数理解析研究所講究録 1291 巻 2002 年 1-10

1

NONCOMMUTATIVE METRIC DIMENSION AND DYNAMICALENTROPY
DAVID KERR

ABSTRACT. We give asummary of some recent work in which notions of dimension anddynamical entropy for metrized C’-algebras were introduced and computed for examples
noncommutative tori. The metric framework here
involving the UHF algebras
is that determined by Rieffel’s concept of aLip-norm, which plays the role of aLipschitz
seminorm on functions over acompact metric space.
$M_{p}\propto \mathrm{a}\mathrm{n}\mathrm{d}$Motivated by the observation of Connes [1, 2, 3] that Dirac operators naturally give rise
to metrics on state spaces, RiefTel initiated aline of investigation $[11, 12]$ that led to thenotion of acompact quantum metric space [13]. Acompact quantum metric space is apair $(A, L)$ consisting of an order-unit space equipped with acertain type of seminorm, called aLip-norm, which is ageneralization of aLipschitz seminorm on functions overacompact metric space. The key requirement in the definition of aLip-norm is that the
metric$A$$L$$\rho_{L}(\mu, \nu)=\sup\{|\mu(a)-\nu(a)| :L(a)\leq 1\}$on the state space of give rise to the weak’ topology. Since the real linear subspace ofself-adjoint elements in aunital C’-algebra can be viewed as an order-unit space, Rieffel’s-algebraic setting, and indeed ’-algebras provide
theory applies in particular to thethe fundamental motivating examples of the subject (see for instance Example 1.6). In[6] we introduced notions of dimension and dynamical entropy for metrized (7’-algebras
within this framework using approximation by finite-dimensional subspaces, and carriedand noncommutaout computations for some examples involving the UHF algebras
metricnoncommutative
ouf
seminorms
tive tori. We also showed that for usual Lipschitz
is toarticlepresent
dimension coincides with the Kolmogorov dimension. The aim of thegive adescription of these results and an indication of the techniques involved in their
proofs. In Section 1we define our metric C’-algebraic framework in precise terms andpresent some examples, and then devote Sections 2and 3to dimension and “product$A$$\mathrm{C}$$\mathrm{C}^{*}$$M_{p}\infty$

entropy,” respectively.

1. cLlP-NORMS ON UNITAL$\mathrm{C}$’-ALGEBRAS

Recall that an order-unit space is areal partially ordered vector spaceguished element , called the order unit, such that, for every $a\in A$ ,with $a\leq re$ , and(1) there exists anthen $a\leq 0$ .
(2) if $a\leq re$ for all
Under the norm$e$$r\in \mathbb{R}$$r\in \mathbb{R}_{>0}$$||a||= \inf\{r\in \mathbb{R} :-re\leq a\leq re\}$$A$

with distin-

2
becomes anormed vector space, and we can recover the order from the norm using thefact that $0\leq a\leq e$ if and only if $||a||\leq 1$ and $||e-a||\leq 1$ . Astate on
is defined aslinearfunctionalonwhose dual norm and value on are both 1(thisanorm-bounded
automatically implies positivity). We denote by $S(A)$ the state space of . The space ofself-adjoint elements in aunital C’-algebra is aprime example of an order-unit space, anin fact every order-unit space can be realized as aunital subspace of self-adjoint operators
on aHilbert space (see [13, Appendix 2]). ALip-norm on an order-unit spaceis aseminorm on such that
, and(1) for all $a\in A$ we have $L(a)=0$ if and only if(2) the metricdefined on $S(A)$ by
$A$

$A$

$A$

$e$

$A$

$A$

$L$

$A$

$a\in \mathbb{R}e$$\rho_{L}$$\rho_{L}(\mu, \nu)$$= \sup${ $|\mu(a)-\nu(a)|$ : $a\in A$ and $L(a)\leq 1$ }gives rise to the weak’ topology.
Acompact quantum metric space is apair $(A, L)$ consisting of an order-unit spaceequipped with aLip-norm .
In particular we can consider aLip-norm on areal unital subspace of self-adjoint elements in aunital C’-algebra, but in this case we wish our generalized Lipschitz seminorm
to be meaningfully defined on the C’-algebra as avector space over the complex numbers,
and so we introduce the notion of acLip-norm.

