SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF
∗-ALGEBRAS ON C∗-ALGEBRAS
JA A JEONG† AND GI HYUN PARK‡
Abstract. If a finite group action α on a unital C∗-algebra M is saturated,
the canonical conditional expectation E : M → Mα onto the fixed point
algebra is known to be of index finite type with Index(E) = |G| in the sense
of Watatani. More generally if a finite dimensional Hopf ∗-algebra A acts on
M and the action is saturated, the same is true with Index(E) = dim(A).
In this paper we prove that the converse is true. Especially in case M is
a commutative C∗-algebra C(X) and α is a finite group action, we give an
equivalent condition in order that the expectation E : C(X) → C(X)α is of
index finite type, from which we obtain that α is saturated if and only if G acts
freely on X. Actions by compact groups are also considered to show that the
gauge action γ on a graph C∗-algebra C∗(E) associated with a locally finite
directed graph E is saturated.
1. Introduction
It is known [17] that if α is an action by a compact group G on a C∗-algebra M ,
the fixed point algebra Mα is isomorphic to a hereditary subalgebra e(M ×αG)e of
the crossed product M ×αG for a projection e in the multiplier algebra of M ×αG.
If e(M ×α G)e is full in M ×α G (that is, e(M ×α G)e generates M ×α G as a
closed two-sided ideal), the action α is said to be saturated (the notion of saturated
action was introduced by Rieffel [14, Chap.7]). Every action α with a simple crossed
product M ×α G is obviously saturated.
On the other hand, an action of a finite dimensional Hopf ∗-algebra A on a unital
C∗-algebra M is considered in [18] and it is shown that if the action is saturated,
the canonical conditional expectation E : M → MA onto the fixed point algebra
MA is of index finite type in the sense of Watatani [19] and Index(E) = (dimA)1.
The main purpose of the present paper is to prove that the converse is also true.
We see from our result that for an action α by a finite group G, α is saturated
if and only if the canonical expectation E : M → Mα is of index-finite type with
index Index(E) = |G|.
Besides, we consider actions by compact groups to study the saturation property
of a gauge action γ on a C∗-algebra C∗(E) associated with a locally finite directed
graph E with no sinks or sources. This paper is organized as follows.
In section 2, we review the C∗-basic construction from [19] and finite dimensional
Hopf ∗-algebras from [18] setting up our notations. Then we prove in section 3 that
if A is a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M such
that E : M → MA is of index finite type with Index(E) = (dimA)1, then the
action is saturated (Theorem 3.3).
In section 4, we deal with the crossed product M ×α G by a finite group in
detail and give other equivalent conditions in order that α be saturated. From the
Research supported by KRF-ABRL-R14(2003-2008)† and Hanshin University Research Grant‡.
http://arxiv.org/abs/0704.1549v1
2 JA A JEONG AND GI HYUN PARK
conditions one easily see that an action with the Rokhlin property [7] is always
saturated. Also we shall show that if M has the cancellation, an action with the
tracial Rokhlin property [12] on M is saturated.
Note that even for an action α by the finite group Z2, the expectation E :
M → Mα may not be of index finite type in general [19, Example 2.8.4]. For a
commutative C∗-algebra C(X) and a finite group action α, we give a necessary and
sufficient condition that E : C(X) → C(X)α is of index finite type (Theorem 4.10)
and provide a formula for Index(E). Then as a corollary we obtain that α is
saturated if and only if G acts freely on X .
In section 5, we consider a compact group action α and investigate the ideal
Jα of M ×α G generated by the hereditary subalgebra e(M ×α G)e. Then we
apply the result on Jα to the gauge action on a graph C∗-algebra in section 6. As
a generalization of the Cuntz-Krieger algebras [5], the class of graph C∗-algebras
C∗(E) associated with directed graphs E has been studied in various directions by
considerably many authors (for example see the bibliography in the book [15] by
Raeburn). In [9], Kumjian and Pask show among others that if γ is the gauge action
on C∗(E), then C∗(E)γ is stably isomorphic to the crossed product C∗(E) ×γ T,
which was done by hiring the notions of skew product of graphs and groupoid C∗-
algebras. In Theorem 6.3 we shall directly show that the gauge action is actually
saturated (this implies that C∗(E)γ and C∗(E)×γ T are stably isomorphic).
2. Preliminaries
Watatani’s index theory for C∗-algebras. In [19], Watatani developed the
index theory for C∗-algebras, and here we briefly review the basic construction
C∗(B, eA). Let B be a C
∗-algebra and A its C∗-subalgebra containing the unit of
B. Let E : B → A be a faithful conditional expectation. If there exist finitely
many elements {vi}ni=1 in B satisfying the following
E(bvi)v
viE(v
i b), for every b ∈ B,
E is said to be of index-finite type and {(vi, v∗i )}ni=1 is called a quasi-basis for E.
The positive element
i is then the index of E, Index(E), which is known to
be an element in the center of B and does not depend on the choice of quasi-bases
for E ([19, Proposition 1.2.8]). Let B be the completion of the pre-Hilbert module
B0 = {η(b) | b ∈ B} over A with an A-valued inner product
〈η(x), η(y)〉 = E(x∗y), η(x), η(y) ∈ B0.
Let LA(B) be the C∗-algebra of all (right) A-module homomorphisms on B with
adjoints. For T ∈ LA(B), the norm ‖T ‖ = sup{‖Tx‖ : ‖x‖ = 1} is always bounded.
Each b ∈ B is regarded as an operator Lb in LA(B) defined by Lb(η(x)) = η(bx)
for η(x) ∈ B0. By eA : B → B we denote the projection in LA(B) such that
eA(η(x)) = η(E(x)), η(x) ∈ B0. Then the C∗-basic construction C∗(B, eA) is the
C∗-subalgebra of LA(B) in which the linear span of elements LbeALb′ (b, b′ ∈ B) is
dense.
Finite dimensional Hopf ∗-algebras. As in [18], a finite dimensional Hopf ∗-
algebra is a finite matrix pseudogroup of [20]. We review from [18] the definition
and some basic properties of a finite dimensional Hopf ∗-algebra which we need in
the following section.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 3
Definition 2.1. ([18, Proposition 2.1]) A finite dimensional C∗-algebra is called a
finite dimensional Hopf ∗-algebra if there exist three linear maps,
∆ : A → A⊗A, ǫ : A → C, and S : A → A
which satisfy the following properties
(i) ∆(comultiplication) and ǫ(counit) are ∗-homomorphisms, and S(antipode)
is a ∗-preserving antimultiplicative involution,
(ii) ∆(1) = 1⊗ 1, ǫ(1) = 1, S(1) = 1,
(iii) (∆⊗ id)∆ = (id⊗∆)∆,
(iv) (ǫ ⊗ id)∆ = ∆(ǫ⊗ id),
(v) m(S⊗id)(∆(a)) = ǫ(a)1 = m(id⊗S)(∆(a)) for a ∈ A, wherem : A⊗A → A
is the multiplication.
Proposition 2.2. ([18], [20]) Let A be a finite dimensional Hopf ∗-algebra. Then
the following properties hold.
(i) For a ∈ A, with the notation ∆(a) =
aLi ⊗ aRi , we have
ǫ(aLi )a
i = a =
ǫ(aRi )a
aLi S(a
i ) = ǫ(a)1 =
S(aLi )a
aRi S(a
i ) = ǫ(a)1 =
S(aRi )a
(ii) There is a unique normalized trace (called the Haar trace) τ on A such that
τ(aLi )a
i = τ(a)1 =
τ(aRi )a
i , a ∈ A.
(iii) There exists a minimal central projection e ∈ A (called the distinguished
projection) such that ae = ǫ(a)e, a ∈ A. We have
ǫ(a) = 1, S(e) = e, and τ(e) = (dimA)−1.
3. Actions by finite dimensional Hopf ∗-algebras
Throughout this section A will be a finite dimensional Hopf ∗-algebra. An action
of A on a unital C∗-algebra M is a bilinear map · : A × M → M such that for
a, b ∈ A, x, y ∈ M ,
1 · x = x,
a · 1 = ǫ(a)1,
ab · x = a · (b · x),
a · xy =
(aLi · x)(aRi · y),
(a · x)∗ = S(a∗) · x∗.
4 JA A JEONG AND GI HYUN PARK
Then the crossed product M ⋊A is the algebraic tensor product M ⊗A as a vector
space with the following multiplication and ∗-operation:
(x⊗ a)(y ⊗ b) :=
x(aLi · y)⊗ aRi b,
(x⊗ a)∗ :=
(aLi )
∗ · x∗ ⊗ (aRi )∗.
Identifying a ∈ A with 1⊗ a and x ∈ M with x⊗ 1, we see [18] that
M ⋊A = span{xa | x ∈ M, a ∈ A}.
For the definition of saturated action of A on M , refer to section 4 of [18].
Proposition 3.1. ([18]) Let MA = {x ∈ M | a · x = ǫ(a)x, for all a ∈ A} be the
fixed point algebra for the action of A on a unital C∗-algebra M .
(i) The action is saturated if and only if
M ⋊A = span{xey | x, y ∈ M},
where e ∈ A is the distinguished projection.
(ii) The map E : M → MA, E(x) = e · x, is a faithful conditional expectation
onto the fixed point algebra such that
E((a · x)y) = E(x(S(a) · y)), a ∈ A, x, y ∈ M.
(iii) The linear map F : M ⋊A → M , F (xa) = τ(a)x, is a faithful conditional
expectation onto M .
Recall that M0 := M is an MA-valued inner product module by
〈η(x), η(y)〉MA = E(x∗y)
(here we use the convention in [19] for the inner product as in section 2). Since
every norm bounded MA-module map on M0 extends uniquely to the Hilbert MA-
moduleM, we may identify the ∗-algebra End(M0) (in [18]) of norm bounded right
MA-module endomorphisms ofM0 having an adjoint with the C∗-algebra LMA(M)
explained in section 2.
Remark 3.2. ([19, Proposition 1.3.3]) If E : M → MA is of index-finite type, then
C∗(M, eMA) = span{LxeMALy | x, y ∈ M} = LMA(M).
In fact, we see from the proof of [19, Proposition 2.1.5] that C∗(M, eMA) contains
the unit of LMA(M). Thus the ideal span{LxeMALy | x, y ∈ M} which is dense in
C∗(M, eMA) must contain the unit of LMA(M).
Theorem 3.3. Let A be a finite dimensional Hopf ∗-algebra acting on a unital
C∗-algebra M . Then the following are equivalent:
(i) The action is saturated.
(ii) E : M → MA is of index finite type with Index(E) = (dimA)1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 5
Proof. (i)=⇒ (ii) is shown in [18, Proposition 4.5].
(ii)=⇒ (i). By Remark 3.2, C∗(M, eMA) = span{LxeMALy | x, y ∈ M}. Consider
a map ϕ : C∗(M, eMA) → M ⋊A given by
LxieMALyi) =
xieyi.
