Hereditary subalgebras of operator algebras

TL;DR

This paper generalizes hereditary subalgebras in C*-algebras, solves a longstanding Morita equivalence problem for operator algebras, and extends Hilbert C*-modules to nonselfadjoint algebras, with implications for noncommutative peak set theory.

Contribution

It introduces a bijective correspondence between open projections and ideals, generalizes hereditary subalgebras, and resolves a decade-old Morita equivalence problem.

Findings

Generalization of hereditary subalgebras of C*-algebras

Solution to Morita equivalence problem for operator algebras

Extension of Hilbert C*-modules to nonselfadjoint algebras

Abstract

In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C*-algebra containing an operator algebra A, and certain one-sided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C*-algebra, and the solution of a ten year old problem on the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C*-modules to nonselfadjoint algebras. We show that an `ideal' of a general operator space X is the intersection of X with an `ideal' in any containing C*-algebra or C*-module. Finally, we discuss the noncommutative variant of the classical theory of `peak sets'.