数理解析研究所講究録
1077 巻 1999 年 46-51

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Noncommutative Topological Entropy
Nathanial P. Brown 1Purdue

1$\mathrm{U}\mathrm{n}\mathrm{i}_{\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{s}}\mathrm{i}\mathrm{t}\mathrm{y}/\mathrm{U}\mathrm{n}\mathrm{i}_{\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{s}}\mathrm{i}\mathrm{t}\mathrm{y}$of Tokyo

Introduction-algebra and $\alpha\in Aut(A)$ is
is aRecall that a pair $(A, \alpha)$ , wherean automorphism, is called a -dynamical system. In the classical (i.e.
abelian) setting we would have $A=C(X)$ for a compact Hausdorff space $X$
and would be induced by a homeomorphism : $Xarrow X$ via the formula. An important invariant in the study of classical dynamicalsystems in entropy. Roughly speaking, entropy is a measure of how mucha homeomorphism : $Xarrow X$ “mixes up” the space $X$ (i.e. a measure ofergodicity).
$A$$C^{*}$$C^{*}$$\alpha$$\varphi$$\alpha(f)=f\circ\varphi^{-1}$$\varphi$

In the classical setting, there are two notions of entropy. With classicaltopological entropy one attempts to count the minimal number of open setsrequired to cover the (compact) space $X$ (see [Wal]). With classical measurable entropy one is given a -invariant probability measure on $X$ and one
takes a certain weighted measure of a partition (see [Wal]). At first glance,
it is not apparent that these notions are particularly well related to eachother. But indeed they are and the following variational principle provides
the bridge: If $\varphi:Xarrow X$ is a homeomorphism of a compact metric space $X$anddenotes the topological entropy of then$\mu$$\varphi$$h_{Top}(\varphi)$$\varphi$$h_{Top}( \varphi)=\sup_{\mu}h_{\mu}(\varphi)$,

wheredenotes the measurable entropy of with respect tosupremum is taken over all -invariant probability measures .
$h_{\mu}(\varphi)$$\varphi$$\mu$

and the$\mu$$\varphi$

There are dozens of papers dealing with various notions of noncommutative measurable entropy (i.e. notions of entropy of automorphisms (or endomorphisms) of noncommutative-algebras taken with respect to anand$C^{*}$$W^{*}$

1Supported by an NSF Dissertation Enhancement Award

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invariant state or trace). However, there has been relatively little work donein noncommutative generalizations of topological entropy. Unfortunately,
noncommutative entropy is a highly technical subject and it requires a fairamount of diligence just to understand some of the definitions. Another difficult aspect is the large number of competing definitions (which probably best
illustrates the mathematical infancy of the subject). Indeed, an incomplete
list of notions of noncommutative entropy is [CS], [CNT], [ST], [Ch2], [Th],[Hu] and [Vo]. (It should be remarked that some of these various definitions
are known to agree when restricted to suitably nice classes of operator algebras.) However, of these papers, only [Th], [Hu] and section 4 of [Vo] reallydeal with noncommutative. topological entropy.
In this note we will briefly survey Voiculescu’s approximation approachto topological entropy (cf. [Vo], [Brl]) and try to point out some of the openquestions which we believe to be of particular interest. This approach also
yields notions of measurable entropy (see [Vo]) and has the added feature ofbeing conceptually simpler than some of the other approaches listed above.

The Approximation Approach

2-algebra and $\alpha\in Aut(A)$ then the topologicalis a unital nuclear
If. This definition has
entropy of (in the sense of [Vo]) is denoted byrecently been modified so as to apply to automorphisms of arbitrary exact-algebras as well. As previously mentioned, this approach is relatively
simple (compared to [CNT], for example) but is still too technical to recall
the precise definition (please see [Brl]).
$C^{*}$$A$$ht(\alpha)$$\alpha$$C^{*}$

The class of exact -algebras is the largest class for which the approximation approach yields a reasonable definition. It is also not too hard toshow that all of the properties which were known in the nuclear case extend to the exact setting as well. For example, we have the following basic
properties.
$C^{*}$

Proposition 2.1 (cf. $[Vo$ , Prop. 4.2], $[Brl$ , Prop. 2.5.]) If.
$\alpha\in Aut(A)$ then $ht(\alpha^{k})=|k|ht(\alpha)$ for all$A$is exact and$A$is exact and$k\in \mathbb{Z}$Proposition 2.2 (cf. $[Vo$ , Prop. 4.3], $[Brl$ , Prop. 2.6]) If48$\{\omega_{\lambda}\}_{\lambda\in\Lambda}$span

is a net of finite sets (partially ordered by inclusion) such thatis dense in A then$\mathrm{U}_{\lambda\in\Lambda,n\in}\mathbb{Z}\alpha(n\omega_{\lambda})$$ht( \alpha)=\sup..ht(\alpha\lambda’.\omega_{\lambda})$Proposition 2.3 (cf. $[Vo$ , Prop. 4.9], $[Brl$ , Prop. 2.7]) If
are exact,$ht(a_{1}\otimesa_{2})\leqht(\alpha_{1})+ht(\alpha_{2})$. Ifarefor $i=1,2$ thenunital then we also have.
$A_{i}$$\alpha_{i}\in Aut(A_{i})$$A_{1},$ $A_{2}$$ht( \alpha_{1}\otimes\alpha_{2})\geq\max\{ht(\alpha_{1}), ht(a_{2})\}$

It is still an open problem whether or not the equality
always holds. This equality is roughly equivalent to provingthat the best way to approximateis to use algebras of the form
$ht(\alpha_{1}\otimes\alpha_{2})=$$ht(\alpha_{1})+ht(\alpha_{2})$$A_{1}\otimes_{\min}A_{2}$$M_{n_{1}}(\mathbb{C})\otimes M_{n_{2}}(\mathbb{C})$.

