Injective envelopes and local multiplier algebras of some spatial continuous trace C*-algebras

ARGERAMI, MARTIN; FARENICK, DOUGLAS; MASSEY, PEDRO

QUARTERLY JOURNAL OF MATHEMATICS

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2012 vol. 63 p. 1 - 20

0033-5606

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 herein 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 $A$.