Joint spectrum and infinite dihedral group

Abstract

For a tuple $A=(A_1,A_2,...,A_n)$ of elements in a unital Banach algebra $\mathcal{B}$, its {\em projective joint spectrum} $P(A)$ is the collection of $z\in {\mathbb{C}}^n$ such that the multiparameter pencil $A(z)=z_1{A}_1+z_2{A}_2+\cdots +z_n{A}_n$ is not invertible. If $\mathcal{B}$ is the group $C^{\ast }$-algebra for a discrete group $G$ generated by $A_1,A_2,...,A_n$ with respect to a representation $\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\rho$. This paper computes the joint spectrum of $(1,a,t)$ for the infinite dihedral group $D_{\infty }=<a,t|a^2=t^2=1>$ with respect to the left regular representation ${\lambda }_D$, and gives an in-depth analysis on its properties. A formula for the Fuglede-Kadison determinant of the pencil $R(z)=1+z_{1}a+z_{2}t$ is obtained, and it is used to compute the first singular homology group of the joint resolvent set $P^c(R)$. The joint spectrum gives new insight into some earlier studies on groups of intermediate growth, through which the corresponding joint spectrum of $(1,a,t)$ with respect to the Koopman representation $\rho$ (constructed through a self-similar action of $D_{\infty }$ on a binary tree) can be computed. It turns out that the joint spectra with respect to the two representations coincide. Interestingly, this fact leads to a self-similar realization of the group $C^{\ast }$-algebra $C^{\ast }(D_{\infty })$. This self-similarity of $C^{\ast }(D_{\infty })$ is manifested by some dynamical properties of the joint spectrum.

58 pages

References in corpus (1)

• On the Spectrum of Hecke Type Operators related to some Fractal Groups