Vol. 27, No. 1, 1968

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Tensor products of W-algebras

Donald John Charles Bures

Vol. 27 (1968), No. 1, 13–37

This paper deals primarily with a characterization of the tensor products of a family of W-algebras (abstract von Neumann algebras). It is especially concerned with infinite tensor products; the results, however, apply and have interest in the finite case.

A tensor product for a family (𝒜i) of W-algebras is defined to be a W-algebra 𝒜 together with injections αi of 𝒜i into 𝒜 satisfying four conditions: the first two are that the αi(𝒜i) commute and generate 𝒜; the last two are conditions on the set of positive normal functionals of 𝒜 which are products with respect to the αi(𝒜i). A local tensor product is defined to be a tensor product satisfying a fifth condition—that its tail reduce to the scalars. It is shown that the local tensor products of (𝒜i) are precisely the incomplete direct products (𝒜ii), and that every tensor product is a direct sum of local tensor products which are not product isomorphic.

Mathematical Subject Classification
Primary: 46.65
Received: 27 October 1967
Published: 1 October 1968
Donald John Charles Bures