Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Regularity for operator algebras on a Hilbert space

Author: John Froelich
Journal: Proc. Amer. Math. Soc. 119 (1993), 1269-1277
MSC: Primary 47D25; Secondary 46L99
MathSciNet review: 1181164
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Four notions of regularity for operator algebras are introduced. An algebra $ A$ is called $ 1$-regular if for any two linearly independent vectors $ x,y \in H$ there is an $ a \in A$ such that $ ax = 0$ and $ ay \ne 0$. We show that the only weakly closed transitive $ 1$-regular algebra is $ B(H)$, thus providing a natural generalization of the Rickart-Yood density theorem. We construct an example of a $ 1$-regular algebra which contains no nonzero compact operators. This example is related to the "thin set" phenomena of classical harmonic analysis.

References [Enhancements On Off] (What's this?)

  • [1] W. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433-532. MR 0365167 (51:1420)
  • [2] B. Blackadar, $ K$-theory for operator algebras, Math. Sci. Res. Inst. Publ., vol. 5, Springer-Verlag, New York, 1986. MR 859867 (88g:46082)
  • [3] K. Davidson, Nest algebras, Pitman Res. Notes Math. Ser., vol. 191, Longman Sci. Tech., Harlow, 1988. MR 972978 (90f:47062)
  • [4] H. Dym and H. P. McKean, Fourier series and integrals, Academic Press, New York and London, 1972. MR 0442564 (56:945)
  • [5] J. Froelich, Compact operators, invariant subspaces and spectral synthesis, J. Funct. Anal. 81 (1988), 1-37. MR 967889 (90b:47078)
  • [6] C. C. Graham and O. C. McGehee, Essays in commutative harmonic analysis, Springer-Verlag, New York, 1979. MR 550606 (81d:43001)
  • [7] Y. Katznelson, An introduction to harmonic analysis, Wiley, 1968; reprinted by Dover. MR 0248482 (40:1734)
  • [8] S. Power, Analysis in nest algebras, Surveys of recent results in operator theory (J. Conway and B. Morrel, eds.), Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988. MR 976846 (90b:47079)
  • [9] H. Radjavi and P. Rosenthal, Invariant subspaces, Ergeb. Math. Grenzgeb. (3), vol. 77, Springer, Berlin and New York, 1973. MR 0367682 (51:3924)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47D25, 46L99

Retrieve articles in all journals with MSC: 47D25, 46L99

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society