£ Jan. 1B32

UWThPh-1991-62 . December 13, 1991

Quantum Topological Entropy: First Steps of a "Pedestrian" Approach

Thomas Hudetz* Institut für Theoretische Physik

Universität Wien

Abstract

We introduce a notion of topological entropy for automorphisms of arbitrary (non-commutative, but unital) nuclear <7*-algebras A, generalizing the "classical" topological entropy for a homeomorphism T : X —* X of an arbitrary (possibly connected) com­pact Hausdorif space X, where the generalization is of course understood in the sense that the latter topological dynamical system (with 2Z-action) is equivalently viewed as the C*-dynamical system given by the T-induced automorphism of the Abelian C"-algebra A = C(X) of (complex-valued) continuous functions on X. As a simple but basic exam­ple, we calculate our quantum topological entropy for shift automorphisms on AF algebras A associated with topological Markov chains (i.e. "quantum topological" Markov chains); and also a real physical interpretation of our simple "quantum probabilistic" entropy func­tional is discussed (already in the introduction, anticipating the later definitions and results).

*)Expanded version of an invited lecture at the INTSEM seminar on "Entropy and Dynamical Entropy in Mathematical Physic»", Leipzig, August 11-13, 1991.

a)Supported by "Fonds zur Förderung der wissenschaftlichen Forschung in Osterreich" (Proj. P7101-Phy).

1 Introduction and Discussion

1.1 Comparative "mathematical" introduction

The purpose of this paper is to introduce a notion of topological entropy for automor­phisms of arbitrary (non-commutative, but unital) nuclear £7*-algebras A, generalising the "classical" (see (2.1) below) topological entropy for a homeomorphism T : X —* X of an arbitrary (possibly connected) compact Hausdorff space X, where the generalisation is of course understood in the sense that the latter topological dynamical system (with 22-action) is equivalently viewed as the C*-dynamical system given by the T-induced auto­morphism of the Abelian C*-algebra A = C(X) of (complex-valued) continuous functions on X.

In the same sense, also the Connes-Narnhofer-Thirring (CNT for short, [1]) entropy lor a C'-dynamical system (with respect to a single invariant state) is a generalization of the "classical" measure-theoretic (Kolmogorov-Sinai) dynamical entropy; and it was originally designed for this very same class of nuclear C*-algebras, but only recently it has been extended to the class of quasi-local algebras in quantum statistical mechanics (by Park and Shin [2]). In both cases, the key concept is that of completely positive unital maps 7 : B —* A of finite rank (i.e. from any finite-dimensional unital C*-algebra B) into the given C*-algebra A, which is the "natural" extension of the special case of an inclusion «-homomorphism 7 = «s : B —• A of a finite-dimensional C'-auialgebra B C A; and this extension is really necessary for a possible application of the CNT entropy theory to (nuclear) C*-algebras A without sufficiently many finite-dimensional subalgebras, in particular the projectionless (or "connected", cf. [3] and e.g. [4]) <7*-algebras A. The latter class of C*-algebras contains as Abelian algebras A exactly the £7"-algebras A = C(X) of all possible connected compact spaces X; and even then every finite partition of unity a = {J4; 6 r-t+IEt^; = 11 £ A} determines a (completely) positive unital map la '• Ba -•* A from a certain finite-dimensional Abelian (7*-algebra Ba into A (simply defining 7a(e») -- -A> with the minimal projections e; 6 Ba), but on the other hand this partition of unity a uniquely determines an open cover W(a) = {Ui C X\\Ji Ui — X} of X, simply defining the Ui to be the cozero-sets of Ai € A = C(X), i.e. X \ Ui — A^^O). In turn, by Urysohn's lemma, any finite open cover of X has always a subordinate partition a of the unit H € C(X) = A, and thus the set of open covers of the form U(a) is a cofinal subset of the set 0(X) of all finite open covers of X, w.r.t. the natural partial order on 0{X)\ cf. (2.2) below. - This correspondence fa *-» U(a) was used in [5] to "pull back" the classical definition of topological entropy (see still (2.1) below) to the category of Abelian C*-algebras A 3 11 (together with the "auxiliary" Ba) and positive unital maps (in particular the fa : Ba —» A and the -»-automorphisms 6 : A -* A), and we shall once more reformulate this simple redefinition in section (2.3) below.

The challenging extension of this classically equivalent definition to the category of non-Abelian (nuclear) C-algebras A 3 11 and (completely) positive unitaJ maps is of course motivated by the above-mentioned CNT generalization of the measure-theoretic dynamical entropy: Our simple basic idea is to consider the (completely) positive unital

1

maps 7 : B —* A of finite rank also as the "proper" non-commutative generalization of finite open covers for a compact space X\ in much the same way as already the finite-dimensional C*-su6algebras B C A (if any exist) are viewed as the non-Abelian generalization of finite (measurable or clopen Borel-) partitions of X, originally in the CNT- and related work (see e.g. [6, 5] for a review), but even so in Lindbl&d's later counter-proposal to it [7].

From the latter work, however, we take the essential idea of how to define the "joint topological entropy" # (71 , . . • ,7n) of several such positive unital maps fk '• Bk —* A (n 6 IN) for non-commut itive A, generalizing the "topological" entropy H(Ui V . . . VUn) of the common refinement of several open covers W* of a 6pace X (see (2.1) below for its classical definition) in the sense explained above: #(71, • •, 7 n ) will be defined (in (3.8)) by a varia­tional expression, formally simi'ar to the CNT definition of Hu(fi,..., 7») for a particularly chosen state u on A [1]; but now the supremum is taken over (finite) "square root" de­compositions a* C Bk of the identity (in the sense of Lindblad's "operational partitions of unity" = OPUs [7]) in each of the preimage-algebras Bk, respectively, and the functional to be maximized depends only on the n-tuple of "image-OPUs" (7i[ati]j,.. • ,7n[on]a) C -4". Here 7*[»]a : Bk-* A+ denotes the "quantum mechanically" non-linear application of the originally /»near map 7* : Bk —* A (to be defined in Def.(3.6) below), and in the special case of the inclusion *-horoomorphism 7* = tßb : Bk —» A of a finite-dimensional auialgebra Bk C A this coincides again with the usual linear application on the positive cone Bjf. In this case we thus have »sja*]» = tgh(ak) for our OPUs a* C Bf with positive elements, which we can in turn identify with a* = 13,(a*) as OPUs in the sutalgebras Bk C A; and then the "entropy" functional, maximized to give ^(ifl , , . . . ,!©,) = H(B\,. ..,Bn), depends more precisely only on the possible n-fold products (in A) of the elements in a*, always one for each k and ordered in. k. This is exactly the same construction as used by Lindblad [7] to define the "ordered" refinement (or "composition", in his terminology) of the OPUs a* "associated" with the resp. finite-dimensional subalgebras Bk C A; and evi­dently by this construction the functional H{B\,... ,Bn), 01 more generally # ( 7 1 . . -,fn), is not necessa ily symmetric in its arguments Bk resp. 7*, in contradistinction to the CNT functional Hu with state u».

1.2 Physical interpretation and discussion As also Lindblad [7] in his counter-proposal to the latter CNT approach, we regard this intrinsic "time order" as physically natural and even meaningful in the quantum case, in the sense of generalized "instruments" (cf. e.g. [8] and references there) or "operations" (see e.g. [9]) to be represented by the OPUs; but in contradistinction to Lindblad's approach in [7], the off-diagonal elements of the (again generalized) "decoherence functional" matrix (see Gell-Mann and Hartle [10]) for the set of generalized "histories" (in the sense of Griffiths [11], as extended by Omnes [12] and stated in a coherent form - in the "twilight" of quantum cosmology - by Gell-Mann and Hartle [10]) given by the set {{a\,... ,a„)} C An

(in the above latter setting, with our somewhat symbolic notation) do not contribute to our "topological" entropy functional (optimized to) H(B\,...,#„), whereas the diagonal matrix elements of the decoherence functional actually contribute as functionals also on

2

the stale space 5.4 of our C*-algebra A (i.e. taking into account any possible initial state for the sequence of "measurement situations" represented by the OPUs {<*fc}4elN):

For the convenience of the reader and to clarif) our notation and terminology, let us first zecall these notions from [10] but naturally generalized for our "positive" OPUs ak = {AÜV € Bk\Lik(M?)2 = Hfc}. with A^ not necessarily projections in Bh- A history in the set denoted by { ( a i , . . . , a„)} is given by an n-tuple (A\t , . . . , A!*?'); and thus out symbolic "braced" notation simply mean6 that the set braces are "pulled outside" of the tuple brackets: (a» , . . . . a . ) = {{A\%..., {A£>}) VS. { ( a x > . . . , a«)} = {{A£\..., < ) } • Using the short notations In = (-;,... ,*„) € JV„ and Eu = A^ • . . . • A£\ the deco­herence functional matrix Du for given (a t , • . . , ot„) and a fixed state v G SA i« denned by Du(Jn,Jn) = u(E*uEJm) (cf. also [7, p. 185]); and rather obviously by Def. (3.8) of H(Bi, ...,Bn) together with Defs. (3.2,i),(3.1) and (2.5,i) below, we have to take into ac­count in our entropy functional H only the diagonal elements Dv(In, I„) VJ„ for given (c*i... , Q„), but evaluated on any possible state w G S ^ (cf. also (3.16) below).

Of course, the o/f-diagonal elements of the decoherence functional matrix for the set of histories {(ai,... ,On)} will hardly ever vanish identically as functional» on SA, and thus the diagonal matrix elements (now in turn viewed as functional on different histories out of some varying set { (a i , . . ..<»„)}, for varying a* C 5*; & = 1> • • • •«) 'will generally not be n-linearly additive w.r.t. n-tuple elements of { ( « I , . ..,a^)}; or in Gell-Mann & Hartle's physical terminology: As the set of alternative histories { ( a j , . . . ,otn)} will hardly ever "decohere" for every possible initial state, the (at first purely formal) "probabilities" assigned to the individual histories by the diagonal elements of the decoherence functional matrix will generally not be additive w.r.t. "coarse graining" of the histories in the set.

