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

Let ${\text{LL}}_{\mathbf{v}}\subset {\mathbb{Z}}^D$ be a suitable cone semigroup and ${\text{A}}_{\mathbf{v}}$ its reduced semigroup $C^{\ast }$-algebra. In this paper, we compute the ${\text{LL}}_{\mathbf{v}}$-invariant measures in the transversal hull of the semigroup ${\text{LL}}_{\mathbf{v}}$ that exhibit regularity in the boundaries of ${\text{LL}}_{\mathbf{v}}\mathrm{.}$ These measures enable the construction of a trace per-unit hypersurface for observables in ${\text{A}}_{\mathbf{v}}$ supported near the boundaries of ${\text{LL}}_{\mathbf{v}}$, leading to the construction of appropriate Chern cocycles in the “boundary” ideals of ${\text{A}}_{\mathbf{v}}$. 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.