$A$

$L$

Notation 1.1. Given aseminorm on aunital C’-algebra which is permitted to take, we denote the sets $\{a\in A : L(a)<\infty\}$ and $\{a\in A : L(a)\leq r\}$ (for a
the value, respectively. For $r>0$ the norm ball $\{a\in A:||a||\leq r\}$ will begiven $r>0$ ) by and.
denoted by
$A$

$L$

$+\infty$$L$$L_{r}$$A_{r}$

Definition 1.2. AcLip-nom on aunital C’-algebra is aseminorm on , possibly, such thattaking the value(i) $L(a’)=L(a)$ for all $a\in A$ ,(ii) for every $a\in A$ we have $L(a)=0$ if and only if
, and
(iii)
defines ametric on the state space $S(A)$ which gives
rise to the weak’ topology.
We say that a cLip-norm is aLeibniz cLip-norm if it satisfies the Leibniz rule
$A$

$L$

$A$

$+\infty$$a\in \mathrm{C}1$$\rho_{L}(\sigma, \omega)=\sup_{a\in L_{1}}|\sigma(a)-\omega(a)|$$L$$L(ab)\leq L(a)||b||+||a||L(b)$for all , $b\in L$ .
$a$

The Leibniz rule plays acrucial role in our theory of dynamical entropy (Section 3).
On the other hand, our definition of metric dimension (Definition 2.3) applies to any cLipnorm on aunital C’-algebra, and in fact is easily adapted to the general order-unit context
of compact quantum metric spaces.
It is readily seen that if is acLip-norm on aunital C’-algebra then the restriction $L’$of to the order-unit space
is aLip-norm, and the restriction map from $S(A)$ tois aweak’ homeomorphism which is isometric relative to the metrics
anddefined via and $L’$ , respectively.
Theorem 1.8 of [11] implies the following proposition. Our definitions of dimension anddynamical entropy will be based on the precompactness condition (4)
$L$

$L$

$L$

$S(L \cap A_{\mathrm{s}\mathrm{a}})$$\rho L’$$\cap A_{\mathrm{s}\mathrm{a}}$$\rho L$$L$

3
Proposition 1.3. Let
be aseminorm on aunital C’-algebra , possibly taking the. Then acLip-norm if and only if it separates $S(A)$ (equivalently,$\sup_{a\in L_{1}}|\sigma(a)-\omega(a)|$ defines ametric on $S(A))$ and satisfies(1) $L(a^{*})=L(a)$ for all $a\in A$ ,,(2) for every $a\in A$ we have $L(a)=0$ if and only if(3) $\sup$ { $|\sigma(a)-\omega(a)|$ : , $\in S(A)$ and $a\in L_{1}$ } $<\infty$ , and(4) the set.
is totally bounded in for
$A$

$L$

$L$

$\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}+\infty$$\rho_{L}(\sigma, \omega)=$$a\in \mathrm{C}1$$\omega$$\sigma$$A$$L_{1}\cap A_{1}$$||\cdot$$||$

The essential maps between cLip-normed unital C’-algebras are those which satisfy a
Lipschitz condition:, repecDefinition 1.4. Let and $B$ be unital C’-algebras with cLip-norms
andtively. Apositive unital linear map: $Aarrow B$ is said to be Lipschitz if there exists asuch that$A$$L_{B}$$L_{A}$$\phi$$\lambda\geq 0$$L_{B}(\phi(a))\leq\lambda L_{A}(a)$for all $a\in L$ , in which case the least such Ais defined to be the Lipschitz number of .
If is invertible and both andare Lipschitz and positive, then we say that is-Lipschitz, and if furthermore$\phi$$\phi$$\phi^{-1}$$\phi$$\phi$$bi$$L_{B}(\phi(a))=L_{A}(a)$for all $a\in A$ then we say thatis Lipschitz isometric. We denote by AutL(A) thecollection of Lipschitz ’-automorphisms of .
$\phi$$A$$\mathrm{b}\mathrm{i}$It can be shown that if is alower semicontinuous Leibniz cLip-norm on aunital C’algebra $A$ and is aunitary in , then the inner ’-automorphism Ad is Lipschitz, with,
the Lipschitz numbers of Ad and its inverse Ad ’ bounded bywhere diam(S(A)) is the diameter of $S(A)$ under the metricdefined by .
Next we turn to asome examples.
$L$

