A generalisation of W^∗-algebras

George A. Reid

Vol. 15 (1965), No. 3, 1019–1026

DOI: 10.2140/pjm.1965.15.1019

Abstract

Using the theory of double centralisers due to B. E. Johnson, we define a QW^∗-algebra as being a B^∗-algebra, A, such that the algebra of double centralisers of each closed ∗-subalgebra B is contained in a suitable related closed ∗-subalgebra B₀₀.

After obtaining explicit descriptions of the algebras of double centralisers of commutative and noncommutative B^∗-algebras, we prove that in the general noncommutative case a W^∗-algebra is a QW^∗-algebra, and a QW^∗-algebra is an AW^∗-algebra, while in the commutative case the QW^∗ and AW^∗ conditions are equivalent.

We prove that if A is QW^∗ then so are its centre, any maximal commutative *-subalgebra, and any subalgebra of the form eAe for e a projection in A.

Mathematical Subject Classification

Primary: 46.60

Secondary: 46.65

Milestones

Received: 27 August 1964

Published: 1 September 1965