Semi-simplicity of a proper weak \(H^*\)-algebra (Q1187612)

A weak right \(H^*\)-algebra is a Banach algebra \(A\) which is a Hilbert space and which has a dense subset \(D_ r\) such that for each \(x\) in \(A\) there is \(x^ r\) in \(D_ r\) with \((yx,z)= (y,zx^ r)\) for all \(y\), \(z\) in \(A\). The author shows that each proper weak right \(H^*\)-algebra is semi-simple. It follows that \(A\), being a right complemented algebra, is a direct sum of simple weak right \(H^*\)-algebras, each of which is isometrically isomorphic to the algebra of all Hilbert-Schmidt operators on some Hilbert space.