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

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Δ is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show herein to be a Kaplansky–Hilbert module over the abelian AW*-algebra C(Δ). 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.