Vol. 15, No. 3, 1965

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A generalisation of W-algebras

George A. Reid

Vol. 15 (1965), No. 3, 1019–1026

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
Received: 27 August 1964
Published: 1 September 1965
George A. Reid