$L$

$u$

$u$

$u$

$2(1+2L(u)\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(S(A)))$

$u$

$L$

$\rho_{L}$Example 1.5. Let $(X, d)$ be acompact metric space. The Lipschitz seminorm{$L(f)= \sup$ $|f(x)-f(y)|/d(x,$ $y)$ : $x$ , $y\in X$ and $x\neq y$on$\mathrm{C}(X)$}.

yields an example of aLeibniz cLip-norm.

Example 1.6 (ergodic compact group actions). The most important examples for us willbe those arising from ergodic actions of compact groups. Let be an ergodic action of
acompact group $G$ on aunital C’-algebra , with denoting the identity element of $G$ .
Let be alength function on $G$ , that is, acontinuous function :such that, forall , $h\in G$ ,(1) $\ell(gh)\leq\ell(g)+\ell(h)$ ,(2) $\ell(g^{-1})=\ell(g)$ , and(3) $\ell(g)=0$ if and only if $g=e$ .
Prom the length function and the group action we obtain the seminorm$\gamma$$A$$e$$\ell$$\ell$$Garrow \mathbb{R}\geq\circ$$g$$\ell$$\gamma$$L(a)= \sup\{||\gamma_{g}(a)-a||/\ell(g) :g\neq e\}$,. Evidently is adjoint invariant and satisfies the Leibniz rule, while $L(a)=0$ if and. It follows from [11, Thm. 2.3] that the metricgives rise to the weak’
only if
topology on $S(A)$ , and so is Leibniz cLip-norm

on

$A$

$L$

$a\in \mathrm{C}1$$\rho_{L}$$L$4
In [6] we also showed how acLip-norm can be defined on acrossed product by abiLipschitz ’-automorphism by means of the dual action, following the approach suggested
by Example 1.6.

2. METRIC DIMENSION
We begin with some notation to describe our approximation framework.

Notation 2.1. Let $(X, ||\cdot||)$ be anormed linear space (which will be either a -algebra
or aHilbert space in our case). The collection of finite-dimensional subspaces of $X$ will
. Given subsets , $Z\subset X$ and $\delta>0$ we will write
be denoted by
if for every
there exists an $x\in Z$ with $||y-x||<\delta$ . With $\dim$ denoting vector space dimension,
we set, for any subset $Z\subset A$ and $\delta>0$ ,$\mathrm{C}^{*}$$Y$$\mathcal{F}(X)$$Y\subset\delta Z$$y\in \mathrm{Y}${$D(Z, \delta)=\inf$ $\dim X$(or $D$ ( ,$Z$$\delta)=\infty$:$X\in \mathcal{F}(A)$and$Z\subset\delta X$

if the set on the right is empty) and if $\sigma\in S(A)$ then we set$D_{\sigma}(Z, \delta)=\inf${ $\dim X$ :$X\in \mathcal{F}(\Re_{\sigma})$and$\pi_{\sigma}(Z)\xi_{\sigma}\subset\delta X$(or( $Z$ ,
if the set on the right is empty). Here:
representation associated to , with canonical cyclic vector 4,.
$D_{\sigma}$}$\delta)=\mathrm{o}\mathrm{o}$$\pi_{\sigma}$}$Aarrow \mathfrak{B}(\mathcal{H}_{\sigma})$is the GNS$\sigma$

The following proposition provides atool for obtaining lower bounds for $D(Z, \delta)$ (andwhich will be crucial in our computations of both dimension andhence also for
dynamical entropy.
$D_{\sigma}(Z, \delta))$

Proposition 2.2 ([15, Lemma 7.8]). Let $B$ be an orthonormal set of vectors in aHilbert
space
and $\delta>0$ . Then$\Re$$\inf${ $\dim X$ :$X\in \mathcal{F}(\mathrm{H})$and$X\mathrm{c}_{\delta}B$}$\geq(1-\delta^{2})\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(B)$.

