Some exact sequences for Toeplitz algebras of spherical isometries

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.