1)
where we assume that 2n|(k − n).
Let J be the set consisting of all these (t1, . . . , tm) ∈ Nm with 2nt1 + . . . + 2ntm = k − n.
Then the cardinality of the set J is equal to
!(m− 1)!
By Weyl’s theorem in [27] on the distance of unitary orbits of two self-adjoint matrices, for
any two distinct elements
(s1, . . . , sm) and (t1, . . . , tm)
in J and any W in U(k), we have
‖A1 −WA2W ∗‖ ≥ ω,
where
A1 = diag(λ1I2nt1 , λ2I2nt2 , . . . , λmI2ntm , λ1, . . . , λm, . . . , λn)
A2 = diag(λ1I2ns1, λ2I2ns2 , . . . , λmI2nsm, λ1, . . . , λm, . . . , λn)
are two diagonal self-adjoint matrices in Mk(C). Combining with (7.2.1), we have
(top)
R (x; k, ǫ, P1, . . . , Pr),
) ≥ |J | ≥
!(m− 1)!
Hence
lim sup
log o∞(Γ
(top)
R (x; k, ǫ, P1, . . . , Pr),
log k
≥ lim sup
( k−n2n −1)!
( k−n2n −m)!(m−1)!
log k
= m− 1.
7.3. Topological free orbit dimension of one self-adjoint element.
Theorem 7.1. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A and σ(x)
is the spectrum of x in A. Let d(σ(x)) be the packing dimension of the set σ(x) in R. Let
f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
log k
− log ω
for s, k ∈ N, ω > 0. Then
kf(x) = d(σ(x)).
Proof. The result follows directly from Lemma 7.1, Lemma 7.2 and Definition 7.1. �
Theorem 7.2. Suppose that x = x∗ is a self-adjoint element in a unital C∗ algebra A. Let
f : N× N× R+ → R be defined by
f(s, k, ω) =
log s
−k2 logω
for s, k ∈ N, ω > 0. Then
kf(x) = 0.
Proof. The result follows directly from Lemma 7.1 and Definition 7.1. �
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http://arxiv.org/abs/math/0510576
http://arxiv.org/abs/math/0603669
	1. Introduction
	2. Definitions and preliminary
	3. Some technical lemmas
	4. Topological free entropy dimension of one variable
	5. Topological free entropy dimension of n-tuple in unital C* algebras 
	6. Topological free orbit dimension of C* algebras
	7. Topological free orbit dimension of one variable
	References