To see that ϕ is well defined, let
LxieMALyi = 0. Then for each z ∈ M ,
LxieMALyi)(η(z)) =
η(xiE(yiz)) = η(
xi(e · (yiz))) = 0,
hence by the injectivity of η ([19, 2.1]),
xi(e·(yiz)) = 0 inM . Since (a·x)e = axe
for a ∈ A, a ∈ M (see (7) of [18]), we thus have
xi(e · (yiz))e =
(xieyi)ze = 0
in M ⋊A for every z ∈ M , which then implies that
xieyi)(zez
′) = 0, z, z′ ∈ M.
Particulary, (
xieyi)(
xieyi)
∗ = 0, so that
xieyi = 0 (in M ⋊ A). Thus ϕ
is well defined. It is tedious to show that ϕ is a ∗-homomorphism such that the
range ϕ(C∗(M, eMA)) = MeM is an ideal of M ⋊A; if x, y, and z ∈ M and a ∈ A,
(za)(xey) = (z(a · x))ey ∈ MeM.
Hence it suffices to show that ϕ(1) = 1. If {(ui, u∗i )}ni=1 is a quasi-basis for E, then
LuieMALu∗i (η(z)) =
η(uiE(u
i z)) = η(z), z ∈ M,
which means that
i LuieMALu∗i = 1 ∈ LMA(M). Therefore by Proposition 2.2(iii)
and Proposition 3.1(iii)
F (ϕ(1)) = F (
i ) =
τ(e)uiu
i = 1.
Since ϕ is a ∗-homomorphism, ϕ(1) is a projection in M⋊A such that F (1−ϕ(1)) =
0. But F is faithful, and ϕ(1) = 1 follows. �
4. Actions by finite groups
Throughout this section G will denote a finite group. As is well known the group
C∗-algebra C∗(G) generated by the unitaries {λg | g ∈ G} is a finite dimensional
Hopf ∗-algebra with
∆(λg) = λg ⊗ λg, ǫ(λg) = 1, S(λg) = λg−1 for λg ∈ C∗(G).
The Haar trace τ is given by τ(λg) = διg, where ι is the identity of G, and the
distinguished projection is e = 1
g λg.
Let α be an action of G on a unital C∗-algebra M . Then it is easy to see that
λg · x := αg(x) for g ∈ G, x ∈ M,
6 JA A JEONG AND GI HYUN PARK
defines an action of C∗(G) on M . Furthermore M ⋊ C∗(G) is nothing but the
usual crossed product M ×α G = span{xλg | x ∈ M, g ∈ G}, and the expectations
E : M → Mα(= MC∗(G)), F : M ×α G → M of Proposition 3.1 are given by
E(x) =
αg(x) and F (
xgλg) = xι (x, xg ∈ M, g ∈ G). (1)
Note that for each
xhλh ∈ M ×α G and g ∈ G,
‖xg‖ = ‖F ((
xhλh)λg−1)‖ ≤ ‖(
xhλh)λg−1‖ = ‖
xhλh‖. (2)
If Jα denotes the closed ideal ofM×αG generated by the distinguished projection
e, then Proposition 3.1(i) says that α is saturated if and only if Jα = M ×α G. We
will see in Proposition 5.4 that
Jα = span{
xαg(y)λg | x, y ∈ M} = span{
xαg(x
∗)λg | x ∈ M}. (3)
The ∗-homomorphism ϕ : C∗(M, eMα) → M ×α G we discussed in the proof of
Theorem 3.3 can be rewritten as follows.
ϕ(LxeMαLy) =
xαg(y)λg, x, y ∈ M (4)
because ϕ(LxeMαLy) = xey and e =
λg. If {(ui, u∗i )} is a quasi-basis for E,
we see from
LuieMαLu∗i = 1 and (4) that
ϕ(1) =
uiαg(u
i ))λg (5)
is a projection in M ×α G. Recall that ϕ(1) = 1 holds if α is saturated.
Theorem 4.1. Let M be a unital C∗-algebra and α be an action of a finite group
G on M . Then the following are equivalent:
(i) α is saturated, that is, Jα = M ×α G.
(ii) E : M → Mα is of index finite type with Index(E) = |G|.
(iii) E : M → Mα is of index finite type with Index(E) = |G| and
uiαg(u
i ) = 0, g 6= ι (6)
for a quasi-basis {(ui, u∗i )} for E.
(iv) There exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that
(a) αg(b
) = b
, for j = 1, . . . ,m and g, h ∈ G.
bjg(b
)∗ = δgh.
(v) For every ε > 0, there exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1
such that
‖αg(bjh)− b
‖ < ε,
(b) ‖
bjg(b
)∗ − δgh‖ < ε.
Proof. (i) ⇐⇒ (ii) follows from Theorem 3.3.
(i) =⇒ (iii). If {(ui, u∗i )} is a quasi-basis for E, we have from (5) that
uiαg(u
i ) =
0 for g 6= ι since ϕ(1) = 1.
(iii) =⇒ (ii). Obvious.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 7
(i) =⇒ (iv). Suppose Jα = M ×α G. By (3) there exist m ∈ N and bj ∈ M ,
1 ≤ j ≤ m, such that
bjαg(b
λg = 1.
j = 1 and
bjαg(b
j ) = 0 for g 6= ι. (7)
Set bjg := αg(bj). Then
) = αg(αh(bj)) = αgh(bj) = b
bjg(b
αg(bj)αh(b
bjαg−1h(b
= δgh by (7).
(iv) =⇒ (v). Obvious.
(v) =⇒ (i). Let ε > 0 and let {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} satisfy (a) and
(b) of (v). Note that (b) implies ‖bjg‖ < 1 + ε for g ∈ G, 1 ≤ j ≤ m. Indeed from
bjg(b
∥ ≤ ‖
bjg(b
∗−1‖ < ε, we have ‖bjg‖2 ≤ ‖
bjg(b
∗‖ < 1+ε.
αg((b
)∗)λg)− |G| ‖
αg((b
)∗)λg − |G| ‖
)∗ − |G| ‖+ ‖
g 6=ι
αg((b
)∗)λg)‖
)∗ − 1 ‖+
g 6=ι
αg((b
< ε|G|+
g 6=ι
αg((b
)∗)− (bj
g 6=ι
|G|+ |G|2 max
‖bjg‖+ |G|2
|G|+ |G|2(1 + ε) + |G|2
Since
αg((b
)∗)λg) ∈ Jα and ε can be chosen to be arbitrarily small,
we conclude that Jα = M ×α G. �
Example 4.2. Let w =
be a unitary with wn = 1 and define an
automorphism α on M2(C)) by α(a) = waw
∗, a ∈ M2(C). We will show that α is
saturated if and only if z2 = −z1. For this, recall from (3) that α is saturated if
and only if there exist xj ∈ M2(C), 1 ≤ i ≤ m, satisfying
k(x∗j )λk = 1M2(C). (8)
8 JA A JEONG AND GI HYUN PARK
Hence, particularly for k = 0, 1, we have
xjα(x
j ) =
With xj =
aj bj
cj dj
and zi = e
iθi , i = 1, 2, this means
|aj |2 + |bj |2 aj c̄j + bj d̄j
cj āj + dj b̄j |cj |2 + |dj |2
|aj |2 + ei(θ2−θ1)|bj |2 ei(θ1−θ2)aj c̄j + bj d̄j
|aj |2 + ei(θ2−θ1)dj b̄j ei(θ1−θ2)aj c̄j + |dj |2
. (9)
Therefore, by comparing (1,1) entries of each matrices, it follows that if α is satu-
rated, then there exist positive real numbers a (=
|aj |2) > 0, b (=
|bj |2) > 0
such that
a+ b = 1 and a+ ei(θ2−θ1)b = 0. (10)
Note that b 6= 0 since b = 0 implies a = 0 from
|aj |2 + ei(θ2−θ1)dj b̄j
= 0 in
(9). There are three possible cases for θ1, θ2 as follows.
(i) If θ2 − θ1 ≡ 0(mod 2π), that is, α is trivial, then (10) is not possible.
(ii) If θ2 − θ1 ≡ π(mod 2π), then
and x2 =
satisfy (8) with m = 2. Thus α is saturated.
(iii) If θ2− θ1 6= 0, π(mod 2π), then (10) is not possible for any a, b > 0. Hence
α is not saturated.
Remark 4.3. Let α, β ∈ Aut(M) satisfy αn = βn = idM for some n ≥ 1. If
there is a unitary u ∈ M such that β = Ad(u) ◦ α, then α and β are said to
be exterior equivalent, and if this is the case the crossed products are isomorphic,
M×αG ∼= M×βG, [14, p.45]. Example 4.2 says that the property of being saturated
may not be preserved under exterior equivalence. Also the case (iii) of Example 4.2
above with w = diag(λ, λ̄), λ = e
3 (hence θ1− θ2 = 2π3 −
6= π(mod 2π)), shows
that Index(E) < |G| is possible even when E is of index-finite type. In fact, if
and u2 =
, then {(ui, u∗i )}2i=1 forms a quasi-basis for E,
but Index(E) = 2 < |Z3|.
Remark 4.4. Recall that the Rokhlin property and the tracial Rokhlin property
(weaker than the Rokhlin property) are defined as follows and considered intensively
in [7] and [12], respectively:
(a) ([7]) α is said to have the Rokhlin property if for every finite set F ⊂ M ,
every ε > 0, there are mutually orthogonal projections {eg | g ∈ G} in M
such that
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii)
g∈G eg = 1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 9
(b) ([12]) α is said to have the tracial Rokhlin property if for every finite set
F ⊂ M , every ε > 0, every n ∈ N, and every nonzero positive element
x ∈ M , there are mutually orthogonal projections {eg | g ∈ G} in M such
that:
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii) With e =
g∈G eg, the projection 1 − e is Murray-von Neumann
equivalent to a projection in the hereditary subalgebra of M generated
by x.
The following proposition is actually observed in [12, Lemma 1.13], and we put a
proof for reader’s convenience.
Proposition 4.5. Let M be a unital C∗-algebra and α be an action of a discrete
group G on M . Suppose that for every ε > 0 and every finite subset F ⊂ M , there
exist a family of projections {eg}g∈G such that
(1) ‖αg(eh)− egh‖ < ε.
(2) ‖egx− xeg‖ < ε for each x ∈ F .
Then α is an outer action.
Proof. Suppose there is a unitary u ∈ M such that αg(x) = uxu∗ for every x ∈ M
(g 6= ι). Put F = {u} and 0 < ε < 1/4. Then there exist mutually orthogonal
projections {eg}g∈G such that ‖αg(eh)− egh‖ < ε < 1/4. Thus ‖egu− ueg‖ < ε <
1/4. Then ‖αg(eι)− ueιu∗‖ = 0. But
‖αg(eι)− ueιu∗‖ = ‖αg(eι)− eg + eg − eι + eι − ueιu∗‖
≥ ‖eg − eι‖ − ‖αg(eι)− eg‖ − ‖eι − ueι1u∗‖
≥ 1− 1
which is a contradiction. �
Remark 4.6. If M ×αG is simple, α is obviously saturated, and this is the case if G
is a finite group, M is α-simple, and T̃(αg) 6= {1} for all g 6= ι ([8, Theorem 3.1]).
