Close hereditary C-algebras and the structure of quasi-multipliers

TL;DR

This paper investigates the structure of hereditary C*-subalgebras, quasi-multipliers, and their isomorphisms, providing new results, counterexamples, and clarifications in the theory of sigma-unital C*-algebras.

Contribution

It establishes a correspondence between order isomorphisms and invertible bimodule multipliers, and clarifies the generation of quasi-multipliers by multipliers in sigma-unital C*-algebras.

Findings

Complete order isomorphisms decompose into isomorphisms and inner isomorphisms.

Counterexample shows quasi-multiplier space is not linearly generated by multipliers.

In sigma-unital case, quasi-multiplier space is multiplicatively generated by multipliers.

Abstract

We answer a question of Takesaki by showing that the following can be derived from the thesis of N-T Shen: If A and B are sigma-unital hereditary C*-subalgebras of C such that ||p - q|| < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the sigma-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between sigma-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the sigma-unital case. In particular every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between sigma-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersen's "Non-commutative Tietze extension theorem.") We elaborate the relations of the above results with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady (yes for separable fields, no in general). We discuss the relationship of our results to the theory of perturbations of C*-algebras.