On Banach modules II. Pseudodeterminants and traces

Abstract

1. In the paper [Kai] it was observed that if A is a Banach algebra (over R or C) then the dual space is not only an A-A-bimodule, but is also injective as a left (or right) A-module. Furthermore, if M is a left (or right) Banach module over the unital Banach algebra A, then there is a natural bilinear map, there denoted Tr_A, from M × M′ to A′, defined by

formula here

(or 〈Tr_A(m, m′), a〉 = 〈m′, ma〉). The map Tr_A can be extended to the projective tensor product M[otimesas]M′, which is also an A–A–bimodule.It is easy to see that the map Tr_A is a bimodule homomorphism, so that the image is an A–A–submodule of A′. This module was denoted E_A (M) in [Kai] and is in general not closed as a subspace of M′. It does, however, have a natural norm (as a quotient space of M[otimesas]M′) and the unit ball can be used to define a new norm ∥a∥_new = sup {|〈e, a〉 | e∈ the unit ball of E_A(M)} on A, and it is easy to see that this new norm is simply the operator norm of a as an operator on M. The conclusion is that if a′∈A′ is not only continuous with respect to the norm ∥a∥_L(M) (which is of course in general smaller thatn the given norm on A) but also with respect to the weak topology on A given by the set of all functionals of the form (0·1), then a′ has a representation of the form

formula here.