Noncommutative topological entropy of endomorphisms of Cuntz algebras II

A study of noncommutative topological entropy of gauge invariant endomorphisms of Cuntz algebras began in our earlier work with Joachim Zacharias is continued and extended to endomorphisms which are not necessarily of permutation type. In particular it is shown that if H is an N-dimensional Hilbert space, V is an irreducible multiplicative unitary on the tensor product of H with itself and F is the tensor flip, then the Voiculescu entropy of the Longo's canonical endomorphism associated with the unitary VF is equal to log N.Comment: 8 page

Quantum isometry groups of duals of free powers of cyclic groups

We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underline{\epsilon}$/2, where $\epsilon$ is the uniform measure on the s-th roots of unity, and $\epsilon\to\underline{\epsilon}$ is the canonical projection map from complex to real measures. We discuss as well a number of technical versions of this result, notably with the construction of a new quantum group, which appears as a "representation-theoretic limit", at s equal to infinity.Comment: 23 pages, in v2 some proofs are modified and expanded (notably that of Theorem 3.5), a few illustrations of the operations related to the considered categories of partitions added and some typos corrected. The paper will appear in the International Mathematics Research Notice

Two-parameter families of quantum symmetry groups

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O^+(p,q), B^+(p,q), S^+(p,q) and H^+(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of) free quantum groups studied earlier. For H^+(p,q) the situation is different: we show that H^+(p,0) is isomorphic to the quantum isometry group of the C*-algebra of the free group and it can be viewed as a liberation of the classical isometry group of the p-dimensional torus.Comment: 29 page