Joint spectrum and the infinite dihedral group

Abstract

For a tuple A = (A₁,A₂, ... ,A_n) of elements in a unital Banach algebra B, its projective joint spectrum P(A) is the collection of z ∈ ℂⁿ such that the multiparameter pencil A(z) = z₁A₁ + z₂A₂ + ... + z_nA_n is not invertible. If B is the group C*-algebra for a discrete group G generated by A₁,A₂, ... ,A_n with respect to a representation ρ, then P(A) is an invariant of (weak) equivalence for ρ. This paper computes the joint spectrum of R = (1, a, t) for the infinite dihedral group D_∞ = 〈a, t | a² = t² = 1〉 with respect to the left regular representation λ_D∞, and gives an in-depth analysis on its properties. A formula for the Fuglede–Kadison determinant of the pencil R(z) = z₀ + z₁a + z₂t 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 ρ (constructed through a self-similar action of D_∞ 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*-algebra C*(D_∞). This self-similarity of C*(D_∞) manifests itself in some dynamical properties of the joint spectrum.