Representations and positive functionals of involutive \(\ell^1\)-Munn algebras
Given a unital Banach algebra \({\mathcal A}\), and a \(J\times I\) matrix \(P\) over \({\mathcal A}\), with no zero row or column, such that \(P_{j,i}\) is invertible whenever \(P_{j,i}\not=0\), and \(\sup_{j,i}\|P_{j,i}\|\leq 1\), the \(\ell^1\)-Munn algebra \({\mathcal LM}({\mathcal A},P)\) associated to \({\mathcal A}\) and \(P\) is the set of all \(I\times J\) matrices \(A\) over \({\mathcal A}\) such that \(\|A\|_1=\sum_{i,j}\|A_{i,j}\|<+\infty\), equipped with the product \(A\circ B=A P B\). In this paper, \(I\) and \(J\) are assumed to be finite and equal. It is shown that, when \({\mathcal A}\) is involutive and \(P\) is self-adjoint (in the sense that \(P_{i,j}^\ast=P_{j,i}\)), \({\mathcal LM}({\mathcal A},P)\) has a natural isometric involution; the author characterizes the positive functionals \(F\in {\mathcal LM}({\mathcal A},P)^\ast\), and, given a \(\ast\)-representation of \({\mathcal A}\) on a Hilbert space \({\mathcal H}\), he constructs a \(\ast\)-representation of \({\mathcal LM}({\mathcal A},P)\) on \({\mathcal H}^J\); he ends with a connection with semigroup algebras.
scientific article; zbMATH DE number 1506486