Nevertheless, we want to physically interprete any possible complete set of histories {(ai,...,a„)} from above as representing an n-step "measurement situation", which means (still in Gell-Mann & Hartle's language) that each of the possible n-fold ordered operator products £/„ 6 A of the elements in a* (one for each k) is a priori assumed to be ideally "fully correlated" with a "quasiclassical" operator e/„ (not necessarily be­longing to the system's algebra A - as we are not dealing with quantum cosmology -but possibly to some "classical" Abelian algebra C "outside" of A) in a decohering set of "external output configurations"; and then the (originally only formal) probabilities Dv(Inf In) of the non-decohering histories in { ( a i , . . . , a«)} can be consistently assigned to the resp. corresponding (since resp. "fully correlated"!) decohering "output" operators e/„ 6 C. - Note that this (here still somewhat vague and artificial) "correlation" between non-decohering sets of histories { (« i , . . . »a«)} and decohering sets of operators (in par­ticular and typically, mutually orthogonal projections) ej. € C, of course depending on the particularly chosen state u> on A, can be formalized (at least abstractly) by a natural notion of an "Abelian model" map Pu : A —» C (with a state fiu on the Abelian C7*-algebra C, both Pw and \iu uniquely determined by ( o i , . . . , a«) and u>, but generally not such that w = nu o Pu Vu/ € SA), which is again formally similar to the notion of "Abelian models" introduced in the CNT definition of Hv(ii}... ,7„) as cited above [1], see (3.16) below.

On the one hand, this physical interpretation justifies (or better, necessitates) the sup-

3

pression of the possibly non-zero off-diagonal decoherence functional matrix elements in our definition of #(71, . . . , 7 n ) for maps 74 : 8k —» A as discussed before; in contradistinc­tion to Lindblad's construction cited [7, p. 185], defining the entropy of an OPU w.r.t. a state by the "quantum mechanical" (von Neumann) entropy of the resulting decoherence functional matrix as a "second quantized" /mife-di/nensional density matrix. - Note, by the way, that in our (and actually also Lindblad's) generalized setting those off-diagonal ele­ments can be non-vanishing even in tae "classical" case of Abelian A, due to the fact that a* resp. 7fc[a*J2 are generalized OPUs, not necessarily consisting only of mutually orthogonal projections (as in the Gellmann & Hartle approach). Nevertheless, the announced equiva­lent redefinition of the "classical" topological entropy (see set. (2.3) below) uses really our functional #(71, - • • >7r») i n the special case of Abelian A, thus neglecting the off-diagonal decoherence functional matrix elements for the ocurring OPUs (7![11)3,... ,7n[<*n]j)i which would be even impossible to be incorporated in the classically equivalent definition (note further that Lindblad [7] could still "pretend" to eliminate them a posteriori in the com­mutative case, because he considered only finite-dimensional su&algebras B C. A instead of our more general maps 7 : B —> A; a problem which, however, he himself had to realize as that of a generalized OPU, not belonging to a finite-dimensional subalgebra, and then "containing a non-trivial inner dynamics of A" in the terminology of [7]).

On the other hand, this physical interpretation for the set of "histories'' { (7 i [a i ] 2 , . . . ,7n[«n]2)} C An, as ocurring in the definition of # ( 7 1 , . . . , 7 n ) , does of course not remedy the non-additivity of the diagonal probabilities in the decoherence functional matrix (evaluated on SA) w.r.t. elements of the resp. preimo^e-OPUs a* C Bk] a non-linearity which is mathematically due to the non-vanishing but (here!) a priori "physi­cally irrelevant" off -diagonal elements, and which is even aggravated by the "quantum mechanically" necessary but rather formal than really essential non-linear application 7*[»]2 : Bk —> A+. This remnant of the quantum-mechanical non-decoherence resp. non-linearity, being irrelevant by the assumption of a "full correlation" between (the probabili­ties of) the histories in {(7i[ai]2,.. •, Ttjänja)} *od the resp. decohering "output" operators (in C with state fiu, cf. above and (3.16) below), is consequently "wiped out" in the final definition of the "topological" entropy functional / f ( 7 i , . . . , 7 n ) by taking the supremum (actually maximum) over the set of maps 7 »-• 0(7) from the set of all positive unital maps of finite rank 7 (into A, i.e. 7 : By —* A with some finite-dimensional B T ) into the set of OPUs a in any finite-dimensional C'-algebra, but with the restriction that 0(7) C B-, and thus yielding {(01(71),.. . , at n(7 n))|at C Bk V&}. In turn, this could be viewed as one "jus­tification" (or even confirmation) of our basic idea to consider the maps 7 : Bk —» A as the "proper" non-commutative generalization of finite open covers (instead of the more directly related partitions of unity D G A, here the "intermediate" image-OPUt ,«(a*]j C A).

Physically speaking, exp{/f(7i, . . . ,7*)} € IN counts the minimal number of "histories'', definitely chosen from the set ol history-n-tuples {(7i[ai]j, • •. ,fn[<*n}i)} M above, which is necessary in order to have for any possible initial state in SA at least one history (among the chosen ones, which could then be called the "S^-necessary" ones) realized with a positive probability; but in turn this (resp. minimal) number Is still maximized to the extent of performing the optimal "operation" (or choosing the optimal "instrument", in

4

the terminology of [9] resp. [8]) Tkla^jj for each distinct 7* individually, "triggered" by means of the resp. finite-dimensional "apparatus" algebra B^ D a* (and optimally in the sense of an absolutely maximal number of possible and "SU-necessary" histories, or finally outcomes corresponding to mutually orthogonal projections in the "Abelian model" algebra C as mentioned above). For the experimental physicist, this coupled two-fold optimization procedure amounts to the economical rule to "switch off" as many (partially) redundant "channels" (outputs in C) as possible without completely losing control of the individual initial state of the system represented by A, i.e. such that in any event at least one channel has a certain chance to pick up a (possibly not very informative) signal; but to adjust the apparatus (in the final effect, possibly on a microscopic - quantum mechanical - scale, represented by the possibly non-Abelian "triggering" algebras Bk and the non­linear "apparatus implementation" maps lk[*]i '- Sfc —* A+) and correspondingly also the output device (represented by the "Abelian model" map P„ : A —* C Vu> G SA, cf. (3.16) below) in such a way that an absolutely maximal number of (5>-) necessary channels has to be kept in operation (i.e. in order to collect maximal "potential information" about the system with the available apparatus, but always in a "back-burner" operation).

2 Classical Theory and C*-algebraic Reformulation

2.1 The classical theory of topological entropy

For the convenience of the reader and to fix our notation, we first recall the definitions and general properties of the "classical" topological entropy from the original paper by Adler, Konheim and McAndrew [13], which contains already most of the essential parts of the classical theory. Throughout this section we use the standing notation {X, T) for a topological dynamical system, given by a compact Hausdorff space X (not necessarily metric) and a homeomorphism T : X —• X. By definition, any open cover of X possesses a finite subcover, and we can restrict ourselves to the latter from the very beginning, denoting by O(X) or simply O the set of finite open covers of X.

Definition (2.1): For U,V £ O(X) we define their join U V V = {U D V\U 6 U, V G V} G 0{X), and the action of T on W 6 0 by T-\U) = {T~\U)\U G U).

(i) N(U) - min{cardW|W' C U : W G 0{X)} denotes the cardinality of a minimal subcover W of U G O. The "topological" entropy of U G O(X) is defined by H(U) = H N{14).

(ii) The entropy of T w.r.t. U G Ö is defined as h(T,U) = limn.,«, {H{U V T~XU V . . . V T'^U).

(iii) The topological entropy of T is h(T) - s u p W € 0 h{T,U).

A cover V G O is said to be a refinement of a cover U, U -< V, if VV G V 3U 6 U : V C U\ and obviously (O, y) is a partially pre-ordered set, but even a (pre-)directed set with UvV>U,V.

5

Proposition (2.2): The entropy functional (2.1,i-iii) have the following general prop­erties VW, V e O(X),

ad (i) (a) U -< V = * N(U) < N(V) <=* H{U) < H(V).

(b) N(U) • N{V) > N(U VV)<=* H{U V V) < H(U) + H(V)

(c) N{T'lU) = N(U) <=> H^U) = #(W).

ad (Ü) (d) W X V =*> h(T,U) < fc(r, V), foUows from (a).

(e) fc(T,#) < H{U), foUows from (b) & (c).

(f) &(r,W) = h(T~\U), follows from (c).

ad (iii) (g) h(T) = limbec h(T,U) as a net limit (corollary of d); and in particular for a refining sequence {Un £ C?}n6lN °f ° P e n covers, i.e. Un -< W n + l Vn and such that W e C? 3a G IN : V -i Un, as a subnet: &(T) = lim«^«, h{T,Un).

(h) Let $ : X —» X' be a homeomorphism onto a compact (Hausdorff) space JC', then h(T) = h($oTo S" 1), follows from (c).

(i) h(T) = hiT-1), follows from (f).

As used already in [13], (g) simplifies the computation of h(T) for metric spaces (X,d), where each sequence {Vfc 6 Q{X)}keJH w**'1 <ü«nieters <f(V*) = maxy^gv» ^(^i) shrinking to zero, d(Vfc) —* 0, gives a refining sequence {W„|Wn — V*=i Vfc}„eIN ^ v Lebesgue's covering lemma.

Theorem (2.3): In addition, the topological entropy (2.1,iii) has the following special properties refering to the varying (non-conjugate) topological dynamical system (X, T):

(i) h{Tk) = k • h(T) Vfc 6 IN.

(ii) Let T = T\ x Tj be the product homeomorphism of Ti : X{ ~» X* (i = 1,2) acting on X = Xi x X7 (with product topology). Then h(T) = Ä(TI) + h(T7).

(iii) Denote (again generally) by M(X) — M the convex compact space of (Borel) prob­ability measures on X (with the weak-* topology), and by TM denote the induced affin; homeomorphism TM : M —» Af defined by TJ/(M) = M° ^ _ 1 M € M(X). If £(7) > 0, then A(7W) - oo (cf. [14, 15]).

(iv) Denote by M(X, T) C M(X) the (non-empty) convex compact set of Tai-flxed points n = TM(H), i'C- T-invariant probability measures ft on X. Then A(T) = *nPneM{x,T)hf{T)i where Ä.M(T) denotes the measure-theoretic (Kolmogorov-Sinai, KS) entropy of T w.r.t. fi (cf. originally [16, 17]).

