Joint spectrum and the infinite dihedral group

Rostislav Grigorchuk ; Rongwei Yang

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 165-200.

Résumé

For a tuple $A=(A_1,A_2,\dots,A_n)$ 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 $A(z)=z_1A_1+z_2A_2+\dots+z_nA_n$ is not invertible. If $\mathcal B$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,A_2,\dots,A_n$ 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.

Bibliographie

[1] Andersson M., Sjöstrand J., “Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula”, J. Funct. Anal., 210:2 (2004), 341–375 | DOI | MR | Zbl

[2] Arveson W., An invitation to $C^*$-algebras, Grad. Texts Math., 39, Springer, New York, 1981 | MR

[3] Atkinson F. V., Multiparameter eigenvalue problems, v. 1, Matrices and compact operators, Acad. Press, New York, 1972 | MR | Zbl

[4] Bannon J. P., Cade P., Yang R., “On the spectrum of Banach algebra-valued entire functions”, Ill. J. Math., 55:4 (2011), 1455–1465 | MR | Zbl

[5] Bartholdi L., Grigorchuk R. I., “On the spectrum of Hecke type operators related to some fractal groups”, Tr. MIAN, 231, 2000, 5–45 | MR | Zbl

[6] Bekka B., de la Harpe P., Valette A., Kazhdan's property (T), New Math. Monogr., 11, Cambridge Univ. Press, Cambridge, 2008 | MR

[7] Bondarenko I., Grigorchuk R., Kravchenko R., Muntyan Y., Nekrashevych V., Savchuk D., Šunić Z., “Groups generated by 3-state automata over a 2-letter alphabet. I”, São Paulo J. Math. Sci., 1:1 (2007), 1–39 | DOI | MR | Zbl

[8] Cade P., Yang R., “Projective spectrum and cyclic cohomology”, J. Funct. Anal., 265:9 (2013), 1916–1933 | DOI | MR | Zbl

[9] Chagouel I., Stessin M., Zhu K., “Geometric spectral theory for compact operators”, Trans. Amer. Soc., 368:3 (2016), 1559–1582 | DOI | MR | Zbl

[10] Davidson K. R., $C^*$-algebras by example, Fields Inst. Monogr., 6, Amer. Math. Soc., Providence, RI, 1996 | MR

[11] Deninger C., “Mahler measures and Fuglede–Kadison determinants”, Münster J. Math., 2 (2009), 45–64 | MR | Zbl

[12] Dixmier J., Les $C^*$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1969 | MR

[13] Douglas R.G., Yang R., Hermitian geometry on resolvent set, E-print, 2016, arXiv: 1608.05990[math.FA]

[14] Everest G., Ward T., Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer, London, 1999 | DOI | MR | Zbl

[15] Fainshtein A. S., “Taylor joint spectrum for families of operators generating nilpotent Lie algebras”, J. Oper. Theory, 29:1 (1993), 3–27 | MR | Zbl

[16] Fuglede B., Kadison R. V., “Determinant theory in finite factors”, Ann. Math. Ser. 2, 55 (1952), 520–530 | DOI | MR | Zbl

[17] Grigorchuk R. I., “K probleme Bernsaida o periodicheskikh gruppakh”, Funkts. analiz i ego pril., 14:1 (1980), 53–54 | MR | Zbl

[18] Grigorchuk R. I., “Stepeni rosta konechno-porozhdennykh grupp i teoriya invariantnykh srednikh”, Izv. AN SSSR. Ser. mat., 48:5 (1984), 939–985 | MR | Zbl

[19] Grigorchuk R. I., “O ryade Gilberta–Puankare graduirovannykh algebr, assotsiirovannykh s gruppami”, Mat. sb., 180:2 (1989), 207–225 | MR | Zbl

[20] Grigorchuk R., “Solved and unsolved problems around one group”, Infinite groups: Geometric, combinatorial and dynamical aspects, Prog. Math., 248, Birkhäuser, Basel, 2005, 117–218 | DOI | MR | Zbl

[21] Grigorchuk R., Nekrashevych V., “Self-similar groups, operator algebras and Schur complement”, J. Mod. Dyn., 1:3 (2007), 323–370 | DOI | MR | Zbl

