Invariant measures on the transversal hull of cone semigroups and some applications

Let \mathcal{L}_{\mathbf{v}}\subset{\mathbb{Z}}^{D} be a suitable cone semigroup and \mathfrak{A}_{\mathbf{v}} its reduced semigroup C^{*} -algebra. In this paper, we compute the \mathcal{L}_{\mathbf{v}} -invariant measures in the transversal hull of the semigroup \mathcal{L}_{\mathbf{v}} that exhibit regularity in the boundaries of \mathcal{L}_{\mathbf{v}} . These measures enable the construction of a trace per-unit hypersurface for observables in \mathfrak{A}_{\mathbf{v}} supported near the boundaries of \mathcal{L}_{\mathbf{v}} , leading to the construction of appropriate Chern cocycles in the “boundary” ideals of \mathfrak{A}_{\mathbf{v}} . Our approach applies to both finitely and non-finitely generated cone semigroups. Applications for the bulk-defect correspondence of lattice models of topological insulators are also provided.