6

2.2 Motivations for the C*-algebraic formulation

Now, this latter KS entropy was generalized finally by Connes, Narnhofer and Thirring (CNT for short, [1]) to the entropy hw(6) of a «-automorphism 0 o n a (unital) C*-algebra A with invariant state a; = w o 0 £ SA, the state space of A (cf. e.g. [6, 5] for a "historical" review). Remember that, in the most general case, this CNT entropy is defined at first by hw(0) = 8uP7ec?,(.A) ^*(0i7)j where CV\{A) denotes the set of completely positive unital maps 7 : B —• A of finite rank (i.e. from any finite-dimensional unital C-algebra B) into the given C7*-algebra A, and where ku(6,7) is the "entropy" [1] of 9 G * — Aut(.A) w.t.t. 7 6 CVt{A), formally analogous to (2.1,ii & iii) above. Actually, this supremum taken over CVi(A) can already generally be restricted to the set fC%(A) C CV\(A) of all completely positive unital maps 7 : Mn(C) —> A (Vn € IN!) from any matrix algebra into A: Aw(0) = sup 7 € K ; i ^j Aw(0,7); and for an Abelian (7*-algebra A = C(X), X as above, and &x € * - Aut(.A) induced by T : X —• X as above (simply by OT{A) = A o T VA 6 A = C(X)), we can restrict even further as stated in (i) of the following

Proposition (2.4)[5]: Let A = C(X) for some X and 0T 6 * — Aut(.A) for some T : X —» X (as above) be fixed.

(i) Denote by Vi(A) the set of all positive (as well known, here "completely-" is redun­dant!) unital maps from finite-dimensional Abelian C*-aigebras into A. Then the CNT entropy hu{8r) = S UP 76P 1(>) ^«(^TI7)» VU; 6 Sjk with u> = u> 0 fly.

(ii) For /* e Af (X) denote by u„ 6 SA the integral state u>„ = Sx dP- T h e n t t e C N T

entropy HUII(6T) = ^(T 1) V/t 6 Af(X,T), with then obviously invariant uh — u^odj (and the KS entropy on the r.h.s.).

Both (i) and (ii) are not at all surprising, but also not quit« "tautologically" trivial; and in particular (ii) merely points out that, for the whole category of Abelian (7*-algebras and their *-auto(or -isomorphisms, the CNT entropy is really nothing but the "pullback" of the KS entropy (for the category of compact spaces X and homeomorphisms) via the Gelfand isomorphism A : A —• C(X) and Riesz' representation of states on A. On the other hand, however, (i) & (ii) leads to a new "interpretation" (or equivalent redefinition) of this entropy Ä„(T) for T : X ~* X and u £ M{X, T) above, in terms of "fuzzy partitions" (cf. [18, 19]) of the unit U G C{X) and their "entropy" (cf. [5, Appendix] and see [20]), via the following simple construction:

Even for a general (non-commutative) C*-algebra A 3 11, any such map 7 € V\{A) with 7 : B —* A (and hence B Abelian!) as in (2.4.,i) uniquely determines a partition of the unit a-y = {Ai € A+\ £» A{ — B}, simply defining Ai = 7(6*) with the minimal projectors ti £ B (i — 1 , . . . , dimß); and this is the natural choice of a., for the calculation of the entropy hv(6T,f) in (i) (or of the "related", but much better calculable "fuzzy-entropy" [5, App.], [20]), as the latter depends (in the final effect, beside» on u and 0y) only on the set {7(ej) = Ai}, even when "ignoring" that 7 is actually linear. - In turn, again generally, any such partition of the unit a — {Ai € .4 + | t = 1 , . . . ,n : Y%=i M = 11} gives a positive unital map 7,, : Ba ~» A with S a - C © <D © . . . © C (n times; i.e. -ya £ Vi(A) as before),

I

again simply denning 7 a (c.) = Ai with the minimal projectors e* 6 Ba (linearly extended), and thus 7 a is "uniquely" determined by the set (not n-tuple) a up to n-permutation automorphisms of Ba; which then allows to really equivalently rephrase the definition of huißr) for A = C(X) as in (2.4,i & ii) in terms of these "fuzzy partitions" a of 11 G A (see [20]). We henceforth denote by Of (A) generally the set of all such finite positive partitions a of the unit 1 £ A in a C*-algebra A.

But now, on the other hand, for A = C{X) any such a G 0*(A), a — {At € A+l'EiAi = A}, uniquely determines an open cover U(a) G 0(X) as used in (2.1 k 2) above, simply by W(a) = {Ui c X\X \Ui = ^ ( O ) } (i.e. denning the Ui to be the cozero-sets of Ai G C{X)). Of course, this map Of {A) B a i-» U{a) G O(X) is highly non-injective, and we could define a suitable equivalence relation on Of (A) and factor the map accordingly; but for our purpose we may let it be many-to-one. Even worse, for a not metrizable X (i.e. A not separable), the map a *-* U{a) is not even swjective into 0(X) in general, whereas it evidently is onto O(X) for (X,d) metric (as then eoery open set U C X is cozero-set of some A € A+, e.g. take 4(x) = iniyex\u d(x,y), Vx € A', with X \ U = J 4 - J ( 0 ) as required; cf. also e.g. [21]). However, the subset {U(a)\a G Of (A)} C 0(X) is always cofinal w.r.t. the partial order (2.1): W G O(X) 3a G Of {A) such that U{a) >- V (by Urysohn's lemma, cf. e.g. [21]); and thus we can reexpress the topological entropy (2.1,iii) generally as h{T) = supa £o+M) h(T,U{a)), again by (2.2.,d), also for X not necessarily metric.

By the same argument (2.2,d), we know that for 7 6 Vi(A), 7 : B —».4 = C(X) with "associated" cty £ Of (A) as before, and U{ciy) 6 O(X) as just above:

h(T,U{cH))= sup h(T,UW))),

which is clear from the fact that for 6 = {e* G B\i = I , . . . , d i m ß } G 0+(ß) with the minimal projectors e* G ß we have 7(6) = 04 and then obviously U{a^) >- U^ß)) V/3 G Of{B)\ or "symbolically" Ufa) = VVofts) #(->(/?)) as a "supremum" (note that for the finite-dimensional Abelian preimage-algebra B there exist only finitely many different U(i(ß)) G O(X) as ß varies over Of(B), such that the supremum over Of(B) is actually attained as a maximum). In this sense, ou, is again the "natural" choice for computing the entropy (2.1,ii) of T w.r.t. an open cover "associated" with 7 G *Pi{A); and by analogy with (2.4,i & ii) this makes it tempting to define the topological entropy (2.1,iii) of (X,T) formally "algebraically" for A = C(X) and 6r G + — Aut(4) as before, using the above expression as a definition for "A(0r)7)" a n { ^ again defining h(6r) = 9 UP 7ePi(>) M^Ti 7) I such that (obviously from the above discussion) again h(0f) = A(T) would be fulfilled (corresponding to (2.4,ii)).

However, for the desired generalization of this still only formally algebraic "definition" of h{6) to 8 G * — Aut(.4) for non-Abelian C-algebras A (see set. 3), where in addition for 7 € CVi(A) with 7 : ß - • A and also non-commutative pteimage-algebra ß there is no uniquely determined a 7 G C?i"(«4) any more (as there is for 7 € Pi(-4) above), it seems to be necessary (see set. 3) to "mimick" the definition [1] of hw{6,^) as in (2.4,i) even further in the algebraic definition of "£(#,7)" (although still equivalent to the above first

8

attempt in the Abelian case A = C{X) and 7 G V\{A))\ and consequently to consider the maps 7 6 (C)Vi(A) as the "proper" (in section 3, non-Abelian) algebraic generalization 01 the open covers U G 0{X), instead of the more directly related partitions a G Of (A) of 1 G A (as discussed before), using those maps 7 already in the C*-algebraic reformulation (and then non-Abelian generalization) of the entropy (2.1,i) for open covers U G O(X), as it then enters in the definition (2.1 ,ii) of h{T,U).

First, we know from the above discussion of the latter entropy when restricted to all U = U{a) with a 6 Ot{X) "associated" to 7 6 Vi{A\ j:B-*A = C{X) and a = -y(ß) for some ß G Of(B), that a fortiori also for the former /T-entropy:

is valid, with 04 = 7(6) as before (follows from (2.2,a)). Thus the right hand side offers itself as the (again first only formally - ) "algebraic" definition of an entropy H(f) for 7 G (C)Vi(A); and it turns ou+ that this definition can be made "purely" algebraic rather easily, but since in the definition (2.1,i) of H{M(a)) — logN(U(a)) we possibly omit sets Ui G W(a) to get a minimal subcover U(a)' G O(X), corresponding to omitting certain elements A,- G o G Of{A), we first have to weaken the requirement that a be a partition of unity 11 G A.

2.3 C*-algebraic reformulation of the classical theory We do this using the notion of strict positivity in a (general, not necessarily Abelian) unital C*-algebra:

Definition (2.5): Let A 3 11 be a C-algebra, then A G A+ is said to be strictly positive (denoted A > 0 in contradistinction to A > 0), if the following (rather obviously) equivalent conditions (i)-(iii) are fulfilled:

(i) <f>(A)>o \/<f>eSA

(ii) 3 A"1 G A : A A'1 = A~M = n

(iii) 3e > G : A > e • 11.

Note that in particular for an Abelian C*-algebra A, again isomorphic to C(X) for some X as before via Gelfand's isomorphism A : A —» C(X), we have that A > 0 in A iff A{x) > 0 V « e X .

Again for a general C'-algebra A 9 U, clearly A+ 3 A > 0 4=> A' > 0, but as an easy exercise (in elementary C*-theory) one sees that for a set a = {Ai G A+\i = 1 , . . . ,n} also even

( S * ) > O H P ? H and we call such a set a fulfilling these equivalent conditions a "positive operator cover" for A, and 0+(A) D 0*(A) denotes the set of all these a.

9

Now in turn again for Abelian A = C(X), we can directly extend our "definition" (in (2.ii) above) of the map Of(A) 3 a i-» W(a)(e 0(X)\ also to a G 0+(A) but even to a = {Ai G - 4 + } £" C?+(.4), and obviously from above a G & (A) *=» U{a) G 0(X) . Furthermore, for A = C ( * ) and a = {>!*},/3 = {Bj} G 0 + ( 4 ) we can define their "refinement" a V/3 = {A,; • Bj € ,4+}, such that obviously a V0 6 C?+(>1) and W(a Vfl) = U(a)Vli(ß) with the join (2.1) in C?(X). And for T : X -» Jf with induced BT e *-Aut(>t) as in (2.4), we have again obviously: T~*U{a) = U(8T{Q)) Va € 0+(A). Now, we can first "algebraically translate" the classical definition of topological entropy in (2.1) as follows:

Definition (2.8): Let A be an Abelian C-algebra with 0 G * - Aut(.4). For a,ß £ 0+(A) we define their common refinement a V ß G 0+ as just above.