Letbe aunital C’-algebra with cLip-norm . The key to our definition of metricis finite for every $\delta>0$ . This finiteness follows fromdimension is the fact that, while not totally bounded, is the set of translations of athe observation that
totally bounded set by scalar multiples of the identity (see [6, Prop. 3.2]).
$A$

$L$

$D(L_{1}, \delta)$$D(L_{1}, \delta)$

Definition 2.3. The metric dimension of$A$with respect to$\mathrm{M}\dim_{L}(A)=\lim_{\deltaarrow}\sup_{0+}\frac{\log D(L_{1},\delta)}{1\mathrm{o}\mathrm{g}\delta^{-1}}$$L$is defined by.

Note the formal similarity with Kolmogorov dimension, whose definition we recall in thenext paragraph. The metric dimension is invariant under -Lipschitz positive unital maps,
and decreases with respect to quotient cLip-norms under surjective positive unital linear
maps. Also, the metric dimension of adirect sum of two cLip-normed unital C’-algebras
is the maximum of the metric dimensions of the summands.
Let $(X, d)$ be acompact metric space. We denote by $N(\delta, d)$ the minimal cardinality ofacover of $X$ by -balls. The Kolmogorov dimension of $(X, d)$ is defined by$\mathrm{b}\mathrm{i}$$\delta$$\mathrm{K}\dim_{d}(X)=\lim_{\deltaarrow}\sup_{0+}\frac{\log N(\delta,d)}{1\mathrm{o}\mathrm{g}\delta^{-1}}$.

5This also goes by various other names in the literature, such as box dimension and limitd) dewhere
capacity. It can also be expressed as
notes the largest cardinality of a -separated set, or asd) denotes the smallest cardinality of a&-spanning set (see [7, 8]).
where$\lim\sup_{\deltaarrow 0^{+}}\log \mathrm{s}\mathrm{e}\mathrm{p}(\delta, d)/\log\delta^{-1}$$\mathrm{s}\mathrm{e}\mathrm{p}(\delta,$$\lim\sup\deltaarrow 0+\log \mathrm{s}\mathrm{p}\mathrm{n}(\delta, d)/\log\delta^{-1}$$\delta$$\mathrm{s}\mathrm{p}\mathrm{n}(\delta,$

Proposition 2.4. Let $(X, d)$ be acompact metric space with associated Lipschitz seminorm$L(f)= \sup$ { $\mathrm{C}(\mathrm{X})-f(y)|/d(x,$ $y)$ : , $y\in X$ and $x\neq y$ }$x$on$\mathrm{C}(X)$. Then$\mathrm{M}\dim_{L}(\mathrm{C}(X))=\mathrm{K}\dim_{d}(X)$.

To prove Proposition 2.4, we first obtain the inequality
by astraightforward partition of unity argument. The reverse inequality is establishedby means of the following idea. Given a -separated set $E$ of maximum cardinality, we
consider the probability measure uniformly supported on $E$ . We then construct unitaries
with suitably small Lipschitz seminorm which, when viewed inside $L^{2}(X, \mu)$ , forminan orthonormal basis dual to the standard basis. We can then use Proposition 2.2 to get alower bound for $\log D(L_{1}, \delta)/\log\delta^{-1}$ which is asymptotically sharp under taking the limit
supremum.

$\mathrm{M}\dim_{L}(\mathrm{C}(X))\leq \mathrm{K}\dim_{d}(X)$$\delta$$\mu$$\mathrm{C}(X)$(also). Consider the infinite tensor product
Example 2.5 (the UHF algebra
over with the infinite tensor product of) of $p\cross p$ matrix algebras
denoted by
the unique ergodic action ofis
WeylactionWeyl actions. We recall that theon asimple ’-algebra up to conjugacy [9], and it is defined on the clock and shift unitaxy
by the specifications
generators and of$M_{p}^{\otimes \mathbb{Z}}$$M_{p}^{\otimes \mathbb{Z}}$$\mathbb{C}$$M_{p}$$M_{p}\infty$$G=\mathbb{Z}_{p}\cross \mathbb{Z}_{p}$$\mathrm{C}$

$v$

$u$

$M_{p}$$\gamma_{(r,s)}(u)=\rho^{r}u$$\gamma_{(r,s)}(v)=\rho^{s}v$,.

