Joint Projective Spectrum of $D_{\infty h}$

TL;DR

This paper computes the joint spectrum of the group $D_{inite h}$ in the left regular representation, explores cohomology generators of the joint resolvent set, and presents a self-similar realization of its $C^*$-algebra via action on 4-ary trees.

Contribution

It introduces a novel computation of the joint spectrum for $D_{inite h}$ and links it to cohomology and self-similar $C^*$-algebra representations.

Findings

Identified two generators of the De Rham cohomology group of the joint resolvent set.

Established a self-similar realization of the $C^*$-algebra of $D_{inite h}$ on 4-ary trees.

Computed the joint spectrum with respect to the left regular representation.

Abstract

We compute the joint spectrum of $D_{\infty h}$ with respect to the left regular representation, and finds two generators of the De Rham cohomology group of joint resolvent set which is induced by different central linear functionals. Through action of $D_{\infty h}$ on 4-ary trees, we get a self-similar realization of the group $C^*$ algebra of $D_{\infty h}$.