Joint spectrum and the infinite dihedral group

Abstract

For a tuple A = (A 1 , A 2 , ..., A n ) of elements in a unital Banach algebra B, its projective joint spectrum P (A) is the collection of z ∈ C n such that the multiparameter pencil A(z) = z 1 A 1 + z 2 A 2 + · · · + z n A n is not invertible.If B is the group C * -algebra for a discrete group G generated by A 1 , A 2 , ..., 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 | a 2 = t 2 = 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) = z 0 + 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 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 ∞ ) is manifested by some dynamical properties of the joint spectrum.

Hermitian Geometry on Resolvent Set

“…The Hilbert space l 2 (D ∞ ) then can be written as L ⊕ tL, where L = l 2 (H). It was shown in [22] that, up to unitary equivalence, we have the following decomposition:…”

“…For the tuple R = (I, λ(a), λ(t)), we let R * (z) = I + z 1 λ(a) + z 2 λ(t) and P (R * ) be the set of z ∈ C 2 such that R * (z) is not invertible in B D . Then by [22] we have 4) and the Fuglede-…”

“…Let λ be the left regular representation of It was shown in [22] that, up to unitary equivalence, we have the following decomposition:…”

“…where I is the identity, and the tuple (1, a, t) for the infinite dihedral group D ∞ =< a, t | a 2 = t 2 = 1 > with respect to left regular representation ( [22]).…”

Characteristic polynomials and finitely dimensional representations of $\mathfrak{sl}(2, \mathbb{C})$

“…In [12], Yang defines the projective spectrums through the multiparameters pencil z 1 A 1 + • • • + z m A m , and there are many fruitful results in [1], [5] and [6]. For a representation π of a finite group G = {g i } n i=1 , the characteristic polynomial…”