. We can then define the infinite product action
root of unity
where is the.
onof the product groupas, viewing
induced by the Euclidean metric
Let be the length function on
standard,
there is. While this length function may be regarded as
asubgroup of. Perhaps the simplest ones are those obtained byno canonical length function ongeometrically weighting on the factors with respect to aparameter $\lambda\in(0,1)$ :$e^{2\pi i/p}$$\gamma^{\otimes \mathbb{Z}}$$p\mathrm{t}\mathrm{h}$$\rho$$G^{\mathbb{Z}}$$M_{p}^{\otimes \mathbb{Z}}$$\mathbb{R}^{2}$$G$$\ell$$G$$\mathbb{R}^{2}/\mathbb{Z}^{2}$$G^{\mathbb{Z}}$$\ell$$\ell_{\lambda}((g_{j}, h_{j})_{j\in \mathbb{Z}})=\sum_{j\in \mathbb{Z}}\lambda^{|j|}\ell((g_{j}, h_{j}))$We denote by the cLip-norm on
following Example 1.6.
function$L$$M_{p}^{\otimes \mathbb{Z}}$.

which arises from the action$\ell_{\lambda}$

Proposition 2.6. We have$\mathrm{M}\dim_{L}(M_{p}^{\otimes \mathbb{Z}})=\frac{41\mathrm{o}\mathrm{g}p}{1\mathrm{o}\mathrm{g}\lambda^{-1}}$.

$\gamma^{\otimes \mathbb{Z}}$and the length

6
To establish the inequality MdimL
onto the subalgebraofconditional expectation$(M_{p}^{\otimes \mathbb{Z}})\leq 4\log p/\log\lambda^{-1}$$E_{n}$$M_{p^{\Delta}}^{\otimes 7}$

we consider for eachgiven bythe$n$$M_{p}^{\otimes[-n,n]}$,$E_{n}(a)= \int_{G^{\mathrm{Z}\backslash [-n,n]}}\gamma_{g}^{\otimes \mathbb{Z}}(a)dg$

the subgroup ofof elements
andwith $dg$ denoting Haar measure on$a\in
L_{1}$
we estimate
which are the identity at the coordinates in the interval $[-n, n]$ . For, so that we can approximate inside asubalgebra$||E_{n}(a)-a||$ in terms of $L(a)$ andin asufficiently controlled way with respect to that we obtain an asymptotically
sharp upper bound for the metric dimension using the linear dimensions $\dim M_{p}^{\otimes[-n,n]}=$$G^{\mathbb{Z}\backslash [-n,n]}$$G^{\mathbb{Z}}$$G^{\mathbb{Z}}$$a$$\lambda$$M_{p}^{\otimes[-n,n]}$$p^{2(2n+1)}$$n$.

For the reverse inequality we consider for each$U_{n}=${$u^{i_{-n}}v^{j_{-n}}\otimes u^{i_{-n+1}}v^{j_{-n+1}}\otimes\cdots\otimes u^{i_{n}}v^{g_{n}}$:the subset$n$$0\leq i_{k}$, $jk\leq p-1$ for $k=-n$ ,$\ldots$

,

$n$

}

where the factors in the elementary tensors are products of powers of theofas an orthonormal set of vectors in theclock and shift unitaries. We can view eachon
via the GNS construction, so thatunique
the
with
space
associatedHilbertwe can appeal to Proposition 2.2, which, along with the observation that the cLip-norms
are at most $(4n+2)\lambda^{n}$ , yields the desired lower bound for the metricof elements indimension.
$M_{p}^{\otimes[-n,n]}$$U_{n}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$M_{p}\infty$$U_{n}$

Example 2.7 (noncommutative tori). Given an antisymmetric bicharacter, for $1\leq i$ , $j\leq k$ we set,$\rho:\mathbb{Z}^{p}\cross \mathbb{Z}^{p}arrow$$\mathbb{T}$$\rho_{ij}=\rho(e_{i}, e_{j})$, and we defineis the standard basis forwhere, satisfying
C’-algebra generated by unitaries ,$\{e_{1}, \ldots, e_{p}\}$$\mathbb{Z}^{\rho}$$u_{1}$$\ldots$

to be the universal$u_{p}$$u_{j}u_{i}=\rho_{ij}u_{i}u_{j}$.

