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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some exact sequences for Toeplitz algebras of spherical isometries


Author: Bebe Prunaru
Journal: Proc. Amer. Math. Soc. 135 (2007), 3621-3630
MSC (2000): Primary 47L80, 47B35; Secondary 47B20, 46L07
Published electronically: August 1, 2007
MathSciNet review: 2336578
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Abstract: A family $ \{T_{j}\}_{j\in J}$ of commuting bounded operators on a Hilbert space $ H$ is said to be a spherical isometry if $ \sum _{j\in J}T^{*}_{j}T_{j}=1$ in the weak operator topology. We show that every commuting family $ \mathcal{F}$ of spherical isometries is jointly subnormal, which means that it has a commuting normal extension $ \widehat {\mathcal{F}}$ on some Hilbert space $ \widehat {H}\supset H.$ Suppose now that the normal extension $ \widehat {\mathcal{F}}$ is minimal. Then we show that every bounded operator $ X$ in the commutant of $ \mathcal{F}$ has a unique norm preserving extension to an operator $ \widehat {X}$ in the commutant of $ \widehat {\mathcal{F}}.$ Moreover, if $ \mathcal{C}$ is the commutator ideal in $ C^{*}(\mathcal{F}),$ then $ C^{*}(\mathcal{F})/{\mathcal{C}}$ is *-isomorphic to $ C^{*}(\widehat {\mathcal{F}}).$ We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to $ \mathcal{F}.$ We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.


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Additional Information

Bebe Prunaru
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Bebe.Prunaru@imar.ro

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08893-4
PII: S 0002-9939(07)08893-4
Keywords: Toeplitz operators, Toeplitz algebras, spectral inclusion, spherical isometry, completely positive projection, injective operator spaces, strictly pseudoconvex domains
Received by editor(s): April 3, 2006
Received by editor(s) in revised form: August 22, 2006
Published electronically: August 1, 2007
Additional Notes: This research was partially supported by the Romanian Ministry of Education and Research, through the grant CEx05-D11-23/2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society