(i) N(a) = min{card a'|a' C a , a' G C?+(<4)} G IN denotes the cardinality of a minimal "operator subcover" a' of a G 0+(A).

For 7 i , . . . ,7n € Vi (A) with % '• &k —* A (B/, hence a finite-dimensional Abelian C-algebra; k = 1 , . . . ,n), their ent-opy is defined by

#(7i,...,7«)= , " " « , logJV( 7 l(A)V...V7 n(^)),

where the maximum is taken (first as supremum) over the set of n-tuples with el­ements ßk(^k) G Oi(Bk) determined by the resp. map 7* (VA: = l , . . . , n ; i.e. for 7* = 7/ also ßk(lk) - ßi{lt)), a n t l the maximum rather obviously exists (cf. the discussion in (2.2), in paiticular before (2.3), and remember Prop.(2.2,b); see also set. 3 below).

(ii) The entropy of 6 w.r.t. 7 £ V\{A) is defined as Ä(0,7) = limn-.oo -H(j,9 o 7 , . . . , 0 n - 1 o 7), where again the limit obviously exists (cf. (2.2,b k c)).

(iii) The "topological" entropy of 6 is h(6) = s u p 7 € ? j ^ R ( ö , 7 ) .

Summarizing our findings thus far, we have the

Theorem (2.7):(cf. [5]) Let (X,T) be a topological djnamicai system with "associ­ated" C*-dynamical system A = C{X) and 0j € * - Aut(^4), then the definitions (2.1) and (2.6) coincide: h{T) = h(6T)

Corollary (2.8): Let A be a (separable) Abelian C*-algebra with 6 G * — Aut(.4), and denote by SA the set of ^-invariant states w = w o S £ 5^, then cbviously by (2.3,iv), (2,4,ii) and (2.7): h(6) = sup^» hu(0), with the CNT entropy on the right hand side.

In view of the definition (2.6,iii) of h(8) - 6np^Vl^h(0,'y), resp. the result hu(0) — suPiti>i{A) ^w( f 7) °f (2.4,i) for Abelian A, it is tempting to think (cf. [5j) of a direct "purely algebraic" proof of (2.8), simply by showing that h(9,i) — 8up w e 5# hu{9,i), or at least h{0,l) > hu(e,-r) \Ju> e S°A. But this can certainly not be true for all 7 6 V:(A), if

10

ever, for the following reason: As shown in [1], hu(8,')) is norm-continuous in 7 even for non-Abelian A and 7 G CV\(A) with our notation from set. (2.2), what then implies the "non-Abelian KS theorem" for nuclear A (recently generalized for quasi-loc J C*-algebras by Park and Shin [2]). Just to remind the reader and again fix the notation, we state

Prop' ition (2.9)[l]: Let A 3 il be a (general) unital C'-algebra with B G * -Aut(.4), u> G S^.

(i) V 7 E CVi{A), 7 : B -> A, and Ve > 0 3*(dimB,e) > 0 : V7' 6 CV^A), 7 ' : B -> A with (I7 - 7'|| < 5, the CNT entropies M ö > 7 ) - M*>7')l < e -

(ii) If A is nuclear [22], there exists a net of completely positive unital maps av : A —» Bv, r„ : Bv —* A with finite-dimensional C*-algebras (or full matrix algebras) B„ (f G N, a directed set), 6uch that linvetf ||r„ 0 (r„(Ä) — A\\ — 0, VA G A. Then as a corollary of (i), quite obviously the full CNT entropy ku(8) = l i m , ^ hw(0, r„), although {TV\V G JV} C CVI{A). As is well known (and rather easily seen), there exists an "approximating" sequence <rn : A —* Bn, Tn : Bn -* A (n G IN) iff A is separable; and in this case hu[6) = hm n _ 0 O hv(0,Tn).

But for our "topological" entropy £(#,7) of (2.6,ii) with A Abelian and 7 G V\{A) as before, we have unfortunately the following continuity-"no go" result:

Proposition (2.10): Let A 3 11 be an Abeliar C*-algebra with 6 G * - Aut(.A).

(i) For 7 G Vi(A), 7 : ß -v A{B finite-dimensional Abelian), and W > 0, 3 f G Vi{A)t

i :B-^ A such that | [ 7 - 7 ' | | < 6, but h(9,~f') = 0.

(ii) For a„y 7' G 7>i(.A) with h{6,f') = 0 as above, Sefr') > 0 such that h[9,f") = 0 V7" G Pi (A), i'-.B^A with ||7' - 7"|| < t.

We leave the proof to the reader as an easy but tedious exercise (going through the above definitions (2.6,i & ii) and using the correspondence Pi(A) 3 7 »-* a, G (9*(.4) as discussed in (2.2) for computing # ( 7 1 , . . . , 7„) = log # ( 0 4 , , . . . , Oy.) = 0 whenever 3A* G a* : At, > 0 Vfc = 1 , . . . ,n). - Thus the 7' : B —» A with ft(0,7') = 0 are an open dense subset of {7 G V\(A)\i : B —* A} 3 7' in norm; and hence, also in view of the proof [1] of the "non-Abelian KS theorem" (2.9,ii) for hu($), it is clear that the mere existence of an approximating sequence of maps r„ : An —» A, <rn : A -+ An with ||T„ O an(A) — A\\ -» 0 (VA G .4) is not at all sufficient for proving that K($, rn) —» h{6) by the same (norm-approximation) argument. Surprisingly, however, it turns out that in this Abelian case A = C(X) with (X,d) metric (as A is necessarily separable by (2.9,ii)), any "refining" sequence of maps {T„ G "Pi{A)}n^f w i t b H^>r«) ~* W) n & 8 *° ^ e neceoaarily of this form (together with the appropriate maps (rn): Note first that we can define a partial pre-order y on 0+(A) by a y ß for a,ß G 0+, ifVAi G a 3Bi{i) G ß and A< G (0,1]: AiAi < % , ; such that obviously a>-j3 <£=>• W(a) >• £/(/?) G 0{X) as defined in (2.1), with our notation from set. (2.2).

i l

Lemma (2.11): Let (X,d) be a metric compact space and A = C(X). Choose a refiniug sequence {Un £ 0{X)\d(Un) —» 0} n e lN of open covers as in (2.2,g) and corre­sponding partitions of unity {ßn £ ^i+(-^)}„6IN 8 U c n t n a t ^ » ~ ^0»> ^ ^ o u r notation fro* set. (2.2). For ßn = {£' € A+\in = l,...,Nn : E£=i / £ ° - &} define maps rn:B„~ © ^ ( C ) * -» 4 by r„(e£°) = f^] with the minimal projections e£ } £ ß n , such that for r„ G 'Pi(-A) again wrh our notation from set. (2.2): r n = 7^. resp. ßn = a,-.. Then there exists a sequence {<rn : A —» #n} n €]N °f positive unital maps <rn with the approxi­mation property (2.9), i.e. r n 0 an —»Id^ in poin'wise norm; and for any Ö £ * — Aut(.4) we have h{9) = lira n_ 0 O 7i(0, r n).

Proof: Choose points x£* £ X such that / ^ ( « j ? ) > *•' V*« = J» • • •»N»> n € IN, and define <r„(il) = £ * • , «i?'^(«i?). VA € A n £7S. Then obviously rn o<?n(A) = ^ ft? • ^ ( x u ) ~* ^ pointwise on X (i.e. Tn o <rn(/l)|, -» A\g, VA € A x £ X) as U„6JNÄ» separates the points of X (and actually ßn separates "arbitrarily wt••'" fcr n —* 00, because obviously /3 n -< /?„+!: Vn 6 IN, as defined before), which implies by a "Scone-Weierstrass" type argument (actually by Dini's theorem!) that | |T B o <rn(j4) — A\\ —• 0 VA £ A. - In addition, it is evident by (2.2,g) an<? definition (2.6,ii) of ft(0r,T„) = h(T,U(ßn)) for any homeomorphism T : X —» X, that Ä(#r) = linV-.<» Ä(^r,Tn).

This detailed simple proof is an exception of our ruk to skip all of tbem in this paper; but actually (2,11) paves the way out of the "no-go" impasse (2.10) via the following more gene:ai sublemma (with essentially the came proof):

Sublemma (2.11,1): Let .A 3 11 be a non-separable Abelian C*-algebra, and note that Of (A) with the partial pre-order >~ as introduced before is a (pre-)directed set with the common refinement (as in (2.6)) a V j 3 x a,ß £ Ox {A). Considering 'Px(A) 3 7 a

as a net indexed by a £ 0*{A) with our notation from set. (2.2), there exists a net {<ra\a £ Ox (A)} of positive unital r aps <ra from A into the resp. preimage-algebra of 7„ ;

such that l i m a £ 0 + ^ | |7 a o cra(A) — A\\ ~0, VA € A; and for any 6 £ * — Aut(«4) we have h{8) = l i m a 6 0 + ( j ) ft(ö,7a) (as a net limit).

Definition (2.12): Let A 3 ll be an Abelian C'-algebra and 6 £ * - Aut(.A). For an approximating sequence or net r = {T„ £ T\(A)\v £ N} such that r„ o <rv —• Id.4 in pointwise norm (with positive unital maps <r„ as usual), we define the "T-topologJcal" entropy hr{6) — limsup„ c^Ä(0,r u). ~ Note that obviously by definition (2.6,iii) we have generally hr{6) < h(0)) and in particular hr(6) — h(&) if r forms also a subnet of Vx(A) 3 ")a

w.r.t. the net structure from the directed index set Ox(A) 3 a as in (2.11 & ,1) above (i.e. if the directed set N is order isomorphic to a cofinal suu-poset of Ox{A)).

