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 Tr. Mat. Inst. Steklova, 2017, Volume 297, Pages 165–200 (Mi tm3797)

Joint spectrum and the infinite dihedral group

Rostislav Grigorchukab, Rongwei Yangc

a Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
c Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY 12222, USA

Abstract: For a tuple $A=(A_1,A_2,…,A_n)$ of elements in a unital Banach algebra $\mathcal B$, its projective joint spectrum $P(A)$ is the collection of $z\in\mathbb C^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+…+z_nA_n$ is not invertible. If $\mathcal B$ is the group $C^*$-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 $R=(1,a,t)$ for the infinite dihedral group $D_\infty=\langle a,t\mid a^2=t^2=1\rangle$ with respect to the left regular representation $\lambda_{D_\infty}$, and gives an in-depth analysis on its properties. A formula for the Fuglede–Kadison determinant of the pencil $R(z)=z_0+z_1a+z_2t$ is obtained, and it is used to compute the first singular homology group of the joint resolvent set $P^\mathrm c(R)$. 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 $C^*$-algebra $C^*(D_\infty)$. This self-similarity of $C^*(D_\infty)$ manifests itself in some dynamical properties of the joint spectrum.

 Funding Agency Grant Number NSA - National Security Agency H98230-15-1 Swiss National Science Foundation European Research Council AG COMPASP The first author is supported by the NSA grant H98230-15-1, the Swiss National Science Foundation, and ERC AG COMPASP.

DOI: https://doi.org/10.1134/S0371968517020091

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 297, 145–178

Bibliographic databases:

UDC: 517.986+517.984+512.547
MSC: Primary 47A13; Secondary 20E08, 20Cxx

Citation: Rostislav Grigorchuk, Rongwei Yang, “Joint spectrum and the infinite dihedral group”, Order and chaos in dynamical systems, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Dmitry Victorovich Anosov, Tr. Mat. Inst. Steklova, 297, MAIK Nauka/Interperiodica, Moscow, 2017, 165–200; Proc. Steklov Inst. Math., 297 (2017), 145–178

Citation in format AMSBIB
\Bibitem{GriYan17} \by Rostislav~Grigorchuk, Rongwei~Yang \paper Joint spectrum and the infinite dihedral group \inbook Order and chaos in dynamical systems \bookinfo Collected papers. On the occasion of the 125th anniversary of the birth of Academician Dmitry Victorovich Anosov \serial Tr. Mat. Inst. Steklova \yr 2017 \vol 297 \pages 165--200 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3797} \crossref{https://doi.org/10.1134/S0371968517020091} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3695412} \elib{http://elibrary.ru/item.asp?id=29859495} \transl \jour Proc. Steklov Inst. Math. \yr 2017 \vol 297 \pages 145--178 \crossref{https://doi.org/10.1134/S0081543817040095} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000410199700009} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85029169089} 

• http://mi.mathnet.ru/eng/tm3797
• https://doi.org/10.1134/S0371968517020091
• http://mi.mathnet.ru/eng/tm/v297/p165

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This publication is cited in the following articles:
1. He Wei, Wang Xiaofeng, Yang Rongwei, “Projective spectrum and kernel bundle. II”, J. Operator Theory, 78:2 (2017), 417–433
2. Hu Zhiguang, Yang Rongwei, “On the characteristic polynomials of multiparameter pencils”, Linear Algebra Appl., 558 (2018), 250–263
3. Hu Zhiguang, Zhang Philip B., “Determinants and characteristic polynomials of Lie algebras”, Linear Algebra Appl., 563 (2019), 426–439
4. B. Goldberg, R. Yang, “Hermitian Metric and the Infinite Dihedral Group”, Proc. Steklov Inst. Math., 304 (2019), 136–145
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