Some exact sequences for Toeplitz algebras of spherical isometries

Abstract

Abstract. A family {T j } j∈J of commuting bounded operators on a Hilbert space H is said to be a spherical isometry if j∈J T * j T j = 1 in the weak operator topology. We show that every commuting family F of spherical isometries is jointly subnormal, which means that it has a commuting normal extension F on some Hilbert space H ⊃ H. Suppose now that the normal extension F is minimal. Then we show that every bounded operator X in the commutant of F has a unique norm preserving extension to an operator X in the commutant of F . Moreover, if C is the commutator ideal in C * (F ), then C * (F )/C is *-isomorphic to C * ( 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 F . We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.

Inner functions and spherical isometries

“…Generalizing this algebraic condition, Prunaru [14] defines the set of all Toeplitz operators with respect to a spherical isometry T ∈ B(H) n as…”

Toeplitz operators associated to commuting row contractions

“…The following result was proved in [20] for the case of unital mappings, however one can easily see that the proof works as well for non-unital mappings. It is a quite straightforward application of Theorem 2.3.…”

“…A similar approach was used in [20] to study spherical isometries. A particular case of the results of the present paper is an alternate proof of the main result from [5] that every spherical isometry is jointly subnormal.…”

Approximate Factorization in Generalized Hardy Spaces

“…We also show in this section that the dual algebra generated by a spherical isometry with a possibly infinite number of components has the property (A 1 (1)). The proof is based on results obtained in the previous sections together with those obtained in [50]. The corresponding result for spherical isometries with a finite number of terms was proved in [26].…”

“…The following result, which is a particular case of a more general theorem from [50] will be used in the proof of Theorem 4.7 below. Here P H denotes the orthogonal projection of K onto H. In particular the above mapping induces a dual algebras isomorphism between the dual algebra A N generated by {N n } n≥1 and the dual algebra A T generated by {T n } n≥1 .…”