12

3 Quantum Topological Entropy for C*-dynamical Systems

3.1 The general theory for nuclear C*-algebras Definition (3.1): Let A 3 ll be a (first general) non-Abelian C'-algebra. Using the general definition (2.5), we introduce the following notations: 0(A) denotes the set of all finite "operator covers" or = {Ai G «4| Y* A*At > 0}, 02(A) is the set of (not necessarily positive) "operator partitions of unity" (OPUs) ß = {Bj € >l |£j B-Bj = 1}, and Ot(A) denotes the set of positive "square root" OPUs ß = {Bj G A+\ZjB* = 11}. Note that obviously 0$(A) C 07(A) C 0{A) resp. Ot(A) C 0+(A) C 0(A) with 0+ as in (2.iii), and "symbolically" for ß G Op. A) element-wise (ß3) G 0{ [A), resp. for a G Of (A) in turn (a*) G Oa

+(.4). For a,# G 0(A), a = {Ai} and ß = {Bj}, we can well define their "ordered refinement" a\}ß - {BjA} G 0(A), as we have for £,• B)Bj > c • 11 (for some e > 0, using (2.5,iii)) that £ y ^ B / i ? ^ > c Zi AIM > 0.

Obviously, V : O x O —> O is associative (because A is it), but not commutative for nore-Abelian A, and Vs : C?a x C?2 -» C?2 (but \7 : 0 + X 0+ -» C?, with image more than 0 + ! ) . On the other hand, for Abelian A and a,ß G C?+(*4) of course aVV? = aV0with the "classical" refinement as used in Def. (2.6).

Now we can first generalize for non-Abelian A only the first part of Def. (2.6,i), and also tie partial pie-order y on 0+(A) as in (2.11 & ,1), but necessarily also extended to all of 0(A): Definition (3.2): Let again A 3 $ be a general (7*~algebra.

(i) For a G 0(A), N(a) = min{carda'|a' C a,a' e 0(A)} G IN denotes the cardinality of a minimal "operator subcover" a' of a.

(ii) We define a partial pre-order y on 0(A) by a y ß for a, ß G 0 , if V>U € a 3 B i W € /3 and A,- G (0,1] : A,v4,-4 < % % , .

Note that these are really generalizations of the "classical" definitions, although formally different for Abelian A; remember that generally a € 0+(A) <=> •*' € 0+(A) (element-wise!), resp. for A,B € A+ : XA < B = » A^A^ < B> (being equivalent for Abelian A).

Proposition (3.3): For these preceding definitions, we have the following rather ob­vious properties: Va,ß,f £ O(A),

ad (ii) (a) a -< a\?ß, but not necessarily ß -< aüß (i.e. the "directed join" v does not render (O, y) a directed set any more).

(b) ß y 7 =*• a1?/? >• a\?7, but not necessarily also ß$a >- 7v*a!

13

ad (i) (c) N(a) < N(a'Jß) < N(a) • N(ß), but not necessarily also N(ß) < N(*\Jß), as follows from (a) (this generalizes (2.2,a & b) for N partly).

(d) N(8(a)) = N(a) VA G * - Aut(.4) (generalizes (2.2,c) for JV).

(e) N(a$aV*... tfa) = N(a) Va € 0+(A) and for arbitrary ^-"powers" of o (generalizes N(U) = N{U V U) in (2.1,i), ab follows from U V U >- U and UvU <UiotUe 0{X) by (2.2,a)), but no*necessarily also JV(aVa) < N{a) for a G O(A) \ 0+{A).

(f) a •< ß ==> N(a) < N(ß) (generalizes (2.2,a) for N).

Lemma (3.4): Let B be a,/imte-dimensional C*-algebra, i.e. »-isomorphic to ($)£_! Mjk = B with full rffc x (ifc-matrix algebras Mdk, *nd define D(B) to be the dimension of a maximal Abelian subalgebra C C B: D{B) - ££=!<&• Then for any a 6 0{B) there exists a ß G 0+{B), ß = {e{ G B +i* = 1 , . . . , D{B) : Y,fLi «• > 0} with mtmmal projectors e, G 5, such that ß X a as defined in (3.2,ii).

Corollary (3.5): Let B again be a finite-dimensional C*-algebra with D(B) defined as in (3.4), then Va G 0[B) : N(a) < D(B), as follows from (3.3,f) & (3.4). Obvi­ously, there exists a ß G Of(B) f) <3j"(B) with again minimal projectors e,-, /# — {e,- G B + | i = 1 , . . . ,D(ß) : YlfLi ei = 1} , sucn that N{ß) = Z)(B); %nd together we have that max a e 0 (B) N(a) = D(ß) - maxße0+{ß)N(ß).

Definition (3.6): Let 7 : B —• A be a positive linear map between C'-algebras B and .4, then we delme the "2-application" 7[#]j : B —• v4+ (as a map, also denoted by "absolute value" |7J) by 7[B] 3 — [^(B'B)]*. Obviously, |-y| : B —* A* is still positive but not linear any more (only still positive homogeneous: 7[AB]j = A • 7[B]j, VA > 0, BE B); and if 7 is a *-homomorphism, than j-y|B+ = 7B+ (Less trivially, this latter statement has a partial converse: If 7 is even completely positive, then 7[B]j = 7(B) for a particdar B G B implies that B is in the "left multiplicative domain" of 7, i.e. 7(B) • 7(C) = 7(5C) VC7 G 5; see [23]). Note that generally j[B}3 = 7(B) at least whenever B = B ' G B + ani 7(B) = 7(B) 2 G X + are projections.

Lemma (3.7):

(i) Let 7 : B —* .Ai and 0 : Ai —» A% be positive linear maps between C'-algebras B,AuA2t then 0[7(B] 2] 3 = (6 o i)\B]i VB £ B (or alternatively, symbolically: \8\ 0 | 7 | = | ö o 7 | ) .

(ii) If 8 is even a "»-homomorphism, then also 0(7[B]2) = {ß°l)[B\i VB G B (or (9o j-y[ = |# 0 7I; note that also for 7 a *-homomorphism and 6 general: \& oj\— \6\ o 7).

Definition (3,8): Let 75, . . . , 7 n be positive unital (linear) maps 7* : B), —• A from finite-dimensional C'-ahjebras Bu into the C*- plgebra A '3 B, then their entropy is defined by

Ä(7i , . . . ,7n) = / , . max log JV(7l [aj], V*.. . ^ [ a , ] , ) ,

14

where the maximum is taken (first as supremum) over the set of n-tuples with elements <*k G OfiBk) determined by the respective map 7* (i.e. for 7* = 7* also 04(74) = 0/(7/)), and the maximum exists because of (3.11,ii) and (3.10) below; and where of course 7fc[afc]3 G Oj(A) is understood element-wise. For Abelian A and 71,...,f n G ^(-^Ji i-e- * l s o Bk Abelian VA, this definition rather obviously coincides again with (2.6,i); and on the other hand for non-Abelian A but the inclusion *-homomorphism 74 = isk • Bu -* A of finite-dimensional •-suSalgebras Bu C A, we see from (3.6) that »ajafcfc = t0k(a*) Vatfc € O^iBh), and thus we can denote

H(Blt...,Bn) = H(xBl,...,tBm)= m « l o g % \ / . J a „ ) . {(a, (B t ).•».«».(*.))]

Lemma (3.9): Let 7 : B —» A be a positive unital (linear) map from a finite-dimensional C*-algebra B into the (7*-algebra A 3 11, then

(i) Va E C?+(B) " '(7[a]j) < N(a), where we should emphasize at this occasion that we understand it as defined by the arguments of N (or H) to which algebra they refer (here A on l.h.s., B on r.h.s.), and

(ii) if even 7(a) E 0+{A), then also JV(7[a]a) < #(7(0;)). In particular, for a positive operator cover ß of B, ß = \j>i = £ G B+\HiPi > 0} with projections pi, always

7(/3) € 0+(A) and even N{i\ß]>) = *W)Y

Sketch of Proof: The inequalities (i) resp. (ii) follow from the implications

(i) E.-4- > 0 = > E n ( A ? ) > 0 (odious by (2.5,iii)), resp.

(») Hi 7 ( A ) ' > 0 ==> S»7(^i) > 0 (for which Schwarz positivity of 7 is not even necessary, as i(A2) > 7(A) 2 VA = A" by Kadison's ancient result [24]);

which obviously both cannot be reversed in general. But if in (ii) A? = A,- V», we have again our well-known equivalence (cf. after (2.5)).

Corollary (3.10): Let 7 : B - • A be as in (3.9), then £ (7 ) < logD(B) = H(B), as follows from (3.9,i),(3.5) and the definitions. Thus # ( 7 ) = max a e 0 +/^j log JV(7(a]j) is really a maximum; but even # (7 ) = max o € o J ( B ) log iV^KatJj) does not give more, because for a = {Ai G B\Zi A\Ai = H} G 03(B) we have that (symbolically) \a\ = {(A;A,-)* € B+} G Oi(B), and obviously by Def. (3.G): 7[a] a = 7[|a|]j. By the same argument, we can take a„(7„) € Oi(Bn) in Def. (3.8), but not for k = 1 , . . . ,(n — 1) if we want to keep to the equivalent expression for «ufcalgebras Bk C A in (3.8).

Proposition (3.11): Let A 9 11 be a (first general) C*-algebra, then the entropy functional (3.8) has the following general properties:

(i) For ft* : 0* ~» AH and 7* : Ah —* A positive unital (linear) maps with finite-dimensional C'-aigcbras Ak,Bk (VJfe = l , . . . , n ) : #(71 o $x,... , 7 , o 8n) < H(*tu • • • i7n), which follows from 6k[Ot{Bh)]i C Ol(Ak) using (3.7,i) (and general­izes (2.2,a) for H).

15

(ii) For 7k : Ah -» A as in (i),

#(7n.• • -,lu) < #(7i> • • • ,7«) < ^(7i •• • ,7m) + #(7m+i,•••,7»)

V£,m < 1 and 1 < »1 < »2 < . . . < it < n (where litterally the order of (1 , . . . ,n) has to be respected), which follows from (3.3,c) and generalizes (2.2,a & b) for H.

(iii) For 71 : A\ ~» A as before, -#(71,... ,71) = #(71) for arbitrarily many repetitions of only one single 71 (as follows from (3.3,e) because of the definition (3.8) for H, and generalizes H(UVU) = H(U) in (2.1,i) as noted in (3.3,e)).

(iv) For 8 e * - Aut(4) and 7* : .4* -• A as before, Ä(öo7 l t . . . ,6cy n ) = #(71, . . . ,7„), which follows from (3.3,d) using (3.7,ii) (and generalizes (-.2,c) for If).

(v) If there exists a finite-dimensional C*-subalgebra B C A such that Im(7*) C B VJfc = l , . . . ,n , then #(7i , . . . ,7„) < log£>(5) = H(B) with the "Abelian di­mension" D[B) as introduced in (3.4); which follows again from (3.4) applied to TiMaN?... vSnKJi 6 C,(£) in Def. (3.8), and (3.3,f).

