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

TL;DR

This paper computes invariant measures on cone semigroups' transversal hulls, enabling boundary trace constructions in semigroup C*-algebras, with applications to topological insulators.

Contribution

It introduces a method to compute invariant measures on cone semigroups' hulls, applicable to both finitely and non-finitely generated cases, and links these to boundary Chern cocycles.

Findings

Computed invariant measures for cone semigroups.

Constructed boundary traces and Chern cocycles.

Applied results to topological insulator models.

Abstract

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.