Tensor products overH∗-algebras

Throughout, A, B, and C denote (semi-simple) iϊ*-algebras whose respective decompositions into minimal closed ideals are A = Σ © A a , B = Σ 0 B β , and C = Σ © C y .It is assumed that A is a right C-module and B is a left C-module.We define a tensor product A® 0 B that is again an iϊ*-algebra, and show that it is isometric and isomorphic with an ideal in A®B®C.As a corollary, A® G B 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 ίP-algebra, with ordinary multiplication as the module action, then A® 0 B is shown to be isomorphic with the direct sum of all the one-dimensional ideals in A n B n C. When A = L\ B = L\ and C = L\ for suitable related compact groups G, H, and K, then the module actions are defined, and A®oB can be constructed.When G = H = K 9 it is shown that A® G B = L/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 Π H.