Maximal two-sided ideals in tensor products of Banach algebras

Abstract

In [1 ] and [2] a construction of a bijection M3+->M1 X M2 is given, where MZi is the set of maximal modular (two-sided) ideals in the Banach algebraAi (i=1, 2, 3) and where A3=Al0,A2 is the greatest cross-norm tensor product of A1 and A2. In a recent correction [3] it is shown that there is indeed a closed 1-1 mapping MlXM2--+M3 when hull-kernel topologies are used. However, it is an open question when this mapping is surjective. In this note we show that the mapping is onto when one of the Banach algebras A1 and A2 is commutative. Also we give a correct proof of a theorem in [5], the original proof depended on [2]. The methods employed are adaptions of those in [6]. Suppose A1 and A2 are Banach algebras and suppose A1 is commutative. Let Sti be the set of maximal modular (two-sided) ideals. Each hEz-Tl is a continuous C-valued homomorphism and induces a homomorphismn