(vi) More generally (but less applicable): If there exists a positive unital injective (linear) map 7 : B —» A with a finite-dimensional C*-algebra B such that for 74 : A\ —» .4 as before, the set

{7i(Bi)i-7>(Ba)^....7„(5n)-..-7a(B,)*-7i(B,)*)|VB*€>C, |j£*|| < 1} C 7 ( ß + ) ,

then H(-yi,... ,7«) < #(7) (< #(B) by (3.10) above); which follows directly from Def. (3.8), and is a different generalization of (2.2,a), than that obtained by combining (i) k (ii).

Corollary (3.12): ad (i) If 7k : A* —» A and Ak C Bk with conditional expectations Ek'-Bi,-* Ah (such that the inclusions uk : Ah~*Bh h&v* &k *» W* o-inverse: Eu OM» = Id>ik), then #(71 0 iJj,. . . ,7„ 0 £7n) = #(71, • • • i7n)- Thus we may again assume (as in [1, (111.5,1)]) for the computation of H{fi,. • • ,7 n) that the 7* are maps from resp. full matrix algebras M4 into A; or 74 € K-i(A) (with our notation at the beginning of set. (2.2)) if we restrict to completely positive unital maps 7* G CP\{A).

Definition (3.13): We de^ne n positive unital (linear) maps 7* : Bk —• A (k — l , . . . ,n) to be independently covering (for the C"-algebra A 3 fl), if for Bk 6 ßjf but 7fc(#*) not invertible {ik{Bk) / 0, V*) 3u> 6 £4 : wo 7,(5*) = 0 V* = 1,,. . ,n. By (2.5,i), this condition is equivalent to the following reformulations:

V{B*GBA

+ |7*(5 4)^0 V * = l , . . . , » } = > E 7 * W > 0

<*: V{9 f c€Ö f c

+ |* = l , . . . , n : £ > ( £ f c ) > 0 } = * 3 * : 7 * W > 0 .

16

«

We say that two maps 71 and 7a as before are "commuting" (denoted symbolically [71,7a] = 0) if (7i(B l ),7 3 (B,)] = 0 VB* G Bk (k = 1,2).

Proposition (3.14): If n positive unital (linear) maps 7* : B* —» .4 from finite-dimensional C-algebras B* into A3 11 (k = 1 , . . . , 1*) are independently covering and pairwise commuting, [74,7^ = 0 Vi ^ ; G { 1 , . . . , n } , then # ( 7 1 , . . . ,7«) = 7 7 = 1 tffrfc); i.e. the optimal upper bound of (3.11) is attained. In particular., this is the case if A = A\ ® . . . ® A , with nuclear unital C*-algebras ,4* 3 IL, and 74 : B* -» Hi ® . . . ® A ® • • • <8 Bn (V* = 1,. . ^n) . Sketch of Proof: Let ah G 0}(Bi.) be such that #(7*) = log JV(7*[afc]a) Vfc = 1 , . . . ,n, by (3.10). Then, for the corresponding ^-minimal" subcovers 7k[a*]a 6 0+(Ä) (a'h C Qk with carda k = exp/f(7k)), clearly 71 [a'^^..-^ikloQi is always a subcover of the "unprimed" n-fold ^-refinement of all 7k[atfc]a; but it is even minimal by the assumption (3.13) for {7fc|fc = l , . . . , n } , as otherwise at least one a'k could not be 74-minimal (an elementary C"-exercise for the reader). Thus by Def. (3.8), H(f , . . . , 7 „ ) > £*_! H(T*Y,

and (3.11,ii) gives the converse. The rest is obvious.

Lemma (3.15): Let A\,Aj C A be unital C*-5u6algebras of A 3 11 which are com­muting ([-4i, A}] = 0), then *.*, and tAj (the «-homomorphic inclusions) are independently covering iff A\ and Ai are C - independent in the sense of [25]; with proof of the nontrivial implication similar to that of (3.2,(1)=»(2)) in [25].

Remark (3.16): For an n-tuple of positive unital (linear) maps 74 : B* —» A (k = 1 , . . . ,n), an "Abelian model" is determined by an n-tuple (ait..., a„) of a* G O^ißu) *• before, and is defined as follows: For a f c = {B|*> G B{ \ih = 1,. • •, Nh : E R i ^ i ? ) 1 = M> choose the Abelian C*-algebra C = ®fc = i(©i t l i(C). k ), with corresponding minimal projectors e^, in) G C. Using the notations /„ = (*i,... , t n ) G IN" and Ein — 7„[.9,-Ja • . . . • 7I[BJ,]J, define a positive unital linear map (for fixed state u> G S*) by

u and a state /z.„ G Sc by /'-«(e/,,) = u>(E}nEin). Then, for fixed ( 0 1 , . . . ,a„), u •-• Pu is a map from the state space S* of .4 into the set V\ (A, C) of positive unital maps from A to C; and w »—> /iw is a weak-* continuous affine map M : Sj, —» Sc, whose adjoint M* : C —» A is given by the positive unital (linear) map Af*(ej„) = EjnEjK. Now, it is not true in general that /tw o Pu = u; Vw G 0.4 (only if u; is "tTacial" at least for Ein, Vi n); but Cj. : UJ •-» A'w(cO = « " M*(eiu) is a weak-* continuous (affine) function on $.4, C/. G ^(S^), and for B M = {Cin\In} G C?j+(C(5>,)) we have % ! , . . . , 7 n ) = m a x { J , = v ( Q l ( . . . r a . ) } log #(/?*), by definitions (3.8), (3.3,i) resp. (2.6,i), and (2.5,i). Note, finally, that for the pure states e}m G Sc C C*, defined by e} (ej,) = 5/.J., we get the "reduced" states e'Jn 0 PU(A) = w(#£ • A • £j . ) • nu{ej„)-\ V4 G A

Definition (3.17): For a positive unital map of finite rank 7 : B —> .4 into a C-algebra A 3 H, and for 0 G * — Aut(.4), the "entropy" of 0 w.r.t 7 is defined by h(0,j) =

17

lim,,-,«, ^H(f,0 o 7 , . . . , 0" ' o 7), where the limit existß by (3.11,ii & iv), and actually is the infimwn.

Proposition (3.18): The "entropy" (3.17) has the following easily provable properties (as corollaries of (3.11)):

(0 ft(*>7) < #(7)» generalizes (2.2,e).

(ii) For 73 : B 2 —» A and 71 : Bi -* S 2 with finite-dimensional Bu (k = 1,2): Ä(ö,72 o 7 l ) < a(ö,7a); generalizes (2.2,d).

(iii) h(a 0 6 o cTx,a o 7) = £(£,7) Vc fc * - Aut(.A).

(iv) ^(^,7) < \n\ -%{Qti) Vn e 2Z (generalizes (2.2,f), if n = -1); but fc(0,7) < K{tr,7) is not provable here.

(v) For nuclear A = A\ ® -4j, 71 : B\ -> .4i ® flj and 72 : B 2 -* A' ® .Aj, and 6 = 0i ® 63

with 0* 6 * - Aut(.4fc): ^( ,71 ® 72) > £(0i,7i) + ^7,7a), M follows from Lemma (3.19) below; but the converse inequality is unfortunately improvable thus far (cf. [26, Lemma(3.4)] for the CNT entropy).

(vi) If there exists a sequence of positive unit J injective maps 7 n : Bn -» A of finite rank, such that for given 7 : B —» A:

{7(A)* .«07(A)* . . . . • ^ - 1 o 7 ( B B ) . . . . . 7 l ( ^ | V B * 6 S + , | | 2 y < l } C 7 n ( B + ) ,

then ft(*,7) < liminf n e I N i % B ) .

Lemma (3.19): Actually, we need for the proof of (v) the following corollary of the- proof of (3.14): For nuclear A = AiQAt, let ßkZO*{At,), and <*i =0i®Hj,a 3 = lli®£ 3 6 Ot{A), then ^V(ai\7a2) = JV(ati) • N(a2) (again by much the classical argument [13], using (3.13) for Ak).

Definition (3.20): Let A3 the a. nuclear C"-algebra and choose an approximating net of completely positive unital maps r = {T„ € CV\(A)\v e N), TV : A„ -* A (with corresponding av : A -* A*) as defined in (2.9,ii). We define the "r-topological" entropy of 0 6 * - Aut(.4) by repeating (2.12): hr{0) = ]im»npveNh(ß,Tv). For <r € » — Aut(.A), we use the notation <r(r) = {aOT„ 6 CV\{A)\v £ JV}, which is again an approximating net (with corresponding av 0 <r_1 : A —> AJ).

Corollary (3.21): The T-topological entropy has the following easily provable prop­erties (by (3.18) before):

(i) K(r){<r o0o a~x) = hr(e), V<r e * - Aut(.4); generalizes (2.2,h).

(ii) AT(0") < |n| • hJB), Vn € IN (unfortunately, the converse inequality is unprovable thus far; cf. [27, Corollary(2.9)] for tue CNT entropy); generalizes (2.3,i) partly.

18

(iii) For A = A ® At with nuclear A and 8k 6 * - Aut(Ak), & = 0i ® 0j e * - Aut(-4); and for approximating nets T* = \rVk : A » —* A\v± G JVfc} of -4*, also r = {T^ ® iVj : A*, ® A , -» >tf(^!, i^) 6 M x JVj} is an approximating net of A. Then by (3.18,v): hr{9) >hn{0l) + ^(fij); generalizes (2.3,ii) partly.

Problem (3.22): Characterize those approximating nets r of A as in (3.20), for which the (first "formal") supremum h(ß) ~ s u p ^ ^ ^ ^ o ^ ) = »up^/dC^M^) (& (3.12)) is attained: K(0) = W V* € * - Aut(>4); cf. (2.12) for Abelian A. Is there any relation still to the partial pre-order (3.2,ii) for non-Abelian At Note that it is possibly not true in general that h0[r){ß) = ^(0), contrasting (3.214) above.

