The Maximal C*-Algebra of Quotients as an Operator Bimodule

TL;DR

This paper characterizes the maximal C*-algebra of quotients of a unital C*-algebra as a direct limit of operator bimodule homomorphisms, revealing its structure and invariance properties under Morita equivalence.

Contribution

It provides a new description of the maximal C*-algebra of quotients using operator bimodule homomorphisms and demonstrates its invariance under strong Morita equivalence.

Findings

Describes the maximal C*-algebra of quotients as a direct limit of bimodule homomorphisms.

Shows invariance of the construction under strong Morita equivalence.

Connects the structure to the Haagerup tensor product.

Abstract

We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra $A$ as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product $A\otimes_h A$ labelled by the essential closed right ideals of $A$ into $A$. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.