In particular, α is saturated if M is simple and α is outer.
But for a nonsimple M , this may not hold. In fact, if α is an outer action of Zn
on M and u is a unitary in M with un = 1 such that the action Ad(u) on M is not
saturated (as in Example 4.2), then the action α ⊕ Ad(u) on M ⊕M is outer but
not saturated.
Now we show that if α satisfies the Rokhlin property (or satisfies the tracial Rokhlin
property and M has cancellation) then α is saturated. For this we first review the
cancellation property of C∗-algebras. For projections p, q in a C∗-algebra, we write
p ⊥ q if pq = 0, and p ∼ q if they are Murrey-von Neumann equivalent.
Definition 4.7. A unital C∗-algebra M has the cancellation if, whenever p, q, r
are projections in Mn(M) for some n, with p ⊥ r, q ⊥ r, and (p + r) ∼ (q + r),
then p ∼ q.
10 JA A JEONG AND GI HYUN PARK
Remark 4.8. (1) If M has the cancellation and p, q are projections in M such
that (1− p) ∼ (1− q), then p ∼ q ([4, V.2.4.14]).
(2) It is well known that every C∗-algebra with stable rank one has the cancel-
lation ([4, V.3.1.24]).
Proposition 4.9. Let α be an action of a finite group G on a unital C∗-algebra
M . Then α is saturated if one of the following holds.
(i) α has the Rokhlin property.
(ii) α has the tracial Rokhlin property and M has the cancellation.
Proof. (i) For an ε > 0, there exist mutually orthogonal projections {eg}g such that
g eg = 1 and ‖αg(eh)− egh‖ < ε. Then, with m = 1, the elements b1g := eg satisfy
(v) of Theorem 4.1.
(ii) Now suppose α has the tracial Rokhlin property and M has the cancellation.
We shall show that Jα contains the unit of M ×α G. Let 0 < ε < 1. For each
g ∈ G, choose mutually orthogonal projections {eg
}h∈G such that
‖αk(egh)− e
‖ < ε
2|G|2 ,
and put eg :=
h∈G e
. If eg = 1, for some g, then b1h := e
(h ∈ G) will satisfy
(v) of Theorem 4.1 as in (i). If eg 6= 1 for every g ∈ G, then by the tracial Rokhlin
property of α there exist mutually orthogonal projections {fg
}h∈G in M such that
‖αk(fgh)− f
‖ < ε
2|G|2
) ∼ (eg)′ < eg
for a subprojection (eg)′ of eg. Put fg :=
. Then since M has cancellation, it
follows that fg ∼
1−(eg)′
> (1−eg). Let vg ∈ M be a partial isometry satisfying
v∗gvg = f
g, vgv
g = 1− (eg)′,
and set
xg :=
(ekhαg(e
h) + (1 − ek)vkfkhαg(fkhαg−1 (v∗k))
, g ∈ G.
Now we show that the element x :=
xgλg ∈ Jα satisfies ‖x − 1‖ < ε. In fact,
for g 6= ι,
‖xg‖ ≤
‖(ekhαg(ekh) + (1− ek)vkfkhαg(fkhαg−1(v∗k))‖
≤ 1|G|
‖(ekhαg(ekh)‖ + ‖fkhαg(fkh )‖
≤ 1|G|
‖ekh(αg(ekh)− ekgh)‖ + ‖ekhekgh‖+ ‖fkh (αg(fkh )− fkgh)‖+ ‖fkhfkgh‖
≤ 1|G|
2|G|2 +
2|G|2 )
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 11
|G| (
(1− ek)vkfkv∗k)
(1 − ek)(1 − (ek)′)
For the rest of this section we consider a finite group action on a commutative
C∗-algebra C(X). If G acts on a compact Hausdorff space X , it induces an action,
say α, on C(X) by
αg(f)(x) = f(g
−1x), f ∈ C(X).
For each x ∈ X , let Gx = {g ∈ G : gx = x} be the isotropy group of x and for a
subgroup H of G (H < G), put
XH = {x ∈ X : Gx = H}.
It is readily seen that XH and XH′ are disjoint if H 6= H ′, and X is partitioned as
Theorem 4.10. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action of G on C(X), the following are equivalent:
(i) E : C(X) → C(X)α is of index finite type.
(ii) XH is closed for each H < G.
Moreover, if this is case the index of E is Index(E) =
χXH , where χXH
is the characteristic function on XH .
Proof. (i) =⇒ (ii). If E is of index-finite type and {(ui, u∗i )}ki=1 is a quasi-basis for
E, then
uiE(u
i f) = f,
that is,
ui(x)
u∗i (g
−1x)f(g−1x)
= f(x), (11)
for f ∈ C(X) and x ∈ X . For each x ∈ X , choose a continuous function fx ∈ C(X)
satisfying fx|Gx\{x} ≡ 0 and fx(x) = 1. Then (11) with fx in place of f gives
ui(x)
u∗i (g
−1x)fx(g
= fx(x), (12)
and so we have
ui(x)u
i (x) = 1. (13)
12 JA A JEONG AND GI HYUN PARK
To show that each XH is closed, let {xn ∈ XH : n = 1, 2, . . .} be a sequence of
elements in XH with limit x ∈ XH′ . Then (11) gives
fx(xn) =
ui(xn)
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
u∗i (g
−1xn)fx(g
−1xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
|H |u∗i (xn)fx(xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
Taking the limit as n → ∞, we have
fx(x) =
ui(x)
|H |u∗i (x)fx(x) +
g 6∈H
u∗i (g
−1x)fx(g
ui(x)
|H |u∗i (x)fx(x) + |H ′ \H |u∗i (x)fx(x)
|H |+ |H ′ \H |
ui(x)u
i (x)fx(x).
Therefore, comparing with (13), we obtain
|H ′| = |H |+ |H ′ \H |
since Gx = H
′ and fx(x) = 1. Hence
H ⊂ H ′.
On the other hand, since Gxn = H , again by (13),
ui(xn)u
i (xn) = 1 with
the limit
ui(x)u
i (x) = 1 as n → ∞. But also
ui(x)u
i (x) = 1 by
(13), and thus |H | = |H ′|. Consequently we have
H = H ′
because H ⊂ H ′. This shows that XH is closed.
(ii) =⇒ (i). Assume that XH is closed for every subgroup H of G. Then XH
is open since there are only finitely many such subsets. Let UH = {UH,iH : iH =
1, 2, . . . , nH} be an open covering of XH such that
x ∈ UH,iH =⇒ g−1x 6∈ UH,iH or g−1x 6∈ XH whenever g−1x 6= x.
Let {vH,iH} be a partition of unity subordinate to UH . We understand that the
domain of vH,iH is X by assigning 0 to x 6∈ XH . Let uH,iH =
vH,iH .
We claim that
|H | uH,iH ,
|H | u
: H < G, iH = 1, 2, . . . , nH
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 13
is a quasi-basis for E. For f ∈ C(X) and x ∈ X , let F < G and 1 ≤ j ≤ nF be
such that x ∈ XF and x ∈ UF,j. Then
|H | uH,iHE
|H | u
|H |uH,iH (x)
|H |u
(g−1x)f(g−1x)
|F |uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)|F |u∗F,iF (x)f(x)
vF,iF (x)f(x)
= f(x),
as claimed.
Recall that an action G on X is free if gx 6= x for g ∈ G, g 6= 1, and x ∈ X
Corollary 4.11. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action on C(X), the following are equivalent.
(i) G acts freely on X.
(ii) E : C(X) → C(X)α is of index-finite type with Index(E) = |G|.
(iii) α is saturated.
Proof. (i) =⇒ (ii) is proved in [19, Proposition 2.8.1].
To show (ii) =⇒ (i), let E be of index-finite type with Index(E) = |G|. Then
from Theorem 4.10, we have
|G| = Index(E) =
|H |χXH ,
which implies that H = {ι} is the only subgroup of G such that XH 6= ∅. Hence
X = X{ι}, that is, G acts freely on X . (ii) ⇐⇒ (iii) comes from Theorem 4.1. �
5. Saturated actions by compact groups
Notation 5.1. Let M be a C∗-algebra and α be an action of a compact group G on
M . For x, y ∈ M , define continuous functions fx,y, fx,1, f1,y ∈ C(G,M) from G to
14 JA A JEONG AND GI HYUN PARK
M as follows:
fx,y(t) = xαt(y),
fx,1(t) = x, f1,y(t) = αt(y) for t ∈ G.
Then it is easily checked that fx,y = fx,1 ∗ f1,y and f∗x,y = fy∗,x∗ .
Recall that C(G,M) is a dense ∗-subalgebra of M ×α G with the multiplication
and involution defined by
f ∗ g(t) =
f(s)αs(g(s
−1t))ds,
f∗(t) = αt(f(t
−1)∗),
where dg is the normalized Haar measure ([13, 7.7], [6, 8.3.1]). Hence if G is a finite
group, fx,y can be written as fx,y =
xαg(y)λg .
If M̃ denotes the smallest unitization of M (so M̃ = M if M is unital), the
function
e : G → M̃, e(s) = 1, for every s ∈ G
is a projection of the multiplier algebra of M ×α G ([17]).
Proposition 5.2. ([17]) Let α be an action of a compact group G on a C∗-algebra
M . Then identifying x ∈ Mα and the constant function in C(G,M) with the value
x everywhere we see that
x 7→ fx,1 : Mα → e(M ×α G)e
is an isomorphism of Mα onto the hereditary subalgebra e(M×αG)e of the crossed
product M ×α G.
The notion of saturated action is introduced by Rieffel for a compact group action
on a C∗-algebra, and we adopt the following equivalent condition as the definition.
Definition 5.3. (Rieffel, see [14, 7.1.9 Lemma]) Let M be a C∗-algebra and α be
an action of compact group G on M . α is said to be saturated if the linear span of
{fa,b | a, b ∈ M} is dense in M ×α G (see Notation 5.1). We denote
Jα = span{fa,b | a, b ∈ M}.
Proposition 5.4. Let α be an action of a compact group G on a C∗-algebra M .
Then Jα is the ideal of M×αG generated by the hereditary subalgebra e(M ×αG)e.
Moreover Jα = span{fa,a∗ ∈ C(G,M) | a ∈ M}.
Proof. We first show that Jα is an ideal ofM×αG. Let x ∈ C(G,M) and a, b ∈ M .
Then x ∗ fa,b ∈ Jα. Indeed,
(x ∗ fa,b)(t) =
x(s)αs(fa,b(s
−1t))ds
x(s)αs(a)αt(b)ds
x(s)αs(a)ds
αt(b),
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 15
hence x∗fa,b = fc,b ∈ Jα, where c =
x(s)αs(a)ds ∈ M . Also f∗a,b = fb∗,a∗ implies
that Jα = Jα∗ is an ideal of M ×α G.
