Semicrossed products and reflexivity (Q2891151)

Given a \(w^*\)-closed unital algebra \(A\) acting on \(H_0\) and a contractive \(w^*\)-continuous endomorphism \(\beta\) of \(A\), there is a \(w^*\)-closed (non-selfadjoint) unital algebra \(\mathbb Z_+ \overline\times_\beta A\) acting on \(H_0\otimes\ell^2(\mathbb Z_+)\), called the \(w^*\)-semicrossed product of \(A\) with \(\beta\).NEWLINENEWLINE The author proves that \(\mathbb Z_+ \overline\times_\beta A\) is a reflexive operator algebra provided that \(A\) is reflexive and \(\beta\) is unitarily implemented, and that \(\mathbb Z_+ \overline\times_\beta A\) is the commutant of a \(w^*\)-semicrossed product and is its own bicommutant if and only if the same holds for \(A\). He also considers the semicrossed product of a commutative \(C^*\)-algebra \(C(K)\) with a continuous map \(\varphi:K\to K\) and shows that the representations induced by a character of \(C(K)\) suffice to obtain that the norm of the semicrossed product as well as the \(w^*\)-closure of such a representation is always reflexive.