Vol. 15, No. 3, 1965

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Tensor products over H∗-algebras

Larry Charles Grove

Vol. 15 (1965), No. 3, 857–863

Throughout, A, B, and C denote (semi-simple) H-algebras whose respective decompositions into minimal closed ideals are A = Σ Aα, B = Σ Bβ, and C = Σ Cγ. It is assumed that A is a right C-module and B is a left C-module. We define a tensor product ACB that is again an H-algebra, and show that it is isometric and isomorphic with an ideal in A B C. As a corollary, A CB is strongly semi-simple if A, B, and C are each strongly semi-simple. The converse to the corollary is shown to be false. When A, B, and C are closed ideals in some H-algebra, with ordinary multiplication as the module action, then A CB is shown to be isomorphic with the direct sum of all the one-dimensional ideals in A B C. When A = L2(G), B = L2(H), and C = L2(K), for suitable related compact groups G, H, and K, then the module actions are defined, and A CB can be constructed. When G = H = K, it is shown that A CBL2(G∕N), where N is the closure of the commutator subgroup of G. A conjecture is stated that would generalize this result to the case where K is a closed subgroup of G H.

Mathematical Subject Classification
Primary: 46.60
Received: 19 February 1965
Revised: 21 April 1965
Published: 1 September 1965
Larry Charles Grove