Let J := (M ×α G)e(M ×α G) be the closed ideal generated by e(M ×α G)e.
Now we show that Jα ⊂ J . From
(fa,b ∗ e)(t) =
fa,b(s)αs(e(s
−1t)) ds =
aαs(b) ds = a
αs(b)ds,
we have fa,b ∗ e = faE(b),1, where E(b) =
αs(b)ds ∈ Mα. Hence for a, b, c, and d
in M , we have
fa,b ∗ e ∗ fc,d = (fa,b ∗ e) ∗ (fd∗,c∗ ∗ e)∗
= faE(b),1 ∗ (fd∗E(c∗),1)∗
= faE(b),1 ∗ f1,E(c∗)d
= faE(b),E(c∗)d,
which means that fax,yd ∈ J for any a, d ∈ M and x, y ∈ Mα. Since Mα contains
an approximate identity for M , it follows that fa,b ∈ J for a, b ∈ A.
For the converse inclusion J ⊂ Jα, note that if x ∈ C(G,M), then (x ∗ e)(t) =
x(s)ds for t ∈ G. With notations x′ =
x(s)ds and x′′ :=
αs(x(s
−1))ds (∈ M),
we see that
(x ∗ e ∗ y)(t) =
(x ∗ e)(s)αs(y(s−1t))ds
αs(y(s
−1t))ds
= x′αt(
αs(y(s
−1))ds)
= x′αt(y
= fx′,y′′(t)
belongs to Jα for x, y ∈ C(G,M).
Finally the following polarization identity proves the last assertion.
aαt(b) =
ik(b + ika∗)∗αt(b+ i
ka∗).
6. The gauge action γ on a graph C∗-algebra
By a (directed) graph E we mean a quadruple E = (E0, E1, r, s) consisting of
the vertex set E0, the edge set E1, and the range, source maps r, s : E1 → E0. If
each vertex of E emits only finitely many edges E is called row finite and a row
finite graph E is locally finite if each vertex receives only finitely many edges. By
En we denote the set of all finite paths α = e1 · · · en (r(ei) = s(ei+1), 1 ≤ i ≤ n−1)
of length n (|α| = n). Each vertex is regarded as a finite path of length 0. Then
E∗ = ∪n≥0En is the set of all finite paths and the maps r and s naturally extend
to E∗. A vertex v is called a sink if s−1(v) = ∅ and a source if r−1(v) = ∅.
If E is a row finite graph, we call a family {se, pv | e ∈ E1, v ∈ E0} of operators a
Cuntz-Krieger(CK) E-family if {se}e are partial isometries and {pv}v are mutually
16 JA A JEONG AND GI HYUN PARK
orthogonal projections such that
s∗ese = pr(e) and pv =
s(e)=v
e if s
−1(v) 6= ∅.
It is now well known that there exists a C∗-algebra C∗(E) generated by a universal
CK E-family {se, pv | e ∈ E1, v ∈ E0}, in this case we simply write C∗(E) =
C∗(se, pv). For the definition and basic properties of graph C
∗-algebras, see, for
example, [1, 2, 10, 11, 15] among others. If α = α1α2 · · ·α|α| (αi ∈ E1) is a finite
path, by sα we denote the partial isometry sα1sα2 · · · sα|α| (sv = s∗v = pv, for v ∈
We will consider only locally finite graphs and it is helpful to note the following
properties of graph C∗-algebras.
Remark 6.1. (i) Let C∗(E) = C∗(se, pv) be the graph C
∗-algebra associated
with a row finite graph E, and let α, β ∈ E∗ be finite paths in E. Then
s∗αsβ =
s∗µ, if α = βµ
sν , if β = αν
0, otherwise.
Therefore C∗(E) = span{sαs∗β | α, β ∈ E∗}.
(ii) Note that sαs
β = 0 for α, β ∈ E∗ with r(α) 6= r(β).
(iii) If α, β, µ, and ν in En are the paths of same length,
β)(sµs
ν) = δβ,µsαs
Thus for each n ∈ N and a vertex v in a locally finite graph E, we see that
span{sαs∗β | α, β ∈ En and r(α) = r(β) = v}
is a ∗-algebra which is isomorphic to the full matrix algebraMm = (Mm(C)),
where m =
∣{α ∈ En | r(α) = v}
Recall that the gauge action γ of T on C∗(E) = C∗(se, pv) is given by
γz(se) = zse, γz(pv) = pv, z ∈ T.
γ is well defined by the universal property of the CK E-family {se, pv}. Since
γz(sαs
β)dz =
z|α|−|β|(sαs
β)dz = 0, |α| 6= |β|,
one sees that
C∗(E)γ = span{sαs∗β | α, β ∈ E∗, |α| = |β|}.
If Z denotes the following graph:
• • • • •______ // ______ // ______ //______ // · · · ,· · ·Z :
−2 −1 0 1 2
then C∗(Z) is isomorphic to the C∗-algebra K of compact operators on an infinite
dimensional separable Hilbert space, hence C∗(Z) is itself a simple AF algebra. But
C∗(Z)γ coincides with the commutative subalgebra span{sαs∗α | α ∈ Z∗} which is
far from being simple, and thus we know that the simplicity of C∗(E) does not
imply that of C∗(E)γ in general.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 17
In [9], the Cartesian product of two graphs E and F is defined to be the graph E×
F = (E0 ×F 0, E1 ×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)).
Since the graph Z × E has no loops for any row-finite graph E, we know that
C∗(E)γ is an AF algebra ([10]) by the following proposition.
Proposition 6.2. ([9]) Let E be a row finite graph with no sources. Then the
following hold:
(a) C∗(E)γ is stably isomorphic to C∗(E)×γ T.
(b) C∗(E)×γ T ∼= C∗(Z× E).
Now we show that a gauge action is saturated. For this, note that the linear
span of the continuous functions of the form
t 7→ f(t)x, f ∈ C(G), x ∈ A
is dense in C(G,A) [13, 7.6.1]. Hence by Remark 6.1(i), one sees that
C∗(E)×γ T = span{znsαs∗β | α, β ∈ E∗ n ∈ Z}. (15)
Theorem 6.3. Let E be a locally finite graph with no sinks and no sources. Then
the gauge action γ on C∗(E) is saturated.
Proof. We show that Jγ = C∗(E)×γ T. By (15) it suffices to see that
znsαs
β ∈ Jγ for all α, β ∈ E∗, n ≥ 0
(because z−nsαs
β = (z
∗ for n ≥ 0).
Now fix α, β ∈ E∗ and n ≥ 0. Put l = n− (|α| − |β|). There are two cases.
(i) l ≥ 0: One can choose a path µ such that |µ| = l and r(µ) = s(α). Then
znsαs
β = z
l+|α|−|β|s∗µsµsαs
β = s
µγz(sµαs
β) = fs∗µ,sµαs∗β (z),
where the function fs∗µ,sµαs∗β belongs to Jγ .
(ii) l < 0: Choose a path ν with |ν| = |β|+n and r(ν) = r(α). With a = sαs∗ν ,
b = sνs
β , we have fa,b ∈ Jγ and
znsαs
β = sαs
νγz(sνs
β) = fa,b(z).
Acknowledgements. The first author would like to thank Hiroyuki Osaka and Ta-
motsu Teruya for valuable discussions.
References
[1] T. Bates, J. H. Hong, I. Raeburn and W. Szymanski, The ideal structure of the C∗-algebras
of infinite graphs, Illinois J. Math., 46(2002), 1159–1176.
[2] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C∗-algebras of row-finite graphs, New
York J. Math. 6(2000), 307–324.
[3] T. Bates and D. Pask, Flow equivalence of graph algebras, Ergod. Th. & Dynam. Sys. 24(2004),
367–382.
[4] B. Blakadar, Operator Algebras, Theory of C∗-algebras and von Neumann algebras, Encyclo-
pedia of mathematical sciences 122, Springer, 2006.
[5] J. Cuntz and W. Krieger, A class of C∗-algebras and topological Markov chains, Invent. Math.
56(1980), 251–268.
[6] P. A. Fillmore, A user’s guide to operator algebras, Canadian Math. Soc. Series of Monographs
and Adv. Texts, John Wiley & Sons, Inc. 1996.
18 JA A JEONG AND GI HYUN PARK
[7] M. Izumi, Finite group actions on C∗-algebras with the Rohlin property, I, Duke Math. J.
122(2004), no. 2, 233–280.
[8] A. Kishimoto, Outer automorphisms and related crossed products of simple C∗-algebras,
Comm. Math. Phys. 81(1981), no. 1, 429–435.
[9] A. Kumjian and D. Pask, C∗-algebras of directed graphs and group actions, Ergodic Theory
Dynam. Systems, 19(1999), 1503–1519.
[10] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J.
Math. 184(1998), 161–174.
[11] A. Kumjian, D. Pask, I. Raeburn and J. Renault Graphs, groupoids, and Cuntz-Krieger
algebras, J. Funct. Anal. 144(1997), 505–541.
[12] H. Osaka and N. C. Phillips, Stable and real rank for crossed products by automorphisms
with the tracial Rokhlin property, Ergod. Th. & Dyn. Sys., to appear.
[13] G. K. Pedersen, C∗-algebras and their automorphism groups, London Math. Soc. Monographs
No.14, Academic Press 1979.
[14] N. C. Phillips, Equivariant K-theory and freness of group actions on C∗-algebras, LNM 1274,
Springer, 1987.
[15] I. Raeburn, Graph algebras, CBMS. 103, 2005, AMS.
[16] J. Renault, A groupoid approach to C∗-algebras, LNM 794, Springer, Berlin, 1980.
[17] J. Rosenberg, Appendix to O. Bratteli’s paper on ”crossed products of UHF algebras, Duke
Math. J. 46(1979), 25–26.
[18] W. Szymański and C. Peligrad, Saturated actions of finite dimensional Hopf ∗-algebras on
C∗-algebras, Math. Scand. 75(1994), 217–239.
[19] Y. Watatani, Index for C∗-subalgebras, Memoirs of the Amer. Math. Soc. 424, AMS, 1990.
[20] S. L. Woronowicz, Compact matrix psedogroups, Comm. Math. Phys. 111(1987), 613–665.