as anoncommutative -torus. Let 7:
We refer to
ergodic action determined by$p$$A_{\rho}$$A_{\rho}$$\mathrm{T}^{p}\cong \mathbb{R}^{p}/(2\pi \mathbb{Z})^{p}arrow \mathrm{A}\mathrm{u}\mathrm{t}(A_{\rho})$be the$\gamma_{(t_{1},\ldots,t_{\rho})}(u_{j})=e^{it_{j}}u_{j}$(see [9]), and consider the length function arising from the Euclidean metric, viewing. Following Example 1.6 we thereby obtain acLip-norm on
as the quotient$\mathrm{T}^{p}$$L$$\mathbb{R}^{p}/(2\pi \mathbb{Z})^{p}$$A_{\rho}$.

Proposition 2.8. We have$\mathrm{M}\dim_{L}(A_{\rho})=p$.

This computation is formally similar to that of Proposition 2.6, and indeed the inequality
is established in the same way, only now with respect to sets of productsof the generating unitaries, which form orthonormal sets in the Hilbert space associatedto the canonical$\mathrm{M}\dim_{L}(A_{\rho})\leq p$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\tau(a)=\int_{-\pi}^{\pi}\cdots\int_{-\pi}^{\pi}\gamma(t_{1}, ,t_{p})(a)dt_{1}\cdots$ $dt_{p}$7invia the GNS construction. For the reverse inequality the conditional expectations
aCes\‘aromean,
andtakingofcase must now be replaced by the operationthesome Fourier analysis is required to obtain the desired bounds for $||a-\sigma_{n}(a)||$ for $a\in L_{1}$as afunction of .
We also showed in [6] that, with respect to anatural cLip-norm, the metric dimensionof acrossed product of anoncommutative -tori by a Lipschitz ’-automorphism is $p+1$ .
$E_{n}$$\sigma_{n}$$M_{p}\infty$$n$$\mathrm{b}\mathrm{i}$$p$3. PRODUCT ENTROPY
The “product entropy” to which this section is devoted has two versions, one topologicaland the other measure-theoretic. The appropriate framework for our definitions is thatof unital C’-algebras with Leibniz cLip-norms, with the dynamics given by bi-Lipschitz’-automorphisms. Our approximation approach is formally similar to that of Voiculescu[15], but the algebraic structure enters the picture here in avery different way. Productentropy can be viewed as an analytic version of entropy for discrete Abelian groups [10].
We may thus think of it as being “dual” to Voiculescu entropy in some rough sense.
We begin with some notation (see also Notation 1.1 and 2.1).
Notation 3.1. Given aset $X$ , we denote by $Pf(X)$ the collection of finite subsets of $X$ .
to refer to, we write,, ,If is aC’-algebra with subsets
the set, }.
:for each $i=1$ ,{$A$$X_{2}$$X_{1}$$a_{1}a_{2}\cdots a_{n}$$\ldots$$X_{1}\cdot$$X_{n}$$a_{i}\in X_{i}$$\ldots$$X_{2}\cdot\cdots\cdot$ $X_{n}$$n$Definition 3.2. Let be aunital C’-algebra with Leibniz cLip-norm . Let(see Definition 1.4). For $\Omega\in Pf(L \cap A_{1})$ and $\delta>0$ we set
$L$

$A$

$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}_{L}(A)$,$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha, \Omega, \delta)=\lim\sup\log D(\Omega\cdot\alpha(\Omega)\cdot\alpha^{2}(\Omega)\cdots\cdot\cdot\alpha^{n-1}(\Omega), \delta)\underline{1}$$n$$narrow\infty$,$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha, \Omega)=\sup_{\delta>0}\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha, \Omega, \delta)$$\mathrm{E}\mathrm{n}\mathrm{t}_{\mathrm{P}L}(\alpha, \Omega)$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha)=\sup\Omega\in Pf(L\cap A_{1})$.

as the product entropy of .
It is readily seen that product entropy is invariant under conjugacies by bi-Lipschitz’-isomorphisms.
The following proposition establishes aconnection between product entropy and metricdimension.
is finite. Thenand suppose that
Proposition 3.3. Let