3.2 AF algebra theory and first applications

Let A = Uf-elN A i be a (separable) unital AF algebra, i.e. An C A + i (Vn € IN) with unital inclusions of finite-dimeneional C*-algebras An, and A = A » is the norm comple­tion of the inductive limit A» = U n € lN A», which is a locally semisimple »-algebra (with pre-C7*-norm). A is nuclear with approximating maps (2.9,ii) given by rn — xj^ : An —» A (•-hornomorphic inclusions) and with any chosen ncrm-one projections (conditional ex­pectations) <rn : A —• An', and we can reformulate our general definition (3.20) of "quantum topological" entropy in this AF case a. follows:

Definition (£.23): Let A — A » be a unital AF algebra ae above, then the "A»~ topological" entropy of 0 € * - Aut(.4) is defined by: ^A^Jfi) — s u p ^ ^ K(0,B), where the supremum is taken over all finite-dimensional C'-subalgebras B C A»-This is really a special c»se of our general definition (3.20) because of (i) of the following concretization of (3.18):

Corollary (3.24): The entropy h(6,B) = h(0yiB) with (3.17) for a finite-dimensional suialgebra B Q A has the following properties:

(i) For BrCBtC A Ä ( M i ) < h{e,Bt).

(ii) h(0n,B)<\n\-h(O,B) V n € 2 .

(iii) If there exists an increasing sequence Bn Q Bn+\ of finite-dimensional subalgebras Bn C A such that B,6(B),...,dn'l(B) C Bn (as not necessarily commuting suba'ge-bras) Vn € IN, then h(8,B) < l iroinf n e I N \H{Bn).

Subcorcllary (3.25): For an AF algebra A = ~AZ as in Def. (3.23), hA<K(8) = lim,,...» h(0, An); follows from (i).

Again generally, we can concretize also (3.11) by Corollary (3.26): The entropy H(Bi,...,Bn) with (3.8) for finite-dimensional juialgebras 5* C A (k = 1 , . . . ,n) has the following general properties (from (3.11)):

(i) For BkcAkCA (V*), H{BU...,Bn)< H(AU..., A ) .

19

(n)H(Bix,...,Bit)<H(B1,...,Bn)<H(B1,...ßm) + H(Bm+1,...,Bn), V£,m < n and 1 < t» < t» < . . . < t/ < n.

(iii) H{B,..., B) =• H(B) with arbitrarily many repetitions of only one single B.

(iv) For 0 € * - Aut(.A), tf(0(&), • • •, *(ß„)) = #(*i , - - -,B n).

(v) If Bi,..., Bn are pairwise commuting ([Bit Bj] = 0 V i / j) and independently cover­ing in the sense of (3.14),(3.15), then H{Blt...,Bn) = E*=i H{Bh).

Note that in (v) the condition (3.13) for {Bi,\k ~ 1, . . . ,n} is seemingly stronger than only pairwise C*-independcnce of (Bi,Bj) Vt ^ j in the sense of [25], as it means that also the "multiple correlations" between Bi,...,Bn vanish in the sense of [25, (3.2,4^].

Proposition (3.27): In addition, the entropy (3.8) has the following special properties refering to resp. only one (particular) finite-dimensional autalgebra of A:

(i)FotBl,...,BncA*s in (3.26), H{BX,...,Bn-Uß„,...,Bn) = H{BU...,£?„) with arbitrarily many repetitions of the last entry B„ in H (but not provable for the other entries Bu..., B„_i analogously!), generalizes (3.26,iii) a tiny bit further.

(ii) For Bi,B3 C A, we define E , c ß } ^ VBi G Bx 3B3 £ B3: \\Bi - B3\\ < 6\\lii\\.

Then if Bx C B2 C A with 6 < 10~\ always H(Bj) < H(B3) (but not provable for more than one argument analogously!).

Corollary (3.28): Defining in (ii): \\Bt - £,|| = mi{6 > 0|Bi C B2,B3 C Bi}, we have obviously: 1(0! - B3\\ = 6 with 6 < 10~4 => H{BX) = H(B3). Although this looks like a "first step" towards a norm-continuity of h($, S) analogous to (2.9,i), it reems to be already the "last" step (into an impasse); not to speak about the generalization for £(#,7) (for 7 g CVi{A)), in view of our "no-go" Proposition (2.10) already for Abelian A. Note that the premise of (3.28) is actually nonsense for Abelian A D Bi,B3, a« then rather obviously \\B\ — B3\\ = 1 VZ?j ^ B3; which could be viewed as rendering any non-commutative analogue of (2.9,i) "unnecessary" even for h(8, B) (cf. however [28, Prop.l] and its discussion in (3.35,2) below).

Proof of (3.27):

(i) By Def. (3.8), we have to consider for the l.h.». Nfatiarf... vfahVofev1... tfo«) with ah e Oi{Bk). Now ajaj... Vo„ € 02(Bn) only, but as in (3.10) ßn = KtfctnV"... \7on| <= Of(Bn), and obviously iV(aj\7.. . ^ O n . , ^ ) is the same as the above expression. Thus H{BXl... ,Bn,Bn,... ,Bn) < H(Bll...,Bn-i,Bn), and (3.26,ii) gives the converse.

(ii) By Thm. (5.3) of [29], there exists a unitary U G A (such that \\t - U\\ < \2Qy/6) with UBtU' C B3. Thus by (3.26,i & iv): H{B3) > H{UBiU*) = #(£?,).

20

Now let again .4. = U n e INA» be AF, then .4 is uniquely determined up to *-isomorphism by the norm-dense, locally semisimple *-subalgebra Aoo — U„eIN'^n> o r

equivalently by a corresponding Bratteli diagram [30], which is in turn equivalent to the sequence of inclusion matrices {[A,t —» Ai+i]|Vn £ IN} (cf. [31] for the notation and ter­minology in the following).

Definition (3.20): We say with Choda [31] that the sequence { A » } B 6 J N U »hove »• periodic with period p, if 3no € IN 6uch that Vn > n^

(i) [Aj -* Ai+i] = [Ai+P - • ^,+p+i]

(ii) The (hence necessarily squaie-) matrix Tj = [Aj —* Aj+P] is primitive (•<=>• 3£ 6 IN : {Tj)n, > 0 Vt, k •$=> by the inclusion Aj C Aj+tp each simple direct summand of Aj is "contained" in every simple direct summand of Aj+tp), which implies that Tj has a unique Perron-Frobenius eigenvalue ßj > 0 (which, together with (i), is actually independent of j > no!).

Lemma (3.30): For a periodic sequence {A»} n 6 ]N as above with period p and Perron-Frobenius eigenvalue ß of Tj = [Aj —* <4j+F], we denote the resp. dimension vector of A , = ©#0=i Mdh(u) by o„ = (d f c ( n ) ) , and its 1-norm by \3»\i = £ i $ „ ) = 1 <**(»)• Then Vj > no : |a y + p |x = ß • \Sj\n and lim,,.,« i # ( A , ) = J log/?. Proof: As indicated in [31], for the trace vectors t n = (r(llfc(n)) • tfo^ = **(„)) cf An, corresponding to a chosen trace r € SA on A, we have tj = ß'tj+p (Vj > no). On the other hand, we know that an-tn = £*{„)<**{«) •**(„) = 1; and thus Ej-tj = Sj-ß-tj+p - aj+f>tj+p,

or explicitly £*(*)=*(;+,) (4$) ' ß ~ tffi*) • tf$ = 0, Vr € £4 tracial. Choosing r as determined by the normalized matrix traces on Af^j, D An with f*(„) = |<u: li"

1 (VÄ(n), n € IN), this implies that |OJ|I • ß — |OJ + P | I .

Now, by Def. (3.8) and (3.5), H{Aj±np) = l ogD(A j + n p ) s log |o i + n „| i = n-log^+log 1 ^ , and limn-,«, ^H(An) = lim«-«, ^H(Aj+np) = J log/?.

Definition (3.31): For an AF algebra A = U,,eIN A» = «^«« ^ e * ~ Aut(.4) is said to be "^-shifty" if the following conditions are fulfilled (cf. [31]):

(i) Vj.m € IN 3a € * - Aut(.4) such that (r{Aj+m) D Aj,6(Aj),.. .,V*-\Aj) M (not necessarily commuting) subalgebras, and

(ii) there is a sequence {rij € iN|ny+i > « , } € [ N 8 U C f l t n a t •4i>^ n'(^i)>-••»^* n yMi) we pairwise commuting and independently covering as in (3.26,v), Vfc 6 IN; and such that lim,-.«, ^j1 — 0.

Theorem (3.32): Let A = U^iN A» be an AF algebra with an ".A^-shifty" auto­morphism 0 G * - Aut(,4), then ^ „ ( 0 ) = lim,,-,» £//(.An). Proof:[31] By (3.25), (3.26.Ü k iv) and (3.11,v) (cf. also (3.24,iii)!),

hA„(6) = Urn h{6,Aj) = hm Km l-H{Aj,B{Aj)f. - ^ ( A ) ) <

21

< lim lim -[H(Ah...ien-\Ai))^H{ßn-^\Ai),...,en-\Ai))\< j—«oon—»oo ji

< Hm Um inf -[H{c{An)) + H(Ö n" J + l o *•(>*„•_!))] =

= Um Um inf A[Ä(A) + J ^ y - x ) ] «Km inf ^ ^ . On the other hand, by (3.24,ii) resp. (3.21,ii):

by assumption (ii) and (3.26,v). Hence

which impUes hAv>{0) > Umsupn jjf ~H(An) (again using assumption (ii) on {n^}).

Corollary (3.33): Let A = U„eIN-^n be an AF algebra with a periodic sequence {•^"hielN (with period p and Perron-Frobenius eigenvalue ß of the inclusion matrix [Aj -* Aj+P\ Vj > no), then for any ".4flC-6hiftyB B G * - Aut(4): hAaB(6) = J log/?.

Examples (3.34):

(a) Let A{n) = ® k e B ( A f n ) f c be the n»-UHF algebra and AN(n) = ®*L*(M»)*, A»(n) = UjvglN -4tf(n)l t n e u for 0 n G * - Aut(.4(n)) determined by the unit shift ("n-shift"?) 0n : (Mn)h - (Mn)k+1 (Vk G IN), we get h^n){6n) = logn (Mows from (3.33) with p = 1 and \AN(n) -* Aff+i{n)) = n 6 IN).

(b) Ain[g) = C*(-{e»|i = — N,...,N}) with the relations t\ - e<, e* = 11 and e<e = ej*{-Vf»-m (i ^ j) with g : IN -» {0,1}. Again A„(s) = U ^ E M M * ) and •4« = Ao(5); then for 6g G * — A u t ^ ) determined by 08(ei) = ei + 1 V» G Z5, we get hA.w(0,) = | log 2 if <7(n) = 1 Vn G IN or if 3n 0 : j/(n) = 0 V» > n, (but with g ^ 0, in which case we would have the classical B2-shift" with h{0) = log 2, corresponding to example (a) with 6 restricted to a maximally Abeli&n C'-subalgebr» C({0 ,1} E ) C .4(2)); see [32] for the proof.