Keywords: Finite dimensional Hopf ∗-algebra; saturated action; conditional ex-
pectation of index-finite type.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul
National University, Seoul, 151–747, Korea
E-mail address: jajeong@snu.ac.kr
Department of Mathematics, Hanshin University, Osan, 447–791, Korea
E-mail address: ghpark@hanshin.ac.kr
	1. Introduction
	2. Preliminaries
	3. Actions by finite dimensional Hopf *-algebras
	4. Actions by finite groups
	5. Saturated actions by compact groups
	6. The gauge action  on a graph C*-algebra
	References

  If a finite group action $\alpha$ on a unital $C^*$-algebra $M$ is saturated,
the canonical conditional expectation $E:M\to M^\alpha$ onto the fixed point
algebra is known to be of index finite type with $Index(E)=|G|$ in the sense of
Watatani. More generally if a finite dimensional Hopf $*$-algebra $A$ acts on
$M$ and the action is saturated, the same is true with $Index (E)=\dim(A)$. In
this paper we prove that the converse is true. Especially in case $M$ is a
commutative $C^*$-algebra $C(X)$ and $\alpha$ is a finite group action, we give
an equivalent condition in order that the expectation $E:C(X)\to C(X)^\alpha$
is of index finite type, from which we obtain that $\alpha$ is saturated if and
only if $G$ acts freely on $X$.
  Actions by compact groups are also considered to show that the gauge action
$\gamma$ on a graph $C^*$-algebra $C^*(E)$ associated with a locally finite
directed graph $E$ is saturated.

Introduction
It is known [17] that if α is an action by a compact group G on a C∗-algebra M ,
the fixed point algebra Mα is isomorphic to a hereditary subalgebra e(M ×αG)e of
the crossed product M ×αG for a projection e in the multiplier algebra of M ×αG.
If e(M ×α G)e is full in M ×α G (that is, e(M ×α G)e generates M ×α G as a
closed two-sided ideal), the action α is said to be saturated (the notion of saturated
action was introduced by Rieffel [14, Chap.7]). Every action α with a simple crossed
product M ×α G is obviously saturated.
On the other hand, an action of a finite dimensional Hopf ∗-algebra A on a unital
C∗-algebra M is considered in [18] and it is shown that if the action is saturated,
the canonical conditional expectation E : M → MA onto the fixed point algebra
MA is of index finite type in the sense of Watatani [19] and Index(E) = (dimA)1.
The main purpose of the present paper is to prove that the converse is also true.
We see from our result that for an action α by a finite group G, α is saturated
if and only if the canonical expectation E : M → Mα is of index-finite type with
index Index(E) = |G|.
Besides, we consider actions by compact groups to study the saturation property
of a gauge action γ on a C∗-algebra C∗(E) associated with a locally finite directed
graph E with no sinks or sources. This paper is organized as follows.
In section 2, we review the C∗-basic construction from [19] and finite dimensional
Hopf ∗-algebras from [18] setting up our notations. Then we prove in section 3 that
if A is a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M such
that E : M → MA is of index finite type with Index(E) = (dimA)1, then the
action is saturated (Theorem 3.3).
In section 4, we deal with the crossed product M ×α G by a finite group in
detail and give other equivalent conditions in order that α be saturated. From the
Research supported by KRF-ABRL-R14(2003-2008)† and Hanshin University Research Grant‡.
http://arxiv.org/abs/0704.1549v1
2 JA A JEONG AND GI HYUN PARK
conditions one easily see that an action with the Rokhlin property [7] is always
saturated. Also we shall show that if M has the cancellation, an action with the
tracial Rokhlin property [12] on M is saturated.
Note that even for an action α by the finite group Z2, the expectation E :
M → Mα may not be of index finite type in general [19, Example 2.8.4]. For a
commutative C∗-algebra C(X) and a finite group action α, we give a necessary and
sufficient condition that E : C(X) → C(X)α is of index finite type (Theorem 4.10)
and provide a formula for Index(E). Then as a corollary we obtain that α is
saturated if and only if G acts freely on X .
In section 5, we consider a compact group action α and investigate the ideal
Jα of M ×α G generated by the hereditary subalgebra e(M ×α G)e. Then we
apply the result on Jα to the gauge action on a graph C∗-algebra in section 6. As
a generalization of the Cuntz-Krieger algebras [5], the class of graph C∗-algebras
C∗(E) associated with directed graphs E has been studied in various directions by
considerably many authors (for example see the bibliography in the book [15] by
Raeburn). In [9], Kumjian and Pask show among others that if γ is the gauge action
on C∗(E), then C∗(E)γ is stably isomorphic to the crossed product C∗(E) ×γ T,
which was done by hiring the notions of skew product of graphs and groupoid C∗-
algebras. In Theorem 6.3 we shall directly show that the gauge action is actually
saturated (this implies that C∗(E)γ and C∗(E)×γ T are stably isomorphic).
2. Preliminaries
Watatani’s index theory for C∗-algebras. In [19], Watatani developed the
index theory for C∗-algebras, and here we briefly review the basic construction
C∗(B, eA). Let B be a C
∗-algebra and A its C∗-subalgebra containing the unit of
B. Let E : B → A be a faithful conditional expectation. If there exist finitely
many elements {vi}ni=1 in B satisfying the following
E(bvi)v
viE(v
i b), for every b ∈ B,
E is said to be of index-finite type and {(vi, v∗i )}ni=1 is called a quasi-basis for E.
The positive element
i is then the index of E, Index(E), which is known to
be an element in the center of B and does not depend on the choice of quasi-bases
for E ([19, Proposition 1.2.8]). Let B be the completion of the pre-Hilbert module
B0 = {η(b) | b ∈ B} over A with an A-valued inner product
〈η(x), η(y)〉 = E(x∗y), η(x), η(y) ∈ B0.
Let LA(B) be the C∗-algebra of all (right) A-module homomorphisms on B with
adjoints. For T ∈ LA(B), the norm ‖T ‖ = sup{‖Tx‖ : ‖x‖ = 1} is always bounded.
Each b ∈ B is regarded as an operator Lb in LA(B) defined by Lb(η(x)) = η(bx)
for η(x) ∈ B0. By eA : B → B we denote the projection in LA(B) such that
eA(η(x)) = η(E(x)), η(x) ∈ B0. Then the C∗-basic construction C∗(B, eA) is the
C∗-subalgebra of LA(B) in which the linear span of elements LbeALb′ (b, b′ ∈ B) is
dense.
Finite dimensional Hopf ∗-algebras. As in [18], a finite dimensional Hopf ∗-
algebra is a finite matrix pseudogroup of [20]. We review from [18] the definition
and some basic properties of a finite dimensional Hopf ∗-algebra which we need in
the following section.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 3
Definition 2.1. ([18, Proposition 2.1]) A finite dimensional C∗-algebra is called a
finite dimensional Hopf ∗-algebra if there exist three linear maps,
∆ : A → A⊗A, ǫ : A → C, and S : A → A
which satisfy the following properties
(i) ∆(comultiplication) and ǫ(counit) are ∗-homomorphisms, and S(antipode)
is a ∗-preserving antimultiplicative involution,
(ii) ∆(1) = 1⊗ 1, ǫ(1) = 1, S(1) = 1,
(iii) (∆⊗ id)∆ = (id⊗∆)∆,
(iv) (ǫ ⊗ id)∆ = ∆(ǫ⊗ id),
(v) m(S⊗id)(∆(a)) = ǫ(a)1 = m(id⊗S)(∆(a)) for a ∈ A, wherem : A⊗A → A
is the multiplication.
Proposition 2.2. ([18], [20]) Let A be a finite dimensional Hopf ∗-algebra. Then
the following properties hold.
(i) For a ∈ A, with the notation ∆(a) =
aLi ⊗ aRi , we have
ǫ(aLi )a
i = a =
ǫ(aRi )a
aLi S(a
i ) = ǫ(a)1 =
S(aLi )a
aRi S(a
i ) = ǫ(a)1 =
S(aRi )a
(ii) There is a unique normalized trace (called the Haar trace) τ on A such that
τ(aLi )a
i = τ(a)1 =
τ(aRi )a
i , a ∈ A.
(iii) There exists a minimal central projection e ∈ A (called the distinguished
projection) such that ae = ǫ(a)e, a ∈ A. We have
ǫ(a) = 1, S(e) = e, and τ(e) = (dimA)−1.
3. Actions by finite dimensional Hopf ∗-algebras
Throughout this section A will be a finite dimensional Hopf ∗-algebra. An action
of A on a unital C∗-algebra M is a bilinear map · : A × M → M such that for
a, b ∈ A, x, y ∈ M ,
1 · x = x,
a · 1 = ǫ(a)1,
ab · x = a · (b · x),
a · xy =
(aLi · x)(aRi · y),
(a · x)∗ = S(a∗) · x∗.
4 JA A JEONG AND GI HYUN PARK
Then the crossed product M ⋊A is the algebraic tensor product M ⊗A as a vector
space with the following multiplication and ∗-operation:
(x⊗ a)(y ⊗ b) :=
x(aLi · y)⊗ aRi b,
(x⊗ a)∗ :=
(aLi )
∗ · x∗ ⊗ (aRi )∗.
Identifying a ∈ A with 1⊗ a and x ∈ M with x⊗ 1, we see [18] that
M ⋊A = span{xa | x ∈ M, a ∈ A}.
For the definition of saturated action of A on M , refer to section 4 of [18].
Proposition 3.1. ([18]) Let MA = {x ∈ M | a · x = ǫ(a)x, for all a ∈ A} be the
fixed point algebra for the action of A on a unital C∗-algebra M .
(i) The action is saturated if and only if
M ⋊A = span{xey | x, y ∈ M},
where e ∈ A is the distinguished projection.
(ii) The map E : M → MA, E(x) = e · x, is a faithful conditional expectation
onto the fixed point algebra such that
E((a · x)y) = E(x(S(a) · y)), a ∈ A, x, y ∈ M.
(iii) The linear map F : M ⋊A → M , F (xa) = τ(a)x, is a faithful conditional
expectation onto M .
Recall that M0 := M is an MA-valued inner product module by
〈η(x), η(y)〉MA = E(x∗y)
(here we use the convention in [19] for the inner product as in section 2). Since
every norm bounded MA-module map on M0 extends uniquely to the Hilbert MA-
moduleM, we may identify the ∗-algebra End(M0) (in [18]) of norm bounded right
MA-module endomorphisms ofM0 having an adjoint with the C∗-algebra LMA(M)
explained in section 2.
Remark 3.2. ([19, Proposition 1.3.3]) If E : M → MA is of index-finite type, then
C∗(M, eMA) = span{LxeMALy | x, y ∈ M} = LMA(M).
In fact, we see from the proof of [19, Proposition 2.1.5] that C∗(M, eMA) contains
the unit of LMA(M). Thus the ideal span{LxeMALy | x, y ∈ M} which is dense in
C∗(M, eMA) must contain the unit of LMA(M).
Theorem 3.3. Let A be a finite dimensional Hopf ∗-algebra acting on a unital
C∗-algebra M . Then the following are equivalent:
(i) The action is saturated.
(ii) E : M → MA is of index finite type with Index(E) = (dimA)1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 5
Proof. (i)=⇒ (ii) is shown in [18, Proposition 4.5].
(ii)=⇒ (i). By Remark 3.2, C∗(M, eMA) = span{LxeMALy | x, y ∈ M}. Consider
a map ϕ : C∗(M, eMA) → M ⋊A given by
LxieMALyi) =
xieyi.
