Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

Md Amir Hossain, S. Sundar

Abstract

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced $C^{*}$-algebra of a product system. We show that for a semigroup dynamical system $(A, P,α)$, under reasonable hypotheses (e.g., $P$ is abelian and finitely generated), the reduced crossed product $A \rtimes_{red} P$ is exact if and only if $A$ is exact. This strengthens our earlier result (\cite{Amir_Sundar-product-system}), where it was assumed that the action of $P$ on $A$ is by injective endomorphisms. We also compare the groupoid crossed product described in \cite{Amir_Sundar-product-system} and the Fell bundle constructed in \cite{Rennie_Sims} for a product system, and show that they are equivalent as Fell bundles.

Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

Abstract

We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced -algebra of a product system. We show that for a semigroup dynamical system , under reasonable hypotheses (e.g., is abelian and finitely generated), the reduced crossed product is exact if and only if is exact. This strengthens our earlier result (\cite{Amir_Sundar-product-system}), where it was assumed that the action of on is by injective endomorphisms. We also compare the groupoid crossed product described in \cite{Amir_Sundar-product-system} and the Fell bundle constructed in \cite{Rennie_Sims} for a product system, and show that they are equivalent as Fell bundles.

Paper Structure

This paper contains 4 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exactness of reduced $C^*$-algebras of semigroup dynamical systems
  4. Equivalence of Fell bundles

Key Result

Lemma 3.1

Let $(A_i, \alpha_{j,i})_{j\geq i}$, $(B_i, \beta_{j,i})_{j\geq i}$ and $(C_i, \gamma_{j,i})_{j\geq i}$ be three inductive systems of $C^*$-algebras with inductive limits $(A, \alpha_i)_{i\in I}$, $(B, \beta_i)_{i\in I}$ and $(C, \gamma_i)_{i\in I}$. Suppose that for each $i$, there is a short exact of $C^*$-algebras, which is compatible with the connecting maps in the sense that $\beta_{j,i}\circ

Theorems & Definitions (24)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.7
  • Theorem 3.8
  • ...and 14 more