Journal of Operator Theory

Volume 59, Issue 2, Spring 2008 pp. 335-357.

Hereditary subalgebras of operator algebras

Authors: David P. Blecher (1), Damon M. Hay (2), and Matthew Neal (3)
Author institution: (1) Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
(2) Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA
(3) Mathematics and Computer Science Department, Denison University, Granville, OH 43023, USA

Summary: 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 basic theory of hereditary subalgebras of a $C^*$-algebra, and the solution of a ten year old problem concerning 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".

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