(c) .4;v(A) = C'({pi\i - -JV,...,JV*}) with the relations pt = p\ = p], PiPi±tfi = Ap; and \pitPi] = 0 Vi,j : \i ~ j \ > 2; for A G {(4cosJ %)-l\m G IN \ {1,2}}. Again A»(A) = UjvgIN - Jv( ) a n <^ « A = A»(A). Then, for 0* G * — Aut(^) determined by 0*(pi) = Pi+i Vi G K, we get hAaeW{dx) = - f log A. (Note that (c) for A = | is identical with (b) for g(n) = tf„i by Pi = \(ti + 11); and for A ^ | , (c) follows again from (3.33) by the same argument as in [31]).

(d) Let (AA,9A) be the shift 9A on an AF algebra AA associated with a topological Markov chain as treated by Evans [28] with his "AF-imitation" of the topological

22

entropy (via the Connes-St0rmer entropy); but without repeating this lengthy exam­ple here, we just compare with Evans' notation: Our (AA,&A) = (CA,<^o) of Evans, where A is an aperiodic (n x n)-matrix with entries in {0,1}, and our A, = N, (with s c IN) resp. Aoo{A) = U,€IN ^» °f Evans. Then, repeating the proof of the main theorem in [28] with our H instead of Evans' H and using properties (3.26,i & iii) and (3.10) of H (resp. its definition (3.8)), we immediately get the same result: ^t.(ji)(^*) = log A, where A is t ie spectral radius of A (note that we do not need at all even the ingredients of Evans' Prop. 2 for the second part of the rewritten proof, as it amounts in our case to the same estimate M the first part, only from below instead from above).

Concluding Remarks (3.35):

1. Possibly, the main problem (3.22) of our thus "pedestrian" approach could be solved more easily in this special AF case as above, i.e. the problem to characterize [or identify) the norm-generating locally semisimple *-subalgebras A^, of the AF algebra A such that the (first "formal") supremum h(8) = 8up 7 e C j , /^Ä(ß,7) is attained: W ) = W). V0e*-AutM).

2. The advantage of Evans' approach [28] cited in (d) is to "circumvent" this problem (1.) by using the norm continuity (2.9,i) of the Connes-St0rmer entropy (for finite-dimensional subalgebras B C A); but on the other hand, this "AF-imitation" of the topological entropy has the serious drawback of an "AF-limitation" in the follow­ing sense; To extend Evans' definition further for non-AF (nuclear) C'-algebras, one would have to repeat his construction with the CNT entropy as in (2.9,i) for 7 G CVi(A) instead of the Connes-St0rmer entropy; and then it seems to be impos­sible (to show?) that this analogously extended definition of Evans' entropy hs{9) coincides again with the "classical" topological entropy (2.1,iii) for a homeomorphism T : X —• X of a connected compact space X: hefo) — h[T)t (Cf. [5, App.], and contrast with Thm. (2.7) here.)

3. Naively, one could think of a direct "classical" definition of a "quantum" topological entropy hd(6) for a C'-dynaruical system (-4,0), «imply defining hd(6) = A(T#) with the topological entropy (2.1,iii) for the adjoint (affine) weak-* homeomorphism Tt : SA -»• SA (defined by Tt{u) = u> o 6, Vw 6 5>). But by Thm. (2.3,iii) this is no generalization of the topological entropy h(T) ^ h(T\i), and one could only think to regard it as a ncn-Abelian analogue; but also that is definitely ruled out by the following most simple "counter-example": In (3.34,a) already with n = 2, hei{82) - oo; which we leave to the reader as a not quite trivial exercise (see [33]).

4. In all the AF examples (3.34), we always have the desirable generalization of (2.8): HA<J$) — 8UPw6S» ^«(0), ^ h the CNT entropy on the r.h.s.; but without having computed hr(9) for any non--AF nuclear C'-algebra with ("optimal") approximating net r (and also not yet the CNT entropy, for non-trivial w and 0, by the way!), we are

23

far from a conjecture (not to speak about a general proof, cf. already [5] for Abekan A).

Acknowledgements: I want to thank the organizers of the INTSEM seminar in Leipzig (August 1991), and in particular Dr. P. M. Alberti, for providing this stimulating meeting; and also for the patience when awaiting this contribution for the mini-proceedings. Fi­nancial support by "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich" (Proj. P7101-Phy), and a travel grant to Leipzig from the Austrian Ministery of Science, ii> gratefully acknowledged.

References [1] Connes, A., Narahofer, H. and Thirring, W.: "Dynamical Entropy for C* Algebras

and von Neumann Algebras", Commun. Math. Phys. 112, 691-719 (1987).

[2] Park, Y. M. and Shin, H. H.: "Dynamical Entropy of Quasi-local Algebras in Quan­tum Statistical Mechanics", Preprint (Yonsei University, Seoul, Korea), to appear.

[3] Eifros, E. G.: "Why the Circle is Connected: An Introduction to Quantized Topol­ogy", Mathematical Intelligencer 11, 27-34 (1989).

[4] Blackadar, B. E.: "A simple unital projectionless C*-algebra", J. Operator Theory 5, 63-71 (1981).

[5] Hudetz, T.: "Algebraic Topological Entropy", in Nonlinear Dynamics and Quantum Dynamical Systems, G. A. Leonov et al. (eds.), Akademie-Verlag Berlin (1990), p. 27-41.

[6] Hudetz, T.: "Non-linear Entropy Functionals and a Characteristic Invariant of Sym­metry Group Actions on Infinite Quantum Systems", in Selected Topics in QFT and Mathematical Physics, J. Niederle and J. Fischer (eds.), World Scientific, Singapore (1990), p. 110-124.

[7] Lindblad, G.: " Dynamical Entropy for Quantum Systems", in Quantum Probability and Applications HI, L. Accardi and W. von Waldenfels (eds.), Springer LNM 1303, Berlin (1988), p. 183-191.

[8] Lindblad, G.: "Non-Markovian Quantum Stochastic Processes and Their Entropy", Commun. Math. Phys. 65, 281-294 (1979).

[9] Kraus, K.: "General State Changes in Quantum Theory", Ann. Phys. 64, 311-335 (1971).

[10] Gell-Mann, M. and Hartle, J. B.: "Quantum Mechanics in the Light of Quantum Cos­mology" , in Proceedings of the 3rd International Symposium on Quantum Mechanics in the Light of New Technology, S. Kobayashi et al. (eds.), Physical Society of Japan, Tokyo (1990).

24

[11] Griffiths, R. B.: " Consistent Histories and the Interpretation of Quantum Mechanics", J. Stat. Phys. 36, 219-272 (1S84).

[12] Omnes, R.: "Logical Reformulation of Quantum Mechanics I", J. Stat. Phys. 53, 893-932 (1988).

[13] Adler, R. L., Konheim, A. G. and McAndrew, M. H.: "Topological Entropy", Trans. Am. Math. Soc. 114, 309-319 (1965).

[14] Bauer, W. and Sigmund, K.: "Topological Dynamics of Transformations Induced on the Space of Probability Measures", Monatshefte Math. 79, 81-92 (1975).

[15] Sigmund, K.: "Affine Transformations on the Space of Probability Measures", Societe Mathemathatique de France, Asterisque 51, 415-427 (1978).

[16] Dinaburg, E. I.: "On the Relations Among Various Entropy Characteristics of Dynam­ical Systems", Mathematics of the USSR - Izvestiya 5, 337-378 (1971); reprinted in Dynamical Systems, Ya. G. Sinai (ed.), World Scientific, Singapore (1991), p. 97-138.

[17] Goodman, T. N. T.:"Relating Topological Entropy and Measure Entropy", Bull. Lon­don Math. Soc. 3, 176-180 (1971).

[18] Kuriyama, K: "Entropy of a Finite Partition of Fuzzy Sets", J. Math. Anal. Appl. 94, 38-43 (1983).

[19] Malicky, P. and Riecan, B.: "On the Entropy of Dynamical Systems", in Ergodic Theory and Related Topics II, H. Michel et al. (eds.), Teubner, Leipzig (1987), p. 135-138.

[20] Hudetz, T.: "Entropy and Dynamical Entropy for Coatir.uous Fuzzy Partition*", in preparation (1992;.

[21] Gillmann, L. and Jerison, M.: Rings of Continuous Functions, Springer GTM 43, New York (1960).

[22] Choi, M.-D. and Effros, E. G.: "Nuclear C*-algebras and the Approximation Prop­erty", Am. J Math. 100, 61-79 (1978).

[23] Choi, M.-D.: "A Schwarz Inequality for Positive Linear Maps on C*-algebras", Illinois J. Math. 18, 565-574 (1974).

[24] Kadison, R. V.: "A Generalized Schwarz Inequality and Algebraic Invariants for Op­erator Algebras", Ann. Math. 56, 494-503 (1952).

[25] Summers, S. J.: "On the Independence of Local Algebras in Quantum Field Theory", Rev. Math. Phys. 2, 201-247 (1990).

25

[26] Stürmer, E. and Voiculescu, D.: "Entropy of Bogoliubov Automorphisms of the Canoaical Anticommutation Relaticis", Comrmm. Math. Phys. 135, 521-542 (1990).

[27] Hudetz, T.: "Spacetime Dynamical Entropy of Quantum Systems", Lett. Math. Phys. 16,151-161 (1988).

[28] Evans, D. E.: "Entropy of Automorphisms of AF Algebras", Publ. RL\A3 Kyoto Univ. 18,1045-1051 (1982).

[29] Christensen, E.: "Near Inclusions of C'-algebras", Acta Math. 144, 249-265 (1980).

[30] Bratteli, 0 . : "Inductive Limits of Finite Dimensional C'-algebras", Trans. Am. Math. Soc. 171, 195-234 (1972).

[31] Choda, M.: "Entropy for t-Endomorphisms and Relative Entropy for Subalgebras", J. Operator Theory (to appear).

[32] Narnhofer, H. and Thirring, W.: "Chaotic Properties of the Noncommutative 2-shift", to appear in From Phase Transitions to Chaos, Topics in Modern Statistical Physics G. Györgyi et al. (eds.), World Scientific, Singapore (1992).

[33] Hudetz, T.: Ph.D. Thesis, to appear (1992).

26