[22] Grigorchuk R. I., Nekrashevich V. V., Suschanskii V. I., “Avtomaty, dinamicheskie sistemy i gruppy”, Tr. MIAN, 231, 2000, 134–214 | MR | Zbl

[23] Grigorchuk R., Šunić Z., “Schreier spectrum of the Hanoi Towers group on three pegs”, Analysis on graphs and its applications, Proc. Symp. Pure Math., 77, Amer. Math. Soc., Providence, RI, 2008, 183–198 | DOI | MR | Zbl

[24] Halmos P. R., “Two subspaces”, Trans. Amer. Math. Soc., 144 (1969), 381–389 | DOI | MR | Zbl

[25] de la Harpe P., “Fuglede–Kadison determinant: theme and variations”, Proc. Natl. Acad. Sci. USA, 110:40 (2013), 15864–15877 | DOI | MR

[26] de la Harpe P., Skandalis G., “Déterminant associé à une trace sur une algèbre de Banach”, Ann. Inst. Fourier, 34:1 (1984), 241–260 | DOI | MR | Zbl

[27] Harte R. E., “The spectral mapping theorem for quasicommuting systems”, Proc. R. Ir. Acad. A, 73 (1973), 7–18 | MR | Zbl

[28] He W., Yang R., “Projective spectrum and kernel bundle”, Sci. China Math., 58:11 (2015), 2363–2372 | DOI | MR | Zbl

[29] Hörmander L., An introduction to complex analysis in several variables, 3rd ed., North-Holland, Amsterdam, 1990 | MR | Zbl

[30] Li H., “Compact group automorphisms, addition formulas and Fuglede–Kadison determinants”, Ann. Math. Ser. 2, 176:1 (2012), 303–347 | DOI | MR | Zbl

[31] Lu T.-T., Shiou S.-H., “Inverses of $2\times2$ block matrices”, Comput. Math. Appl., 43:1–2 (2002), 119–129 | MR | Zbl

[32] Nekrashevych V., Self-similar groups, Math. Surv. Monogr., 117, Amer. Math. Soc., Providence, RI, 2005 | DOI | MR | Zbl

[33] Nekrashevych V., Periodic groups from minimal actions of the infinite dihedral group, E-print, 2016, arXiv: 1601.01033v1[math.GR]

[34] Pedersen G. K., “Measure theory for $C^*$ algebras. II”, Math. Scand., 22 (1968), 63–74 | DOI | MR | Zbl

[35] Putnam I., Lecture notes on $C^*$-algebras, Preprint, , Univ. Victoria, 2016 http://www.math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf

[36] Raeburn I., Sinclair A. M., “The $C^*$-algebra generated by two projections”, Math. Scand., 65 (1989), 278–290 | DOI | MR | Zbl

[37] Rudin W., Real and complex analysis, McGraw-Hill, New York, 1987 | MR | Zbl

[38] Schmidt K., Dynamical systems of algebraic origin, Birkhäuser, Basel, 1995 | MR | Zbl

[39] Sleeman B. D., Multiparameter spectral theory in Hilbert space, Res. Notes Math., 22, Pitman, London, 1978 | MR | Zbl

[40] Stessin M., Yang R., Zhu K., “Analyticity of a joint spectrum and a multivariable analytic Fredholm theorem”, New York J. Math., 17a (2011), 39–44 | MR | Zbl

[41] Taylor J. L., “A joint spectrum for several commuting operators”, J. Funct. Anal., 6 (1970), 172–191 | DOI | MR | Zbl

[42] Taylor J. L., “A general framework for a multi-operator functional calculus”, Adv. Math., 9 (1972), 183–252 | DOI | MR | Zbl

[43] Vinnikov V., “Determinantal representations of algebraic curves”, Linear algebra in signals, systems and control, Proc. SIAM Conf. (Boston. MA, 1986), SIAM, Philadelphia, PA, 1988, 73–99 | MR | Zbl

[44] Yang R., “Projective spectrum in Banach algebras”, J. Topol. Anal., 1:3 (2009), 289–306 | DOI | MR | Zbl

[45] Zaidenberg M. G., Krein S. G., Kuchment P. A., Pankov A. A., “Banakhovy rassloeniya i lineinye operatory”, UMN, 30:5 (1975), 101–157 | MR | Zbl