A new approach to recent constructions of C⁎C⁎-algebras from modular index theory

We present a new approach to C⁎C⁎-algebras recently constructed in the context of modular index theory by Carey, Phillips, Putnam and Rennie. It turns out that their constructions can be identified with full corners of ideals in semigroup C⁎C⁎-algebras. This new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras.