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On the Ranges of Bimodule Projections

Published online by Cambridge University Press:  20 November 2018

Aristides Katavolos  and
Vern I. Paulsen
Aristides Katavolos
Affiliation:
Department of Mathematics, University of Athens, Athens, Greece e-mail: [email protected]
Vern I. Paulsen
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3476, U.S.A. e-mail: [email protected]
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Abstract

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We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of ${{\text{J}}^{*}}$-algebras. We prove that if $P$ is a normal bimodule idempotent and $\left\| P \right\|\,<\,2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.

Keywords

46L1547L25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 97 - 111
Copyright
Copyright © Canadian Mathematical Society 2005

References

[Arv] Arveson, W. B., Operator algebras and invariant subspaces. Ann. Math. 100(1974), 433–532.Google Scholar
[BCD] Bhatia, R., Choi, M. D. and Davis, C., Comparing a matrix to its off-diagonal part. Oper. Theory Adv. Appl. 40(1989), 151–164.Google Scholar
[Dav] Davidson, Kenneth R.. Nest algebras. Triangular forms for operator algebras on Hilbert space. Pitman Research Notes in Mathematics Series, 191, John Wiley and Sons, New York, 1988.Google Scholar
[DP] Davidson, K. R. and Power, S. C., Isometric automorphisms and homology for nonselfadjoint operator algebras. Quart. J. Math. 42(1991), 271–292.Google Scholar
[EKS] Erdos, J. A., Katavolos, A. and Shulman, V. S., Rank one subspaces of bimodules over maximal abelian selfadjoint algebras. J. Funct. Anal. 157(1998), 554–587.Google Scholar
[Haa] Haagerup, Uffe, Decomposition of completely bounded maps on operator algebras, Unpublished manuscript.Google Scholar
[HW] Hofmeier, H. and Wittstock, G., A bicommutant theorem for completely bounded module homomorphisms. Math. Ann. 308(1997), 141–154.Google Scholar
[KS] Kissin, E. and Shulman, V. S., work in progress (private communication).Google Scholar
[Li] Livshits, L., A note on 0-1 Schur multipliers. Lin. Alg. Appl. 222(1995), 15–22.Google Scholar
[Pa] Paulsen, V. I., Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, 78, Cambridge University Press, Cambridge, 2003.Google Scholar
[Pe] Peller, V. V., Hankel operators in the theory of perturbations of unitary and selfadjoint operators. (Russian) Funktsional. Anal. i Prilozhen. 19(1985), 37–51. English translation: Functional Anal. Appl. 19(1985), 111–123.Google Scholar
[Smi] Smith, R. R., Completely bounded module maps and the Haagerup tensor product. J. Funct. Anal. 102(1991), 156–175.Google Scholar
[So] Solel, B., Contractive projections onto bimodules of von Neumann algebras. J. London Math. Soc. 45(1992), 169–179.Google Scholar
[Tak] Takesaki, Masamichi. Theory of operator algebras. I. Springer-Verlag, New York, 1979.Google Scholar
[W] Wittstock, G., Extensions of completely bounded C*-module morphisms. Monogr. Stud. Math. 18, 1984.Google Scholar