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Joint spectrum and the infinite dihedral group

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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 ∈ ℂn 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 ∞) manifests itself in some dynamical properties of the joint spectrum.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Rostislav Grigorchuk

  2. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Rostislav Grigorchuk

  3. Department of Mathematics and Statistics, University at Albany, State University of New York, 1400 Washington Ave., Albany, NY, 12222, USA

    Rongwei Yang

Authors
  1. Rostislav Grigorchuk
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  2. Rongwei Yang
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Correspondence to Rostislav Grigorchuk.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 165–200.

To the blessed memory of Dmitry Victorovich Anosov

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Grigorchuk, R., Yang, R. Joint spectrum and the infinite dihedral group. Proc. Steklov Inst. Math. 297, 145–178 (2017). https://doi.org/10.1134/S0081543817040095

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