Representing matrices, M-ideals and tensor products of L₁-predual spaces

Abstract

Motivated by Bratteli diagrams of Approximately Finite Dimensional (AF) C*- algebras, we consider diagrammatic representations of separable L₁-predual spaces and show that, in analogy to a result in AF C*-algebra theory, in such spaces, every M-ideal corresponds to directed sub diagram. This allows one, given a representing matrix of a L₁-predual space, to recover a representing matrix of an M-ideal in X. We give examples where the converse is true in the sense that given an M-ideal in a L₁-predual space X, there exists a diagrammatic representation of X such that the M-ideal is given by a directed sub diagram and an algorithmic way to recover a representing matrix of M-ideals in these spaces. Given representing matrices of two L₁-predual spaces we construct a representing matrix of their injective tensor product.