We will refer to$\alpha$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha)$$\mathrm{M}\dim_{L}(A)$$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}_{L}(A)$$\mathrm{E}\mathrm{n}\mathrm{t}_{\mathrm{P}L}(\alpha)\leq \mathrm{M}\dim_{L}(A)\cdot\log\max(\lambda, 1)$

where Ais the Lipschitz number of$\alpha$.

whenever is Lipschitz isometric. In particular,
As acorollary we obtain, which suggests that the appropriate context for our notion of productentropy as ameasure of dynamical growth is that of unital C’-algebraic with cLip-norms
under which the metric dimension is finite.
Astandard argument yields the next proposition, using the fact that is closed undermultiplication by the Leibniz rule$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha)=0$$\alpha$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}\mathrm{L}(\mathrm{a})=0$$\mathrm{L}$8
Proposition 3.4. If

and$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}L(A)$$k\in \mathbb{Z}$then$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha^{k})=|k|$EntpL(a).

Product entropy decreases under taking quotients in the presence of aLipschitz crosssection, and decreases when passing to an invariant cLip-normed C’-subalgebra obtained
via restriction if there is anorm-contractive idempotent linear map onto the subalgebra.
Before coming to examples we introduce aversion of product entropy relative to a
dynamically invariant state. For notation see Notation 1.1, 2.1, 2.1, and 3.1.

Definition 3.5. Let$Pf(L \cap A_{1})$ and $\delta>0$ we set$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}L(A)$and let abe an$\alpha$-invariant state on

$A$

. For

$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\sigma}(\alpha, \Omega, \delta)=\lim\sup\log D_{\sigma}(\Omega\cdot\alpha(\Omega)\cdot\alpha^{2}(\Omega)\cdots\cdot\cdot\alpha^{n-1}(\Omega), \delta)\underline{1}$$narrow\infty n$$\Omega\in$,,$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\sigma}(\alpha, \Omega)=\sup_{\delta>0}\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\sigma}(\alpha, \Omega, \delta)$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\sigma}(\alpha)=$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\sigma}(\alpha, \Omega)$$\Omega\in Pf(L\cap A_{1})\sup$We refer to$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}L,\sigma(\alpha)$as the product entropy of$\alpha$.

with respect to$\sigma$.

Product entropy with respect to an invariant state is invariant under conjugacies by
Lipschitz ’-automorphisms which respect the given invariant states, and it decreases
under taking quotients if there exists aLipschitz cross section. It is easy to see that$\mathrm{b}\mathrm{i}$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}L,\sigma(\alpha)\leq \mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}L(\alpha)$

for any invariant state , and we also have the following analogue of Proposition 3.4.
$\sigma$

Proposition 3.6. Let
we have EntpL $(\alpha^{k})=|k|$ EntpL$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}L(A)$$(\alpha)$.

, and let$\sigma$be an$\alpha$-invariant state on

$A$

. For

$k\in \mathbb{Z}$We illustrate these two product entropies with two fundamental classes of examples,
infinite tensor product shifts and noncommutative toral automorphisms.
Example 3.7 (tensor product shifts). Let
be the shift on the infinite tensor productwith the cLip-norm as defined in Example 2.5 with respect to agiven $\lambda\in(0,1)$ .
Then is -Lipschitz, and it is straightforward to verify that the Lipschitz numbers ofat and its inverse are bounded by A. Let
be the unique (and hence a-invariant).
tracial state on$\alpha$$L$$M_{p}^{\otimes \mathbb{Z}}$$\alpha$$\mathrm{b}\mathrm{i}$$\tau=\mathrm{t}\mathrm{r}_{p}^{\otimes \mathbb{Z}}$$M_{p}^{\otimes \mathbb{Z}}$Proposition 3.8. We have.

