Non-Cocommutative C*-Bialgebra Defined as the Direct Sum of Free Group C*-Algebras

Abstract

Leti ${\Bbb F}$n be the free group of rank n and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C*-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n} (1\leq n<\infty$). We introduce a new comultiplication Δϕ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C*-bialgebra. With respect to Δϕ, the tensor product π⊗ϕπ′ of any two representations π and π′ of free groups is defined. The operation ×ϕ is associative and non-commutative. We compute its tensor product formulas of several representations.

C*-bialgebra Defined as the Direct Sum of UHF Algebras

“…In this subsection, we review a general method to construct C * -bialgebras [18,25], and generalize it in order to prove theorems.…”

“…A monoid is a set M equipped with a binary associative operation M × M ∋ (a, b) → ab ∈ M, and a unit with respect to the operation. We recall the definition of C * -weakly coassociative system in [25].…”

“…A C * -bialgebra is a generalization of bialgebra in the theory of C * -algebras, which was introduced in C * -algebraic framework for quantum groups [26,27]. We have studied C * -bialgebras and their construction method, and computed non-symmetric tensor products of * -representations with respect to non-cocommutative comultiplications [17,18,20,21,24,22,25]. In this paper, we introduce a non-cocommutative C * -bialgebra defined as the direct sum of a certain set of uniformly hyperfinite (=UHF) algebras.…”

Separable representations, KMS states, and wavelets for higher-rank graphs

“…By expanding such fertile families of representations from the Cuntz algebra setting to the k-graph setting, which encompasses a much broader class of non-type I C * -algebras, we anticipate that the research contained in these pages will facilitate further progress in a diverse and extensive range of fields. The following is a sample: branching laws for endomorphisms of fermions, see e.g., [1], Markov measures, transfer operators, wavelets and multiresolutions [2], fractional Brownian motion [3], quasi-crystals, see e.g., [6], substitution dynamical systems and complexity, structure of invariant measures, Markov and more general path-space measures, measurable partitions [7], [9], [10], and [12], noncommutative geometry [19], topological Markov chains [21], martingales [27], fractals and self-similarity [28], spectral triples [36] , renormalization theory (physics) [40], topological orbit equivalence [41], wavelet filters [51], continued fraction expansions [60,61], tilings and tiling space [63], quantum channels [67], group actions on the boundary of triangle buildings [84], and entropy [88,89].…”