To see that ϕ is well defined, let
LxieMALyi = 0. Then for each z ∈ M ,
LxieMALyi)(η(z)) =
η(xiE(yiz)) = η(
xi(e · (yiz))) = 0,
hence by the injectivity of η ([19, 2.1]),
xi(e·(yiz)) = 0 inM . Since (a·x)e = axe
for a ∈ A, a ∈ M (see (7) of [18]), we thus have
xi(e · (yiz))e =
(xieyi)ze = 0
in M ⋊A for every z ∈ M , which then implies that
xieyi)(zez
′) = 0, z, z′ ∈ M.
Particulary, (
xieyi)(
xieyi)
∗ = 0, so that
xieyi = 0 (in M ⋊ A). Thus ϕ
is well defined. It is tedious to show that ϕ is a ∗-homomorphism such that the
range ϕ(C∗(M, eMA)) = MeM is an ideal of M ⋊A; if x, y, and z ∈ M and a ∈ A,
(za)(xey) = (z(a · x))ey ∈ MeM.
Hence it suffices to show that ϕ(1) = 1. If {(ui, u∗i )}ni=1 is a quasi-basis for E, then
LuieMALu∗i (η(z)) =
η(uiE(u
i z)) = η(z), z ∈ M,
which means that
i LuieMALu∗i = 1 ∈ LMA(M). Therefore by Proposition 2.2(iii)
and Proposition 3.1(iii)
F (ϕ(1)) = F (
i ) =
τ(e)uiu
i = 1.
Since ϕ is a ∗-homomorphism, ϕ(1) is a projection in M⋊A such that F (1−ϕ(1)) =
0. But F is faithful, and ϕ(1) = 1 follows. �
4. Actions by finite groups
Throughout this section G will denote a finite group. As is well known the group
C∗-algebra C∗(G) generated by the unitaries {λg | g ∈ G} is a finite dimensional
Hopf ∗-algebra with
∆(λg) = λg ⊗ λg, ǫ(λg) = 1, S(λg) = λg−1 for λg ∈ C∗(G).
The Haar trace τ is given by τ(λg) = διg, where ι is the identity of G, and the
distinguished projection is e = 1
g λg.
Let α be an action of G on a unital C∗-algebra M . Then it is easy to see that
λg · x := αg(x) for g ∈ G, x ∈ M,
6 JA A JEONG AND GI HYUN PARK
defines an action of C∗(G) on M . Furthermore M ⋊ C∗(G) is nothing but the
usual crossed product M ×α G = span{xλg | x ∈ M, g ∈ G}, and the expectations
E : M → Mα(= MC∗(G)), F : M ×α G → M of Proposition 3.1 are given by
E(x) =
αg(x) and F (
xgλg) = xι (x, xg ∈ M, g ∈ G). (1)
Note that for each
xhλh ∈ M ×α G and g ∈ G,
‖xg‖ = ‖F ((
xhλh)λg−1)‖ ≤ ‖(
xhλh)λg−1‖ = ‖
xhλh‖. (2)
If Jα denotes the closed ideal ofM×αG generated by the distinguished projection
e, then Proposition 3.1(i) says that α is saturated if and only if Jα = M ×α G. We
will see in Proposition 5.4 that
Jα = span{
xαg(y)λg | x, y ∈ M} = span{
xαg(x
∗)λg | x ∈ M}. (3)
The ∗-homomorphism ϕ : C∗(M, eMα) → M ×α G we discussed in the proof of
Theorem 3.3 can be rewritten as follows.
ϕ(LxeMαLy) =
xαg(y)λg, x, y ∈ M (4)
because ϕ(LxeMαLy) = xey and e =
λg. If {(ui, u∗i )} is a quasi-basis for E,
we see from
LuieMαLu∗i = 1 and (4) that
ϕ(1) =
uiαg(u
i ))λg (5)
is a projection in M ×α G. Recall that ϕ(1) = 1 holds if α is saturated.
Theorem 4.1. Let M be a unital C∗-algebra and α be an action of a finite group
G on M . Then the following are equivalent:
(i) α is saturated, that is, Jα = M ×α G.
(ii) E : M → Mα is of index finite type with Index(E) = |G|.
(iii) E : M → Mα is of index finite type with Index(E) = |G| and
uiαg(u
i ) = 0, g 6= ι (6)
for a quasi-basis {(ui, u∗i )} for E.
(iv) There exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that
(a) αg(b
) = b
, for j = 1, . . . ,m and g, h ∈ G.
bjg(b
)∗ = δgh.
(v) For every ε > 0, there exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1
such that
‖αg(bjh)− b
‖ < ε,
(b) ‖
bjg(b
)∗ − δgh‖ < ε.
Proof. (i) ⇐⇒ (ii) follows from Theorem 3.3.
(i) =⇒ (iii). If {(ui, u∗i )} is a quasi-basis for E, we have from (5) that
uiαg(u
i ) =
0 for g 6= ι since ϕ(1) = 1.
(iii) =⇒ (ii). Obvious.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 7
(i) =⇒ (iv). Suppose Jα = M ×α G. By (3) there exist m ∈ N and bj ∈ M ,
1 ≤ j ≤ m, such that
bjαg(b
λg = 1.
j = 1 and
bjαg(b
j ) = 0 for g 6= ι. (7)
Set bjg := αg(bj). Then
) = αg(αh(bj)) = αgh(bj) = b
bjg(b
αg(bj)αh(b
bjαg−1h(b
= δgh by (7).
(iv) =⇒ (v). Obvious.
(v) =⇒ (i). Let ε > 0 and let {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} satisfy (a) and
(b) of (v). Note that (b) implies ‖bjg‖ < 1 + ε for g ∈ G, 1 ≤ j ≤ m. Indeed from
bjg(b
∥ ≤ ‖
bjg(b
∗−1‖ < ε, we have ‖bjg‖2 ≤ ‖
bjg(b
∗‖ < 1+ε.
αg((b
)∗)λg)− |G| ‖
αg((b
)∗)λg − |G| ‖
)∗ − |G| ‖+ ‖
g 6=ι
αg((b
)∗)λg)‖
)∗ − 1 ‖+
g 6=ι
αg((b
< ε|G|+
g 6=ι
αg((b
)∗)− (bj
g 6=ι
|G|+ |G|2 max
‖bjg‖+ |G|2
|G|+ |G|2(1 + ε) + |G|2
Since
αg((b
)∗)λg) ∈ Jα and ε can be chosen to be arbitrarily small,
we conclude that Jα = M ×α G. �
Example 4.2. Let w =
be a unitary with wn = 1 and define an
automorphism α on M2(C)) by α(a) = waw
∗, a ∈ M2(C). We will show that α is
saturated if and only if z2 = −z1. For this, recall from (3) that α is saturated if
and only if there exist xj ∈ M2(C), 1 ≤ i ≤ m, satisfying
k(x∗j )λk = 1M2(C). (8)
8 JA A JEONG AND GI HYUN PARK
Hence, particularly for k = 0, 1, we have
xjα(x
j ) =
With xj =
aj bj
cj dj
and zi = e
iθi , i = 1, 2, this means
|aj |2 + |bj |2 aj c̄j + bj d̄j
cj āj + dj b̄j |cj |2 + |dj |2
|aj |2 + ei(θ2−θ1)|bj |2 ei(θ1−θ2)aj c̄j + bj d̄j
|aj |2 + ei(θ2−θ1)dj b̄j ei(θ1−θ2)aj c̄j + |dj |2
. (9)
Therefore, by comparing (1,1) entries of each matrices, it follows that if α is satu-
rated, then there exist positive real numbers a (=
|aj |2) > 0, b (=
|bj |2) > 0
such that
a+ b = 1 and a+ ei(θ2−θ1)b = 0. (10)
Note that b 6= 0 since b = 0 implies a = 0 from
|aj |2 + ei(θ2−θ1)dj b̄j
= 0 in
(9). There are three possible cases for θ1, θ2 as follows.
(i) If θ2 − θ1 ≡ 0(mod 2π), that is, α is trivial, then (10) is not possible.
(ii) If θ2 − θ1 ≡ π(mod 2π), then
and x2 =
satisfy (8) with m = 2. Thus α is saturated.
(iii) If θ2− θ1 6= 0, π(mod 2π), then (10) is not possible for any a, b > 0. Hence
α is not saturated.
Remark 4.3. Let α, β ∈ Aut(M) satisfy αn = βn = idM for some n ≥ 1. If
there is a unitary u ∈ M such that β = Ad(u) ◦ α, then α and β are said to
be exterior equivalent, and if this is the case the crossed products are isomorphic,
M×αG ∼= M×βG, [14, p.45]. Example 4.2 says that the property of being saturated
may not be preserved under exterior equivalence. Also the case (iii) of Example 4.2
above with w = diag(λ, λ̄), λ = e
3 (hence θ1− θ2 = 2π3 −
6= π(mod 2π)), shows
that Index(E) < |G| is possible even when E is of index-finite type. In fact, if
and u2 =
, then {(ui, u∗i )}2i=1 forms a quasi-basis for E,
but Index(E) = 2 < |Z3|.
Remark 4.4. Recall that the Rokhlin property and the tracial Rokhlin property
(weaker than the Rokhlin property) are defined as follows and considered intensively
in [7] and [12], respectively:
(a) ([7]) α is said to have the Rokhlin property if for every finite set F ⊂ M ,
every ε > 0, there are mutually orthogonal projections {eg | g ∈ G} in M
such that
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii)
g∈G eg = 1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 9
(b) ([12]) α is said to have the tracial Rokhlin property if for every finite set
F ⊂ M , every ε > 0, every n ∈ N, and every nonzero positive element
x ∈ M , there are mutually orthogonal projections {eg | g ∈ G} in M such
that:
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii) With e =
g∈G eg, the projection 1 − e is Murray-von Neumann
equivalent to a projection in the hereditary subalgebra of M generated
by x.
The following proposition is actually observed in [12, Lemma 1.13], and we put a
proof for reader’s convenience.
Proposition 4.5. Let M be a unital C∗-algebra and α be an action of a discrete
group G on M . Suppose that for every ε > 0 and every finite subset F ⊂ M , there
exist a family of projections {eg}g∈G such that
(1) ‖αg(eh)− egh‖ < ε.
(2) ‖egx− xeg‖ < ε for each x ∈ F .
Then α is an outer action.
Proof. Suppose there is a unitary u ∈ M such that αg(x) = uxu∗ for every x ∈ M
(g 6= ι). Put F = {u} and 0 < ε < 1/4. Then there exist mutually orthogonal
projections {eg}g∈G such that ‖αg(eh)− egh‖ < ε < 1/4. Thus ‖egu− ueg‖ < ε <
1/4. Then ‖αg(eι)− ueιu∗‖ = 0. But
‖αg(eι)− ueιu∗‖ = ‖αg(eι)− eg + eg − eι + eι − ueιu∗‖
≥ ‖eg − eι‖ − ‖αg(eι)− eg‖ − ‖eι − ueι1u∗‖
≥ 1− 1
which is a contradiction. �
Remark 4.6. If M ×αG is simple, α is obviously saturated, and this is the case if G
is a finite group, M is α-simple, and T̃(αg) 6= {1} for all g 6= ι ([8, Theorem 3.1]).
