Vol. 47, No. 2, 1973

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Centralizers of twisted group algebras

Robert C. Busby

Vol. 47 (1973), No. 2, 357–392

Centralizers (left, right, and double) of rings and algebras have received much attention recently, and seem likely to become an important topic in ring theory. They have proved quite useful in Banach algebra theory, and a good deal of work has been done on the computation of centralizers for various Banach algebras. In this paper we compute the left centralizers of a twisted group algebra, a generalization of the group algebra of locally compact group, which includes as special cases the covariance algebras of quantum field theory, and the group algebras of separable group extensions (explicitly given in terms of the subgroup algebra and quotient group). We give a representation of the algebra of left centralizers of a “locally continuous” twisted group algebra as an algebra of vector-valued measures with “twisted convolution”. This result gives more than explicit computation of centralizers. The form of the result enables us to investigate isometric isomorphisms between twisted group algebras along lines previously pursued for ordinary group algebras. In some cases we get a complete description of possible isomorphism classes in terms of orbits in a cohomology set.

Mathematical Subject Classification 2000
Primary: 43A20
Secondary: 46K05
Received: 20 March 1972
Published: 1 August 1973
Robert C. Busby