\(\mathrm C^*\)-envelopes of semicrossed products by lattice ordered abelian semigroups (Q2197819)

These are nonselfadjoint norm closed algebras of operators on Hilbert space. They include certain non-selfadjoint subalgebras of \(C^*\)-crossed products, and in particular they include the class of operator algebras considered by \textit{W. B. Arveson} [Acta Math. 118, 95--109 (1967; Zbl 0182.18201)] and \textit{W. B. Arveson} and \textit{K. B. Josephson} [``Operator algebras and measure preserving automorphisms II'', J. Funct. Anal. 4, 100--134 (1969)]. The author prove that, when the positive cone of a discrete lattice ordered abelian group acts on a \(C*\)-algebra, the \(C*\)-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. Furthermore, by constructing a \(C*\)-cover that itself is a full corner of a crossed product, and computing the Shilov ideal, is obtained an explicit description of the \(C^*\)-envelope.