In particular, α is saturated if M is simple and α is outer.
But for a nonsimple M , this may not hold. In fact, if α is an outer action of Zn
on M and u is a unitary in M with un = 1 such that the action Ad(u) on M is not
saturated (as in Example 4.2), then the action α ⊕ Ad(u) on M ⊕M is outer but
not saturated.
Now we show that if α satisfies the Rokhlin property (or satisfies the tracial Rokhlin
property and M has cancellation) then α is saturated. For this we first review the
cancellation property of C∗-algebras. For projections p, q in a C∗-algebra, we write
p ⊥ q if pq = 0, and p ∼ q if they are Murrey-von Neumann equivalent.
Definition 4.7. A unital C∗-algebra M has the cancellation if, whenever p, q, r
are projections in Mn(M) for some n, with p ⊥ r, q ⊥ r, and (p + r) ∼ (q + r),
then p ∼ q.
10 JA A JEONG AND GI HYUN PARK
Remark 4.8. (1) If M has the cancellation and p, q are projections in M such
that (1− p) ∼ (1− q), then p ∼ q ([4, V.2.4.14]).
(2) It is well known that every C∗-algebra with stable rank one has the cancel-
lation ([4, V.3.1.24]).
Proposition 4.9. Let α be an action of a finite group G on a unital C∗-algebra
M . Then α is saturated if one of the following holds.
(i) α has the Rokhlin property.
(ii) α has the tracial Rokhlin property and M has the cancellation.
Proof. (i) For an ε > 0, there exist mutually orthogonal projections {eg}g such that
g eg = 1 and ‖αg(eh)− egh‖ < ε. Then, with m = 1, the elements b1g := eg satisfy
(v) of Theorem 4.1.
(ii) Now suppose α has the tracial Rokhlin property and M has the cancellation.
We shall show that Jα contains the unit of M ×α G. Let 0 < ε < 1. For each
g ∈ G, choose mutually orthogonal projections {eg
}h∈G such that
‖αk(egh)− e
‖ < ε
2|G|2 ,
and put eg :=
h∈G e
. If eg = 1, for some g, then b1h := e
(h ∈ G) will satisfy
(v) of Theorem 4.1 as in (i). If eg 6= 1 for every g ∈ G, then by the tracial Rokhlin
property of α there exist mutually orthogonal projections {fg
}h∈G in M such that
‖αk(fgh)− f
‖ < ε
2|G|2
) ∼ (eg)′ < eg
for a subprojection (eg)′ of eg. Put fg :=
. Then since M has cancellation, it
follows that fg ∼
1−(eg)′
> (1−eg). Let vg ∈ M be a partial isometry satisfying
v∗gvg = f
g, vgv
g = 1− (eg)′,
and set
xg :=
(ekhαg(e
h) + (1 − ek)vkfkhαg(fkhαg−1 (v∗k))
, g ∈ G.
Now we show that the element x :=
xgλg ∈ Jα satisfies ‖x − 1‖ < ε. In fact,
for g 6= ι,
‖xg‖ ≤
‖(ekhαg(ekh) + (1− ek)vkfkhαg(fkhαg−1(v∗k))‖
≤ 1|G|
‖(ekhαg(ekh)‖ + ‖fkhαg(fkh )‖
≤ 1|G|
‖ekh(αg(ekh)− ekgh)‖ + ‖ekhekgh‖+ ‖fkh (αg(fkh )− fkgh)‖+ ‖fkhfkgh‖
≤ 1|G|
2|G|2 +
2|G|2 )
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 11
|G| (
(1− ek)vkfkv∗k)
(1 − ek)(1 − (ek)′)
For the rest of this section we consider a finite group action on a commutative
C∗-algebra C(X). If G acts on a compact Hausdorff space X , it induces an action,
say α, on C(X) by
αg(f)(x) = f(g
−1x), f ∈ C(X).
For each x ∈ X , let Gx = {g ∈ G : gx = x} be the isotropy group of x and for a
subgroup H of G (H < G), put
XH = {x ∈ X : Gx = H}.
It is readily seen that XH and XH′ are disjoint if H 6= H ′, and X is partitioned as
Theorem 4.10. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action of G on C(X), the following are equivalent:
(i) E : C(X) → C(X)α is of index finite type.
(ii) XH is closed for each H < G.
Moreover, if this is case the index of E is Index(E) =
χXH , where χXH
is the characteristic function on XH .
Proof. (i) =⇒ (ii). If E is of index-finite type and {(ui, u∗i )}ki=1 is a quasi-basis for
E, then
uiE(u
i f) = f,
that is,
ui(x)
u∗i (g
−1x)f(g−1x)
= f(x), (11)
for f ∈ C(X) and x ∈ X . For each x ∈ X , choose a continuous function fx ∈ C(X)
satisfying fx|Gx\{x} ≡ 0 and fx(x) = 1. Then (11) with fx in place of f gives
ui(x)
u∗i (g
−1x)fx(g
= fx(x), (12)
and so we have
ui(x)u
i (x) = 1. (13)
12 JA A JEONG AND GI HYUN PARK
To show that each XH is closed, let {xn ∈ XH : n = 1, 2, . . .} be a sequence of
elements in XH with limit x ∈ XH′ . Then (11) gives
fx(xn) =
ui(xn)
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
u∗i (g
−1xn)fx(g
−1xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
|H |u∗i (xn)fx(xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
Taking the limit as n → ∞, we have
fx(x) =
ui(x)
|H |u∗i (x)fx(x) +
g 6∈H
u∗i (g
−1x)fx(g
ui(x)
|H |u∗i (x)fx(x) + |H ′ \H |u∗i (x)fx(x)
|H |+ |H ′ \H |
ui(x)u
i (x)fx(x).
Therefore, comparing with (13), we obtain
|H ′| = |H |+ |H ′ \H |
since Gx = H
′ and fx(x) = 1. Hence
H ⊂ H ′.
On the other hand, since Gxn = H , again by (13),
ui(xn)u
i (xn) = 1 with
the limit
ui(x)u
i (x) = 1 as n → ∞. But also
ui(x)u
i (x) = 1 by
(13), and thus |H | = |H ′|. Consequently we have
H = H ′
because H ⊂ H ′. This shows that XH is closed.
(ii) =⇒ (i). Assume that XH is closed for every subgroup H of G. Then XH
is open since there are only finitely many such subsets. Let UH = {UH,iH : iH =
1, 2, . . . , nH} be an open covering of XH such that
x ∈ UH,iH =⇒ g−1x 6∈ UH,iH or g−1x 6∈ XH whenever g−1x 6= x.
Let {vH,iH} be a partition of unity subordinate to UH . We understand that the
domain of vH,iH is X by assigning 0 to x 6∈ XH . Let uH,iH =
vH,iH .
We claim that
|H | uH,iH ,
|H | u
: H < G, iH = 1, 2, . . . , nH
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 13
is a quasi-basis for E. For f ∈ C(X) and x ∈ X , let F < G and 1 ≤ j ≤ nF be
such that x ∈ XF and x ∈ UF,j. Then
|H | uH,iHE
|H | u
|H |uH,iH (x)
|H |u
(g−1x)f(g−1x)
|F |uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)|F |u∗F,iF (x)f(x)
vF,iF (x)f(x)
= f(x),
as claimed.
Recall that an action G on X is free if gx 6= x for g ∈ G, g 6= 1, and x ∈ X
Corollary 4.11. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action on C(X), the following are equivalent.
(i) G acts freely on X.
(ii) E : C(X) → C(X)α is of index-finite type with Index(E) = |G|.
(iii) α is saturated.
Proof. (i) =⇒ (ii) is proved in [19, Proposition 2.8.1].
To show (ii) =⇒ (i), let E be of index-finite type with Index(E) = |G|. Then
from Theorem 4.10, we have
|G| = Index(E) =
|H |χXH ,
which implies that H = {ι} is the only subgroup of G such that XH 6= ∅. Hence
X = X{ι}, that is, G acts freely on X . (ii) ⇐⇒ (iii) comes from Theorem 4.1. �
5. Saturated actions by compact groups
Notation 5.1. Let M be a C∗-algebra and α be an action of a compact group G on
M . For x, y ∈ M , define continuous functions fx,y, fx,1, f1,y ∈ C(G,M) from G to
14 JA A JEONG AND GI HYUN PARK
M as follows:
fx,y(t) = xαt(y),
fx,1(t) = x, f1,y(t) = αt(y) for t ∈ G.
Then it is easily checked that fx,y = fx,1 ∗ f1,y and f∗x,y = fy∗,x∗ .
Recall that C(G,M) is a dense ∗-subalgebra of M ×α G with the multiplication
and involution defined by
f ∗ g(t) =
f(s)αs(g(s
−1t))ds,
f∗(t) = αt(f(t
−1)∗),
where dg is the normalized Haar measure ([13, 7.7], [6, 8.3.1]). Hence if G is a finite
group, fx,y can be written as fx,y =
xαg(y)λg .
If M̃ denotes the smallest unitization of M (so M̃ = M if M is unital), the
function
e : G → M̃, e(s) = 1, for every s ∈ G
is a projection of the multiplier algebra of M ×α G ([17]).
Proposition 5.2. ([17]) Let α be an action of a compact group G on a C∗-algebra
M . Then identifying x ∈ Mα and the constant function in C(G,M) with the value
x everywhere we see that
x 7→ fx,1 : Mα → e(M ×α G)e
is an isomorphism of Mα onto the hereditary subalgebra e(M×αG)e of the crossed
product M ×α G.
The notion of saturated action is introduced by Rieffel for a compact group action
on a C∗-algebra, and we adopt the following equivalent condition as the definition.
Definition 5.3. (Rieffel, see [14, 7.1.9 Lemma]) Let M be a C∗-algebra and α be
an action of compact group G on M . α is said to be saturated if the linear span of
{fa,b | a, b ∈ M} is dense in M ×α G (see Notation 5.1). We denote
Jα = span{fa,b | a, b ∈ M}.
Proposition 5.4. Let α be an action of a compact group G on a C∗-algebra M .
Then Jα is the ideal of M×αG generated by the hereditary subalgebra e(M ×αG)e.
Moreover Jα = span{fa,a∗ ∈ C(G,M) | a ∈ M}.
Proof. We first show that Jα is an ideal ofM×αG. Let x ∈ C(G,M) and a, b ∈ M .
Then x ∗ fa,b ∈ Jα. Indeed,
(x ∗ fa,b)(t) =
x(s)αs(fa,b(s
−1t))ds
x(s)αs(a)αt(b)ds
x(s)αs(a)ds
αt(b),
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 15
hence x∗fa,b = fc,b ∈ Jα, where c =
x(s)αs(a)ds ∈ M . Also f∗a,b = fb∗,a∗ implies
that Jα = Jα∗ is an ideal of M ×α G.
