168 research outputs found
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 /2, where is the
uniform measure on the s-th roots of unity, and
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
Approximation of quantum Levy processes by quantum random walks
Every quantum Levy process with a bounded stochastic generator is shown to
arise as a strong limit of a family of suitably scaled quantum random walks.Comment: 7 page
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