Some exact sequences for Toeplitz algebras of spherical isometries

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.