$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha)=\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\tau}(\alpha)=2\log p$

This computation is made by elaborating the proof of Proposition 2.6, with the additional feature now of having to estimate the growth of products of norm-0ne elements inwith respect to subalgebras of the form.
$M_{p}^{\otimes[m,n]}$$L$

Example 3.9 (noncommutative toral automorphisms). Let
be anoncommutative p,
and canonicaltorus with generators ,(see Example 2.7). We consider
a $p\cross p$ integral matrix $T=(s_{ij})$ with $\det T=\pm 1$ , and we suppose that $T$ defines anautomorphism
of $A$ by specifying$A_{\rho}$$u_{1}$$\ldots$$u_{p}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\tau$$\alpha\tau$$\alpha(u_{j})=u_{1}^{s_{1j}}\cdots u_{p^{pj}}^{s}$9-invariant. For an ordinary torus this yields afundaon the generators. Note that ismental example of an Anosov diffeomorphism. We will also consider the ’-automorphismon thedetermined byfor agiven $t=$
$u\in
L$
.
point
We
out
unitary
agivenforAd’-automorphism
inner
generators, and the(i.e., the case $p=2$
that in [5] it was shown that, for an irrational rotation algebrahere), if the angle 0satisfies ageneric Diophantine property then all ’-automorphisms
elements (i.e., all “diffeomorphisms”) have thepreserving the dense ’-subalgebra ofunitary .
for a
form
and its inverse
Using the remark after Definition 1.4 it can be verified that Adu
byLipschitz
boundedwithnumbers-Lipschitz,
are$\tau$$\alpha\tau$$\gamma_{\mathrm{f}}(u_{j})=e^{it_{j}}uj$$(t_{1}, \ldots, t_{p})\in \mathbb{T}^{p}\cong \mathbb{R}^{p}/(2\pi \mathbb{Z})^{p}$$\gamma_{t}$$u$$A_{\theta}$$\mathrm{C}^{\infty}$$\mathrm{C}^{\infty}$$\mathrm{A}\mathrm{d}u\circ\alpha\tau\circ\gamma t$$u$$\circ\alpha_{T}\circ\gamma_{t}$$\mathrm{b}\mathrm{i}$$2r(T)(1+2L(u)\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(S(A)))$and$2r(T^{-1})(1+2L(u)\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(S(A)))$

respectively, where$r(\cdot)$,

denotes the spectral radius.

Proposition 3.10. We have$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\mathrm{A}\mathrm{d}u\circ\alpha_{T}\circ\gamma_{t})\leq\sum_{|\lambda_{i}|\geq 1}\log|\lambda_{i}|$and$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha_{T}\circ\gamma_{t})=\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\tau}(\alpha_{T}\circ\gamma_{t})=\sum_{|\lambda_{i}|\geq 1}\log|\lambda_{i}|$where$\lambda_{1}$,$\cdots$,$\lambda_{p}$are the eigenvalues of$T$

counted with spectral multiplicity. In particular,$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\alpha_{T})=\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\tau}(\alpha\tau)=\sum_{|\lambda_{i}|\geq 1}\log|\lambda_{i}|$$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\gamma_{t})=\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\tau}(\gamma_{t})=0$, and$\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L}(\mathrm{A}\mathrm{d}u)=\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{p}_{L,\tau}(\mathrm{A}\mathrm{d}u)=0$.

To establish upper bounds for the product entropies in the proposition we can adapt
some classical Fourier analysis to our noncommutative setting (as in the proof of PropOby their Ces\‘aro means, the growth of whose
sition 2.8) to approximate elements in
products we can then estimate with respect to the linear span of sets of products of theunitary generators. For the lower bounds it suffices to consider sets of products of theunitary generators, which form orthonormal bases in the Hilbert space associated to .
Estimating the growth of products of these sets essentially reduces to the computation of.
the Abelian group-theoretic entropy of the action of $T$ on$L$$\tau$$\mathbb{Z}P[10]$Acknowledgements. This work was supported by the Natural Sciences and Engineering
Research Council of Canada. Iwould also like to thank Y. Kawahigashi and the operator
algebra group at the University of Tokyo for their hospitality and the vigorous research
environment they have provided, and T. Sanno for giving me the opportunity to present-algebras.
this material in the January 2002 RIMS conference on$\mathrm{C}^{*}$

10
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$\mathrm{K}\mathrm{U}$GRADUATE SChOOl OF MATHEMATICAL Sciences, UNIVERSITY OF Tokyo, 3-8-1 KOMABA, MEGURO, Tokyo 153-8914, iApAN