Note on the tensor product of Banach algebras

Abstract

Introduction. In [2] the maximal ideal space of the tensor product of two commutative Banach algebras was studied.One of the results was: Let^43 = ^4i<8>i ^be the "greatest cross-norm" [3; 4] tensor product of two commutative Banach algebras Ai and A2.Let SDii, 9DÎ2, ÜJÍ3 be the corresponding spaces of regular maximal ideals.Then SDÎ3 and MiXW2 are "naturally" homeomorphic if the weak* topologies are used in all spaces.In the following, we extend the discussion to the case in which no commutativity is assumed.21. Tensor products.Let Ax and A2 be Banach algebras and let C be the complex number system.We consider [3] a subset F of ÇAiXAt.F = j/| / G C^A\ /(O, x2) = f(xi, 0) = 0, 23 \f(xi,x2) I ||xi||i||x2||2 < <*> where x.G^i, * = 1, 2. Since each / in F is nonzero on a set that is countable or finite, to each / there corresponds a sequence (finite or infinite) {(xn, x2i), (xi2, x22), • • • } consisting of just the pairs (xi, x2) where/(xi, x2) 5^0.Addition of elements of / is defined by addition of functions.Multiplication is defined via a form of convolution: If f, gGF, f*g = h is defined by: h(0, x2) = h(xh 0) = 0, h(xi, x2) = 23 f(au <h)g(h, b2) kibi=Xi;a2bz=X2 if Xi, x27i0.

Locations

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