Which multiplier algebras are W*-algebras?

الأصل · EN

We consider the question of when the multiplier algebra M(A) of a C*-algebra 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 A, the algebra B(E) of adjointable operators on E is a W*-algebra, then A is a C*-algebra of compact operators. Also we show that a unital C*-algebra A which is Morita equivalent to a W*-algebra must be a W*-algebra.