C*-envelopes of semicrossed products by lattice ordered abelian semigroups

A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. We 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. By constructing a C*-cover that itself is a full corner of a crossed product, and computing the Shilov ideal, we obtain an explicit description of the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from $\mathbb{Z}_+^n$ to the class of all discrete lattice ordered abelian groups.
Comment: 36 pages. Updated to reflect published version in JFA. Minor typos fixed throughout and new Corollary 3.18 (nonunital case) and Subsection 6.1 (simplicity of the C*-envelope) added