Some Exact Sequences for Toeplitz Algebras of Spherical Isometries

A family $\{T_j{\}}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if ${\sum }_{j\in J}{T}_j^{\ast }{T}_j=1$ in the weak operator topology. We show that every commuting family $\text{Cal}F$ of spherical isometries has a commuting normal extension $\widehat{\text{Cal}F}$. Moreover, if $\widehat{\text{Cal}F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^(\Cal F)\to C^(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\text{Cal}C$ is the commutator ideal in $C^{\ast }(\text{Cal}F)$. We also show that the space of Toeplitz operators associated to $\text{Cal}F$ is completely isometric to the commutant of the minimal normal extension $\widehat{\text{Cal}F}$. Applications of these results are given for Toeplitz operators on strictly pseudoconvex or bounded symmetric domains.