Multipliers and Hereditary Subalgebras of Operator Algebras

TL;DR

This paper extends classical results to operator algebras, providing a new characterization of open and closed projections via multiplier algebras, and linking hereditary subalgebras to strict closures in multiplier algebras.

Contribution

It generalizes Glicksberg's results to operator algebras and characterizes projections using multiplier algebras, connecting hereditary subalgebras with strict closures.

Findings

Multiplier algebra of an operator algebra can be obtained as the strict closure in the multiplier algebra of the generated C*-algebra.

Open projections in operator algebras correspond to hereditary subalgebras with specific multiplier algebra properties.

The results unify the understanding of ideals and projections in operator algebras with classical theories in uniform algebras.

Abstract

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure of the subalgebra with respect to the strict topology of the multiplier algebra of a naturally associated hereditary C*-subalgebra. This immediately implies that the multiplier algebra of an operator algebra A may be obtained as the strict closure of A in the multiplier algebra of the C*-algebra generated by A.