We should also point out thatreally does extend classical topologicalentropy (cf. [Vo, Prop. 4.8], [Brl, Prop. 1.4]).
$ht(\cdot)$

Proposition 2.4 If : $Xarrow X$ is a homeomorphism of the compact metricspace $X$ and $a\in Aut(C(X))$ is defined bythen $ht(\alpha)=$.
$\varphi$$\alpha(f)=f\mathrm{o}\varphi^{-1}$$h_{Top}(\varphi)$

One advantage of the definition given in [Brl] is the following result (cf.

[Brl, Prop. 2.1] .
$)$Proposition 2.5 (Monotonicity) If is exact, $\alpha\in Aut(A)$ andis a -subalgebra such that $\alpha(A_{0})=A_{0}$ then $ht(a|_{A_{0}})\leq ht(\alpha)$ .
$A$$A_{0}\subset A$$C^{*}$

In general, getting lower bounds foris the more difficult task andthe previous proposition gives one strategy for doing so. Another way to getlower bounds foris to compute some of the current notions of measurable
entropy.
$ht(\cdot)$$ht(\cdot)$

Proposition 2.6 (cf. $[Vo$ , Prop. 4.6],$Thm$ .
nuclear -algebra, $a\in Aut(A)$ and is a state on
then$[Ch\mathit{2},$$C^{*}$$\varphi$$h_{\varphi}(\alpha)\leq ht_{\varphi}(\alpha)\leq ht(\alpha)$where$h_{\varphi}(\alpha)$

is defined in $[CN\tau]$ and$ht_{\varphi}(\alpha)$) If is a unital
such that$\mathit{2}.\mathit{6}.\mathit{1}]$

$A$

$A$

$\varphi 0\alpha=\varphi$,

is defined in$[Ch\mathit{2}]$.

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This result begs the question of whether or not we have a noncommutative
analogue of the classical variational principle. It is known that in general
is not the supremum over all invariant states of the [CNT] entropyand. However, 2.6.2 in [Ch2] provides a variational principle forfor certain shifts on UHF algebras. It is natural to ask whether or
not this holds in general.
$ht(\alpha)$$ht(\alpha)$$h_{\varphi}(a)$$ht_{\varphi}(\alpha)$

Another interesting open question (which would generalize a known result in the classical setting) is whether or not entropy decreases in quotients.
Conceptually, it seems reasonable to conjecture that this should be true.
However, it seems difficult to compare the two entropies without some nuclearity hypothesis. (Perhaps assume that either the ideal or the quotient isa nuclear -algebra.)$C^{*}$The main new result of [Brl] provides a strong bridge between dynamics
and geometry in -algebras. It is our hope that in the future, the following
result will be useful both in applications of dynamics to -algebra theory
and, conversely, applications of -algebra theory to (classical) dynamics.
$C^{*}$$C^{*}$$C^{*}$:-algebra,) If is a unital exact
Theorem 2.7 (cf. $[Brl,$ $Thm$ .
$Garrow Aut(A)$ is a group homomorphism (withdiscrete and abelian) taking
is the unitary implementing the automorphismand$a_{g}\in Aut(A)$ then.
$\mathit{3}.\mathit{5}]$$A$$C^{*}$$\alpha$$G$$g\vdash\Rightarrow\alpha_{g}$$\lambda_{g}\in A$$\mathrm{X}_{\alpha}G$$ht_{A}(\alpha_{g})=ht_{A\cross_{\alpha}}c(Ad\lambda_{\mathit{9}})$is the implementing uniIn particular, if $\alpha\in Aut(A)$ and $u\in A$
tary then $ht(\alpha)=ht(Adu)$ , thus reducing most questions about topologicalentropy to the case of inner automorphisms.
$\mathrm{x}_{\alpha}\mathbb{Z}$The point of this theorem is that we may now investigate how the topological entropy of is related to the relative position of the implementing. For example, in [Br2] we showed that $ht(\alpha)>0$ has
unitarygeometric consequences for any potential inductive limit decomposition of
in terms of subhomogeneous algebras. Conversely, since
is known to have such an inductive limit decomposition for minimal Cantor
to learnsystems (cf. [Pu]), we may hope to use the geometry of
about dynamical properties of .
$\alpha$$u\in A\rangle\triangleleft_{\alpha}\mathbb{Z}$$A$$C(X)\rangle\triangleleft_{\alpha}\mathbb{Z}$$x_{\alpha}\mathbb{Z}$$C(X)\mathrm{x}_{\alpha}\mathbb{Z}$$\varphi$

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