Let $\LL_{ \bf v}\subset \Z^D$ be a suitable cone semigroup and $\A_{ \bf v}$ its reduced semigroup $C^*$-algebra. In this paper, we compute the $\LL_{ \bf v}$-invariant measures in the transversal hull of the semigroup $\LL_{ \bf v}$ that exhibit regularity in the boundaries of $\LL_{ \bf v}.$ These measures enable the construction of a trace per-unit hypersurface for observables in $\A_{ \bf v}$ supported near the boundaries of $\LL_{ \bf v}$, leading to the construction of appropriate Chern cocycles in the "boundary" ideals of $\A_{ \bf v}$. 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.
Categories: math-ph math.MP math.OA