  A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to
be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator
topology. We show that every commuting family $\Cal F$ of spherical isometries
has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ 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 $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also
show that the space of Toeplitz operators associated to $\Cal F$ is completely
isometric to the commutant of the minimal normal extension $\hat{\Cal F}$.
Applications of these results are given for Toeplitz operators on strictly
pseudoconvex or bounded symmetric domains.
