Injective Envelopes and Local Multiplier Algebras of Some Spatial Continuous Trace C*-Algebras

TL;DR

This paper characterizes the injective envelope of certain spatial continuous trace C*-algebras over Stonean spaces using weakly continuous Hilbert bundles and explores the relationship between local multiplier algebras and the injective envelope.

Contribution

It provides a detailed description of the injective envelope for these C*-algebras and establishes that the second-order local multiplier algebra equals the injective envelope.

Findings

Injective envelope described via weakly continuous Hilbert bundles.

Second-order local multiplier algebra equals the injective envelope.

Provides new insights into the structure of local multiplier algebras.

Abstract

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A.