Let J := (M ×α G)e(M ×α G) be the closed ideal generated by e(M ×α G)e.
Now we show that Jα ⊂ J . From
(fa,b ∗ e)(t) =
fa,b(s)αs(e(s
−1t)) ds =
aαs(b) ds = a
αs(b)ds,
we have fa,b ∗ e = faE(b),1, where E(b) =
αs(b)ds ∈ Mα. Hence for a, b, c, and d
in M , we have
fa,b ∗ e ∗ fc,d = (fa,b ∗ e) ∗ (fd∗,c∗ ∗ e)∗
= faE(b),1 ∗ (fd∗E(c∗),1)∗
= faE(b),1 ∗ f1,E(c∗)d
= faE(b),E(c∗)d,
which means that fax,yd ∈ J for any a, d ∈ M and x, y ∈ Mα. Since Mα contains
an approximate identity for M , it follows that fa,b ∈ J for a, b ∈ A.
For the converse inclusion J ⊂ Jα, note that if x ∈ C(G,M), then (x ∗ e)(t) =
x(s)ds for t ∈ G. With notations x′ =
x(s)ds and x′′ :=
αs(x(s
−1))ds (∈ M),
we see that
(x ∗ e ∗ y)(t) =
(x ∗ e)(s)αs(y(s−1t))ds
αs(y(s
−1t))ds
= x′αt(
αs(y(s
−1))ds)
= x′αt(y
= fx′,y′′(t)
belongs to Jα for x, y ∈ C(G,M).
Finally the following polarization identity proves the last assertion.
aαt(b) =
ik(b + ika∗)∗αt(b+ i
ka∗).
6. The gauge action γ on a graph C∗-algebra
By a (directed) graph E we mean a quadruple E = (E0, E1, r, s) consisting of
the vertex set E0, the edge set E1, and the range, source maps r, s : E1 → E0. If
each vertex of E emits only finitely many edges E is called row finite and a row
finite graph E is locally finite if each vertex receives only finitely many edges. By
En we denote the set of all finite paths α = e1 · · · en (r(ei) = s(ei+1), 1 ≤ i ≤ n−1)
of length n (|α| = n). Each vertex is regarded as a finite path of length 0. Then
E∗ = ∪n≥0En is the set of all finite paths and the maps r and s naturally extend
to E∗. A vertex v is called a sink if s−1(v) = ∅ and a source if r−1(v) = ∅.
If E is a row finite graph, we call a family {se, pv | e ∈ E1, v ∈ E0} of operators a
Cuntz-Krieger(CK) E-family if {se}e are partial isometries and {pv}v are mutually
16 JA A JEONG AND GI HYUN PARK
orthogonal projections such that
s∗ese = pr(e) and pv =
s(e)=v
e if s
−1(v) 6= ∅.
It is now well known that there exists a C∗-algebra C∗(E) generated by a universal
CK E-family {se, pv | e ∈ E1, v ∈ E0}, in this case we simply write C∗(E) =
C∗(se, pv). For the definition and basic properties of graph C
∗-algebras, see, for
example, [1, 2, 10, 11, 15] among others. If α = α1α2 · · ·α|α| (αi ∈ E1) is a finite
path, by sα we denote the partial isometry sα1sα2 · · · sα|α| (sv = s∗v = pv, for v ∈
We will consider only locally finite graphs and it is helpful to note the following
properties of graph C∗-algebras.
Remark 6.1. (i) Let C∗(E) = C∗(se, pv) be the graph C
∗-algebra associated
with a row finite graph E, and let α, β ∈ E∗ be finite paths in E. Then
s∗αsβ =
s∗µ, if α = βµ
sν , if β = αν
0, otherwise.
Therefore C∗(E) = span{sαs∗β | α, β ∈ E∗}.
(ii) Note that sαs
β = 0 for α, β ∈ E∗ with r(α) 6= r(β).
(iii) If α, β, µ, and ν in En are the paths of same length,
β)(sµs
ν) = δβ,µsαs
Thus for each n ∈ N and a vertex v in a locally finite graph E, we see that
span{sαs∗β | α, β ∈ En and r(α) = r(β) = v}
is a ∗-algebra which is isomorphic to the full matrix algebraMm = (Mm(C)),
where m =
∣{α ∈ En | r(α) = v}
Recall that the gauge action γ of T on C∗(E) = C∗(se, pv) is given by
γz(se) = zse, γz(pv) = pv, z ∈ T.
γ is well defined by the universal property of the CK E-family {se, pv}. Since
γz(sαs
β)dz =
z|α|−|β|(sαs
β)dz = 0, |α| 6= |β|,
one sees that
C∗(E)γ = span{sαs∗β | α, β ∈ E∗, |α| = |β|}.
If Z denotes the following graph:
• • • • •______ // ______ // ______ //______ // · · · ,· · ·Z :
−2 −1 0 1 2
then C∗(Z) is isomorphic to the C∗-algebra K of compact operators on an infinite
dimensional separable Hilbert space, hence C∗(Z) is itself a simple AF algebra. But
C∗(Z)γ coincides with the commutative subalgebra span{sαs∗α | α ∈ Z∗} which is
far from being simple, and thus we know that the simplicity of C∗(E) does not
imply that of C∗(E)γ in general.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 17
In [9], the Cartesian product of two graphs E and F is defined to be the graph E×
F = (E0 ×F 0, E1 ×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)).
Since the graph Z × E has no loops for any row-finite graph E, we know that
C∗(E)γ is an AF algebra ([10]) by the following proposition.
Proposition 6.2. ([9]) Let E be a row finite graph with no sources. Then the
following hold:
(a) C∗(E)γ is stably isomorphic to C∗(E)×γ T.
(b) C∗(E)×γ T ∼= C∗(Z× E).
Now we show that a gauge action is saturated. For this, note that the linear
span of the continuous functions of the form
t 7→ f(t)x, f ∈ C(G), x ∈ A
is dense in C(G,A) [13, 7.6.1]. Hence by Remark 6.1(i), one sees that
C∗(E)×γ T = span{znsαs∗β | α, β ∈ E∗ n ∈ Z}. (15)
Theorem 6.3. Let E be a locally finite graph with no sinks and no sources. Then
the gauge action γ on C∗(E) is saturated.
Proof. We show that Jγ = C∗(E)×γ T. By (15) it suffices to see that
znsαs
β ∈ Jγ for all α, β ∈ E∗, n ≥ 0
(because z−nsαs
β = (z
∗ for n ≥ 0).
Now fix α, β ∈ E∗ and n ≥ 0. Put l = n− (|α| − |β|). There are two cases.
(i) l ≥ 0: One can choose a path µ such that |µ| = l and r(µ) = s(α). Then
znsαs
β = z
l+|α|−|β|s∗µsµsαs
β = s
µγz(sµαs
β) = fs∗µ,sµαs∗β (z),
where the function fs∗µ,sµαs∗β belongs to Jγ .
(ii) l < 0: Choose a path ν with |ν| = |β|+n and r(ν) = r(α). With a = sαs∗ν ,
b = sνs
β , we have fa,b ∈ Jγ and
znsαs
β = sαs
νγz(sνs
β) = fa,b(z).
Acknowledgements. The first author would like to thank Hiroyuki Osaka and Ta-
motsu Teruya for valuable discussions.
References
[1] T. Bates, J. H. Hong, I. Raeburn and W. Szymanski, The ideal structure of the C∗-algebras
of infinite graphs, Illinois J. Math., 46(2002), 1159–1176.
[2] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C∗-algebras of row-finite graphs, New
York J. Math. 6(2000), 307–324.
[3] T. Bates and D. Pask, Flow equivalence of graph algebras, Ergod. Th. & Dynam. Sys. 24(2004),
367–382.
[4] B. Blakadar, Operator Algebras, Theory of C∗-algebras and von Neumann algebras, Encyclo-
pedia of mathematical sciences 122, Springer, 2006.
[5] J. Cuntz and W. Krieger, A class of C∗-algebras and topological Markov chains, Invent. Math.
56(1980), 251–268.
[6] P. A. Fillmore, A user’s guide to operator algebras, Canadian Math. Soc. Series of Monographs
and Adv. Texts, John Wiley & Sons, Inc. 1996.
18 JA A JEONG AND GI HYUN PARK
[7] M. Izumi, Finite group actions on C∗-algebras with the Rohlin property, I, Duke Math. J.
122(2004), no. 2, 233–280.
[8] A. Kishimoto, Outer automorphisms and related crossed products of simple C∗-algebras,
Comm. Math. Phys. 81(1981), no. 1, 429–435.
[9] A. Kumjian and D. Pask, C∗-algebras of directed graphs and group actions, Ergodic Theory
Dynam. Systems, 19(1999), 1503–1519.
[10] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J.
Math. 184(1998), 161–174.
[11] A. Kumjian, D. Pask, I. Raeburn and J. Renault Graphs, groupoids, and Cuntz-Krieger
algebras, J. Funct. Anal. 144(1997), 505–541.
[12] H. Osaka and N. C. Phillips, Stable and real rank for crossed products by automorphisms
with the tracial Rokhlin property, Ergod. Th. & Dyn. Sys., to appear.
[13] G. K. Pedersen, C∗-algebras and their automorphism groups, London Math. Soc. Monographs
No.14, Academic Press 1979.
[14] N. C. Phillips, Equivariant K-theory and freness of group actions on C∗-algebras, LNM 1274,
Springer, 1987.
[15] I. Raeburn, Graph algebras, CBMS. 103, 2005, AMS.
[16] J. Renault, A groupoid approach to C∗-algebras, LNM 794, Springer, Berlin, 1980.
[17] J. Rosenberg, Appendix to O. Bratteli’s paper on ”crossed products of UHF algebras, Duke
Math. J. 46(1979), 25–26.
[18] W. Szymański and C. Peligrad, Saturated actions of finite dimensional Hopf ∗-algebras on
C∗-algebras, Math. Scand. 75(1994), 217–239.
[19] Y. Watatani, Index for C∗-subalgebras, Memoirs of the Amer. Math. Soc. 424, AMS, 1990.
[20] S. L. Woronowicz, Compact matrix psedogroups, Comm. Math. Phys. 111(1987), 613–665.
Keywords: Finite dimensional Hopf ∗-algebra; saturated action; conditional ex-
pectation of index-finite type.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul
National University, Seoul, 151–747, Korea
E-mail address: jajeong@snu.ac.kr
Department of Mathematics, Hanshin University, Osan, 447–791, Korea
E-mail address: ghpark@hanshin.ac.kr
	1. Introduction
	2. Preliminaries
	3. Actions by finite dimensional Hopf *-algebras
	4. Actions by finite groups
	5. Saturated actions by compact groups
	6. The gauge action  on a graph C*-algebra
	References
