Quantized Vershik–Kerov theory and quantized central measures on branching graphs

Abstract

We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila–Voiculescu's work [24] (or [8]), and give its serious investigation when the system consists of quantum unitary groups Uq(N) with q∈(0,1). The key features of this work are: The “quantized trace” of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect to a certain one-parameter automorphism group related to the so-called scaling group. In this paper we provide a Vershik–Kerov type approximation theorem for extremal quantized characters (called the ergodic method) and also compare our quantized character theory for the inductive system of Uq(N) with Gorin's theory on q-Gelfand–Tsetlin graphs [11].