Let \(\Lambda\) be a solid closed convex cone in a Euclidean vector space X. Let \(C^*(X)\) be the \(C^*\)-algebra of operators on \(L^ 2(X)\) generated by convolutions with \(L^ 1\)-functions. Let \(1_{\Lambda}\) denote the operator of multiplication with the characteristic function of \(\Lambda\). Consider two Wiener-Hopf operator \(C^*\)-algebras: \(C^*(X,\Lambda)\), generated by \(C^*(X)\) and \(1_{\Lambda}\) on \(L^ 2(X)\), and \(C^*(\Lambda)=1_{\Lambda}C^*(X,\Lambda)1_{\Lambda}\) on \(L^ 2(\Lambda)\). It is shown that irreducible representations of these \(C^*\)-algebras are equivalent to their canonical representations in the corresponding Wiener-Hopf \(C^*\)-algebras associated with the conormal cones of \(\Lambda\) for a large class of tangible \(\Lambda\) including smooth polyhedral cones and cones which have finite orbit decompositions under a linear group action. It follows that \(C^*(X,\Lambda)\) and \(C^*(\Lambda)\) are post liminary for such \(\Lambda\) (there are misprints on pages 537, 539 and 555 where they are called ''liminary'' instead).