Tensor Products of Banach Algebras

Abstract

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

$L\sp{1}(I,\,X)$ with order convolution

“…The result about the Carrier space of 7*(7, A") can also be obtained by using a theorem of Grothendieck [9, Chapter 1, p. 58] and a theorem of Gelbaum [7]. However, our method is very simple and straightforward.…”

“…Duchon [4] has introduced the convolution algebra A7(7, X) as follows: Let /be in C (7). Then the function of two variables fixy) is continuous in 7 X 7.…”

Arens regularity of the algebra $A\hat\otimes B$

“…For more information about the tensor product of Banach algebras, the reader may consult Chapter VI of the book [3] and the papers [13,29]. The bilinear form m will be called "(weakly) compact" if U is (weakly) compact.…”

The Powers of a Maximal Ideal in a Banach Algebra and Analytic Structure

“…We have By induction, for any n we may for a dense subset of B0 iterate this process n times at each step replacing each element of (5f)~ in a sum such as (3) or (5) by a sum of products of elements of 5^, each represented as in (3). We may then collect terms as in (4) to obtain a representation b= 2 /v+¿n+l, A"+16(5r1)-.…”

“…Since bk-pk e (By1)' for each k,b-pe (By1)' and it is clear that p satisfies (1) We denote by B^ ®y B$ the completion of the algebraic tensor product of B# with itself in the greatest cross norm. This is defined by ||2«; ® bt\\ =inf {2 \\aj\\ \\bj\\} where the infimum is over all representations of the element of the algebraic tensor product [3]. Let T: B^ 0, B$ -> B$ be the linear mapping which for elements of the algebraic tensor product is defined by P(2 ßy ® bj) = 2 aA-T is normdecreasing and the range of Pcontains B%.…”