Ideals and representations of certain semigroup algebras (Q1882667)

Let \(U\) be a Banach algebra, \(I\) and \(J\) be arbitrary index sets, and \(P=(P_{ji})\) be a \(J\times I \) matrix over \(U\) such that \(\| P\| _{\infty}=\sup\{\| P_{ji}\| : j\in J, i\in I\}\leq 1\). The Banach algebra \({\mathcal LM}(U, P)\) of all \(I\times J\) matrices \(A\) over \(U\) such that \(\| A\| _1=\sum_{i\in I, j\in J}\| A_{ij}\| <\infty\) with \({\ell}^1\)-norm and product \(A\circ B=APB\in {\mathcal LM}(U, P)\) is called the \({\ell}^1\)-Munn \(I\times J\) matrix algebra over \(U\) with sandwich matrix \(P\) or briefly \({\ell}^1\)-Munn algebra. In the paper under review, the author considers the ideal structure of \({\ell}^1\)-Munn algebras and describes various types of ideals, Jacobson radical and strong radical of \({\ell}^1\)-Munn algebras with bounded approximate identities. If there is an involution on \(U\) then by the term ``representation'' the author means a \(\ast\)-representation by bounded operators on a Hilbert space. He considers \(\ast\)-representations of involutive \({\ell}^1\)-Munn algebras with positive sandwich matrices from those of their underlying algebras and shows the relations between these two classes of representations. An application of these results to semigroup algebras is also given.