Noncommutative Topological Entropy of Endomorphisms of Cuntz Algebras

Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura. In [Vo] D.Voiculescu defined noncommutative topological entropy for automorphisms (or completely positive maps) of nuclear C-algebras based on completely positive approximations which naturally extends the classical definition of topological entropy for continuous maps of compact spaces. His definition has been subsequently extended to the larger class of exact C-algebras (by Brown [Br]) and intensively studied in the past decade (we refer to the book [NS] for a comprehensive discussion and many examples). For Cuntz algebras and its various generalisations entropy estimates have only been obtained for canonical endomorphisms which may be regarded as noncommutative extension of classical shift maps. In [Ch] M.Choda showed that the noncommutative topological entropy of the canonical shift endomorphism of the Cuntz algebra ON is equal to logN . Later in [BG] F.Boca and P.Goldstein used a different method which allowed them to compute entropy for the shift-type endomorphisms on arbitrary Cuntz-Krieger algebras. Their techniques were extended in [SZ] to determine the values of noncommutative entropy and pressure for the multidimensional shifts on C-algebras associated with higherrank graphs. In all of these results it turned out that the entropy of the canonical shift endomorphism is the same as the entropy of the corresponding classical shift. In this paper we try to estimate entropies for more general endomorphisms of Cuntz algebras. Endomorphisms of Cuntz algebras have been studied intensivly and have applications in subfactors, quantum field theory and other areas. A particularly interesting class is formed by those which leave FN , the canonical UHF-subalgebra ofON , invariant. Besides the canonical shift this class contains the recently introduced and intensively studied permutative polynomial endomorphisms. We refer to [BJ] and [Ka] for connections with branching function systems and permutative representations and [CS1−2] and [CKS] for connections with subfactors and Mathematical Physics. In particular [CS2] draws an interesting connection between our entropy estimates and indices of endomorphisms on O2. In the first part of this paper we give a general upper bound (in Theorem 2.2) for the entropy of endomorphism which leave FN invariant and which verify a certain Permanent address of the first named author: Department of Mathematics, University of Lódź, ul. Banacha 22, 90-238 Lódź, Poland. 2000 Mathematics Subject Classification. Primary 46L55, Secondary 37B40.