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Which multiplier algebras are $W^*$-algebras? (1304.7453v1)

Published 28 Apr 2013 in math.OA

Abstract: We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C*$-algebra $\mathcal{A}$ is a $ W*$-algebra, and show that it holds for a stable $C*$-algebra exactly when it is a $C*$-algebra of compact operators. This implies that if for every Hilbert $C*$-module $E$ over a $C*$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W*$-algebra, then $\mathcal{A}$ is a $C*$-algebra of compact operators. Also we show that a unital $C*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W*$-algebra must be a $ W*$-algebra.

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