Hereditary subalgebras of operator 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'.

Final version, To appear. 21 pages

References in corpus (1)

• The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

Cited by in corpus (16)

• Real positivity and approximate identities in Banach algebras
• Noncommutative peak interpolation revisited
• Completely contractive projections on operator algebras
• On positivity and roots in operator algebras
• Nuclearity-related properties for nonselfadjoint algebras
• Extensions of operator algebras I
• Ideals and structure of operator algebras
• Operator algebras with contractive approximate identities, III
• Generalization of C*-algebra methods via real positivity for operator and Banach algebras
• Contractive projections and real positive maps on operator algebras
• Operator algebras with contractive approximate identities: Weak compactness and the spectrum
• A characterization and a generalization of W*-modules
• Closed projections and peak interpolation for operator algebras
• Jordan operator algebras revisited
• Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and ${\mathrm{C}}^{\ast }$-liminality
• Jordan operator algebras: Basic theory