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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 A⊗CB 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 ⊗CB≅L2(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.
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