\(C^*\)-module algebras (Q2043560)

The authors introduce the notion of a \(C^*\)-module algebra as an algebra \(E_0\) with involution which is a (two sided) module over a \(C^*\)-algebra \(\mathcal{A}\). They also prove that \(E_b\), the set of bounded elements of the completion \(E\) of \(E_0\), is a Banach \(*\)-algebra. They state the following notion. Let \(\mathcal{A}\) be a von Neumann algebra, a \(C^*\)-module algebra \(E\) over \(\mathcal{A}\) is called a \(W^*\)-module algebra if \(E\) has a predual as a Banach space. They prove that, if \(E_0\) is a symmetric pre \(C^*\)-module algebra over a von Neumann algebra \(\mathcal{A}\), then \(E_0'\) (the set of all bounded two sided \(\mathcal{A}\)-module maps from \(E_0\) to \(\mathcal{A}\)) is a \(W^*\)-module algebra.