Amalgams of Inverse Semigroups and C*-algebras

TL;DR

This paper establishes a connection between amalgams of inverse semigroups and their associated C*-algebras, showing that the C*-algebra of a full inverse semigroup amalgam equals the C*-algebraic amalgam of the individual C*-algebras, with applications to various C*-algebra constructions.

Contribution

It proves that the C*-algebra of a full inverse semigroup amalgam is the C*-algebraic amalgam of the individual C*-algebras, extending to specific examples like graph Toeplitz algebras.

Findings

C*-algebra of a full inverse semigroup amalgam equals the algebraic amalgam of component C*-algebras.

Application to finite-dimensional C*-algebras and Toeplitz algebras of graphs.

Provides a framework for constructing complex C*-algebras from inverse semigroup amalgams.

Abstract

An amalgam of inverse semigroups [S,T,U] is full if U contains all of the idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam of C*(S) and C*(T) over C*(U). Using this result, we describe certain amalgamated free products of C*-algebras, including finite-dimensional C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs.