Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-20T06:39:33.057Z Has data issue: false hasContentIssue false
  • English
  • Français

Operator algebras from the discrete Heisenberg semigroup

Part of: Abstract harmonic analysis Linear spaces and algebras of operators

Published online by Cambridge University Press:  08 November 2011

M. Anoussis ,
A. Katavolos  and
I. G. Todorov
M. Anoussis
Affiliation:
Department of Mathematics, University of the Aegean, Karlovassi, 83200 Samos, Greece
A. Katavolos
Affiliation:
Department of Mathematics, University of Athens, Panepistimioupolis, 15784 Athens, Greece
I. G. Todorov
Affiliation:
Department of Pure Mathematics, Queen's University Belfast, University Road, Belfast BT7 1NN, UK
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study reflexivity and structural properties of operator algebras generated by representations of the discrete Heisenberg semigroup. We show that the left regular representation of this semigroup gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation that gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of .

Keywords

non-self-adjoint operator algebrareflexivityslice mapdiscrete Heisenberg semigroup

MSC classification

Secondary: 47L75: Other nonselfadjoint operator algebras 43A65: Representations of groups, semigroups, etc. 47L99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 1 - 22
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Arias, A. and Popescu, G., Factorization and reflexivity on Fock spaces, Integ. Eqns Operat. Theory 23 (1995), 268–286.CrossRefGoogle Scholar
2.Arveson, W., Operator algebras and invariant subspaces, Annals Math. 100(2) (1974), 433–532.CrossRefGoogle Scholar
3.Azoff, E. A., Fong, C. K. and Gilfeather, F., A reduction theory for non-self-adjoint operator algebras, Trans. Am. Math. Soc. 224 (1976), 351–366.CrossRefGoogle Scholar
4.Brenken, B., Representations and automorphisms of the irrational rotation algebras, Pac. J. Math. 111 (1984), 257–282.CrossRefGoogle Scholar
5.Conway, J. B., A course in operator theory, Graduate Studies in Mathematics, Volume 21 (American Mathematical Society, Providence, RI, 2000).Google Scholar
6.Davidson, K. R., C*-algebras by example, Fields Institute Monographs (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
7.Davidson, K. R., Katsoulis, E. and Pitts, D. R., The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125.Google Scholar
8.Davidson, K. R. and Pitts, D. R., The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Annalen 311 (1998), 275–303.CrossRefGoogle Scholar
9.Davidson, K. R. and Pitts, D. R., Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. Lond. Math. Soc. 78 (1999), 401–430.CrossRefGoogle Scholar
10.Hadwin, D., A general view on reflexivity, Trans. Am. Math. Soc. 344 (1994), 325–360.CrossRefGoogle Scholar
11.Hasegawa, A., The invariant subspace structure of L 2( for certain von Neumann algebras, Hokkaido Math. J. 35 (2006), 601–611.CrossRefGoogle Scholar
12.Kakariadis, E., Semicrossed products and reflexivity, J. Operator Theory, in press.Google Scholar
13.Katavolos, A. and Power, S. C., The Fourier binest algebra, Math. Proc. Camb. Phil. Soc. 122 (1997), 525–539.CrossRefGoogle Scholar
14.Katavolos, A. and Power, S. C., Translation and dilation invariant subspaces of L 2(ℝ), J. Reine Angew. Math. 552 (2002), 101–129.Google Scholar
15.Kraus, J., The slice map problem for σ-weakly closed subspaces of von Neumann algebras, Trans. Am. Math. Soc. 279 (1983), 357–376.Google Scholar
16.Levene, R. H., A double triangle operator algebra from SL2(ℝ+), Can. Math. Bull. 49 (2006), 117–126.CrossRefGoogle Scholar
17.Levene, R. H. and Power, S. C., Reflexivity of the translation-dilation algebras on L 2(ℝ), Int. J. Math. 14 (2003), 1081–1090.CrossRefGoogle Scholar
18.Loginov, A. I. and Shul'man, V. S., Hereditary and intermediate reflexivity of W*-algebras, Izv. Akad. Nauk SSSR 39 (1975), 1260–1273.Google Scholar
19.Popescu, G., Von Neumann inequality for (B(H)n)1, Math. Scand. 68 (1991), 292–304.CrossRefGoogle Scholar
20.Popescu, G., A generalization of Beurling's Theoerm and a class of reflexive algebras, J. Operat. Theory 41 (1999), 391–420.Google Scholar
21.Ptak, M., On the reflexivity of pairs of isometries and of tensor products for some reflexive algebras, Studia Math. 37 (1986), 47–55.Google Scholar
22.Sarason, D., Invariant subspaces and unstarred operator algebras, Pac. J. Math. 17 (1966), 511–517.CrossRefGoogle Scholar
23.Tomiyama, J., Tensor products and projections of norm one in von Neumann algebras, Lecture Notes, University of Copenhagen (1970).Google Scholar
24.Wermer, J., On invariant subspaces of normal operators, Proc. Am. Math. Soc. 3 (1952). 270–277.CrossRefGoogle Scholar