Some exact sequences for Toeplitz algebras of spherical isometries
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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.References
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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
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3621-3630
- MSC (2000): Primary 47L80, 47B35; Secondary 47B20, 46L07
- DOI: https://doi.org/10.1090/S0002-9939-07-08893-4
- MathSciNet review: 2336578