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Completeness of eigenvectors of group
representations of operators whose
Arveson spectrum is scattered

Author: Sen-Zhong Huang
Journal: Proc. Amer. Math. Soc. 127 (1999), 1473-1482
MSC (1991): Primary 47A67, 47A10
Published electronically: January 29, 1999
MathSciNet review: 1621945
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish the following result.

Theorem. Let $\alpha:G\to {\mathcal L}(X)$ be a $\sigma(X,X_*)$ integrable bounded group representation whose Arveson spectrum $\operatorname{Sp}(\alpha)$ is scattered. Then the subspace generated by all eigenvectors of the dual representation $\alpha^*$ is $w^*$ dense in $X^*.$ Moreover, the $\sigma(X,X_*)$ closed subalgebra $W_\alpha$ generated by the operators $\alpha _t$ ($t\in G$) is semisimple.

If, in addition, $X$ does not contain any copy of $c_0,$ then the subspace spanned by all eigenvectors of $\alpha$ is $\sigma(X,X_*)$ dense in $X.$ Hence, the representation $\alpha$ is almost periodic whenever it is strongly continuous.

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  • 1. ARVESON, W., On groups of automorphisms of operator algebras. J. Funct. Anal. 15 (1974), 217-243. MR 50:1016
  • 2. BASKAKOV, A.G., Spectral criteria for almost periodicity of solutions of functional equations. Math. Notes of Acad. Sci. USSR 24 (1978), 606-612.
  • 3. BRATTELI, O. AND ROBINSON, D.W., Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, New York-Heidelberg-Berlin (1979). MR 81a:46070
  • 4. CONNES, A., Une classification des facteurs de type III. Ann. Sci. l'École Norm. Sup. 6 (1973), 133-245. MR 49:5865
  • 5. COROJOAR, I. AND FOIA, C., Theory of Generalized Spectral Operators. Gordon and Breach, New York, 1968. MR 52:15085
  • 6. DELEEUW, K. AND GLICKSBERG, L., Applications of almost periodic compactifications. Acta Math. 105 (1961), 63-97. MR 24:A1632
  • 7. D'ANTONI, C., LONGO, C., AND ZSIDÓ, L., A spectral mapping theorem for locally compact groups of operators. Pacific J. Math. 103 (1982), 17-24. MR 84e:47058
  • 8. DUGUNDJI, J., Topology, 4th. ed. Allyn and Bacon Inc., Boston, 1968.
  • 9. FELDMAN, G.M., The semisimplicity of an algebra generated by an isometric operator, Funct. Anal. Appl. 8(1974), 93-94. MR 50:14245
  • 10. GELFAND, I.M., Ideale und primäre Ideale in normierten Ringen. Mat. Sb. 9 (1941), 41-47.
  • 11. HEWITT, E. AND ROSS, R., Abstract Harmonic Analysis I. Springer-Verlag, Berlin-Heidelberg-New York, 1963. MR 28:158
  • 12. HILLE, E. AND PHILLIPS, R. S., Functional Analysis and Semi-Groups, 3rd ed. Rhode Island, Amer. Math. Soc. Colloq. Publ. XXXI, 1968. MR 54:11077
  • 13. HUANG, S.-Z., Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems. Studia Math. 116 (1995), 23-41. MR 96i:47068
  • 14. HUANG, S.-Z., Spectral Theory for Non-Quasianalytic Representations of Locally Compact Abelian Groups. Thesis, Universität Tübingen, 1996. A complete summary is given in ``Dissertation Summaries in Mathematics'' 1 (1996), 171-178.
  • 15. HUANG, S.-Z., VAN NEERVEN, J. AND RÄBIGER, F., Ditkin's condition for certain Beurling algebras. Proc. Amer. Math. Soc. 126 (1998), 1397-1407. CMP 97:11
  • 16. KRENGEL, U., Ergodic Theorems. de Gruyter, Berlin, New York (1985). MR 87i:28001
  • 17. LOOMIS, L. H., The spectral characterization of a class of almost periodic functions. Ann. Math. 72 (1960), 362-368. MR 22:11255
  • 18. LYUBICH, YU. I., Introduction to the Theory of Banach Representations of Groups. Birkhäuser-Verlag, Basel (1988). MR 90i:22001
  • 19. LEVITAN, B.M. AND ZHIKOV, V.V., Almost Periodic Functions and Differential Equations. Cambridge Univ. Press, Cambridge (1982). MR 84g:34004
  • 20. MURAZ, G. AND V\~{U}, QUÔC PHÓNG, Semisimple Banach algebras generated by strongly continuous representations of locally compact abelian groups. J. Funct. Anal. 126 (1994), 1-6. MR 95k:43006
  • 21. RUDIN, W., Fourier Analysis on Groups. Interscience, New York, 1962. MR 27:2808
  • 22. SINCLAIR, A.M., The Banach algebra generated by a hermitian operator, Proc. London Math. Soc. 24 (1972), 681-691. MR 46:4198

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Additional Information

Sen-Zhong Huang
Affiliation: Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany
Address at time of publication: Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, 18055 Rostock, Germany

Keywords: Spectrum of group representation, almost periodicity
Received by editor(s): September 1, 1997
Published electronically: January 29, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

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