Approximation entropies in operator algebras
by
George Popescu
University of Craiova

The paper studies the approximation entropies introduced by D. Voiculescu in 1995. We consider all four cases, i.e. approximation by subalgebras and by completely positive maps, in both the C ^* -algebra and W ^* -algebra case. Let M be a C ^* -algebra and \alpha:M --> M an endomorphism. Then let Pf(M) be the collection of all finite subsets of M and F(M) the collection of all finite dimensional C ^* -subalgebras of M. We define the \delta-rank of \omega in Pf(M) as: r(\omega, \delta)=inf{\funcrank(A) | A in F(M) , \omega subset _\delta A} , where \omega subset _\delta A means for allx in \omega , existsa in A so that   x-a  < \delta. The pattern of defining the approximation entropies is the following:

h(\alpha, \omega, \delta)=\stackundern --> \inftylimsup\frac 1nlogr( \cup _k=0^n-1\alpha^k(\omega), \delta)

h(\alpha, \omega)=\stackunder\delta > 0suph(\alpha, \omega, \delta)

h(\alpha)=\stackunder\omega in Pf(M)suph(\alpha, \omega) ,

h is called an approximation entropy.

We study the invariance of these entropies towards (strong) equivalence of endomorphisms, unitary equivalence and approximative unitarily equivalence. We consider the ''reduced'' entropies, and show that they are invariant towards these equivalences (under suitable conditions).

Then we study the case when these entropies are null. There are two cases when these entropies could be null :

(A) the sequence of the \delta-ranks associated to the orbits \omega, \alpha(\omega), ..., \alpha^n-1(\omega) is bounded independent of n= ''the length of the orbits''. (B) the sequence of the \delta-ranks is ''slowly growing'', for instance r( \cup _k=0^n-1\alpha^k(\omega), \delta) <= n^p where p could be the cardinal of \omega.

The main problems arising are the following:

P1) \stackundern --> \inftylimsup\frac 1nlogr(\omega \cup \alpha(\omega)...\alpha^n-1(\omega), \delta)=\stackundern --> \inftylimsup\frac 1nlogr(\alpha(\omega) \cup \alpha²(\omega)...\alpha^n-1(\omega), \delta)

P2) if \omega₀ subset ker\alpha then h(\alpha, \omega \cup \omega₀)=h(\alpha, \omega) for any \omega in Pf(M)

P3) for any \omega₀ and \omega we have \stackundern --> \inftylimsup\frac 1nlogr(\omega \cup \alpha(\omega)...\alpha^n-1(\omega), \delta)=\stackundern --> \inftylimsup\frac 1nlogr(\omega₀ \cup \omega \cup \alpha(\omega)...\alpha^n-1(\omega), \delta)

P4) h(\alpha, \omega₁)=0 and h(\alpha, \omega₂)=0 ===> h(\alpha, \omega₁ \cup \omega₂)=0

P5) h(\alpha, \omega₀)=0 ===> h(\alpha, \omega \cup \omega₀)=h(\alpha, \omega)

P6) h(\alpha, {x})=0 for any x in \omega ===> h(\alpha, \omega)=0.

The main idea is to identify those finite subsets which have null entropy and then prove that these may be neglected when computing approximation entropies.

We give sufficient conditions for the approximation entropies to be null in the AF  C ^* -algebra case.

Date received: June 8, 2000