zbMATH — the first resource for mathematics

Group representations with empty residual spectrum. (English) Zbl 1220.47005
Integral Equations Oper. Theory 67, No. 1, 95-107 (2010); erratum ibid. 69, No. 1, 149-150 (2011).
Let \(\Gamma\) be a discrete group acting on a Banach space \(X\). Elements of the group algebra \(\mathbb{C}\Gamma\) are therefore represented as bounded linear operators in \(X\). The main result in this paper proves that, if \(X\) is either the reduced group C*-algebra \(C^*_r(\Gamma)\) or a noncommutative \(L^p(M)\) (where \(M\) is the group von Neumann algebra of \(\Gamma\), with respect to the natural trace in \(M\)), with \(\Gamma\) acting on \(X\) by left translation, then the spectrum of every operator \(T\) in \(\mathbb{C}\Gamma\) consists of approximate eigenvalues.

47A10 Spectrum, resolvent
47C10 Linear operators in \({}^*\)-algebras
47C15 Linear operators in \(C^*\)- or von Neumann algebras
Full Text: DOI arXiv
[1] Choi M.D.: The full C*-algebra of the free group on two generators. Pac. J. Math. 87(1), 41–48 (1980) · Zbl 0463.46047
[2] Choi, Y.: Injective convolution operators on {\(\Gamma\)}) are surjective. Can. Math. Bull. (in press) http://arXiv.org/abs/math.FA/0606367 · Zbl 1211.43001
[3] Curtis P.C. Jr, Loy R.J.: The structure of amenable Banach algebras. J. Lond. Math. Soc. (2) 40(1), 89–104 (1989) · Zbl 0698.46043
[4] Dixmier J.: Formes linéaires sur un anneau d’opérateurs. Bull. Soc. Math. France 81, 9–39 (1953) · Zbl 0050.11501
[5] Folland G.B.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995) · Zbl 0857.43001
[6] Gottschalk, W.: Some general dynamical notions. In: Beck, A.(ed.) Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., vol. 318, pp. 120–125. Springer, Berlin (1973)
[7] Halperin I.: On a theorem of Sterling Berberian. C. R. Math. Rep. Acad. Sci. Can. 3(1), 33–35 (1981) · Zbl 0448.47001
[8] Herz C.: The theory of p-spaces with an application to convolution operators. Trans. Am. Math. Soc. 154, 69–82 (1971) · Zbl 0216.15606
[9] Herz C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23(3), 91–123 (1973) · Zbl 0257.43007
[10] Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Pure and Applied Mathematics, vol. 100. Academic Press, Orlando (1986) · Zbl 0601.46054
[11] Laursen K.B., Neumann M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series, vol. 20. The Clarendon Press Oxford University Press, New York (2000) · Zbl 0957.47004
[12] Montgomery M.S.: Left and right inverses in group algebras. Bull. Am. Math. Soc. 75, 539–540 (1969) · Zbl 0174.31204
[13] Pisier, G., Xu, Q.: Non-commutative L p -spaces. In: Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003) · Zbl 1046.46048
[14] Rudin W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, New York (1991) · Zbl 0867.46001
[15] Runde V.: Local spectral properties of convolution operators on non-abelian groups. Proc. Edinb. Math. Soc. (2) 39(1), 237–250 (1996) · Zbl 0853.43009
[16] Willis G.A.: Translation invariant functionals on L p (G) when G is not amen-able. J. Aust. Math. Soc. Ser. A 41(2), 237–250 (1986) · Zbl 0611.43001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.