C * -subalgebras generated by partial isometries

Abstract

We prove a structure theorem for a finite set G of partial isometries in a fixed countably infinite dimensional complex Hilbert space H. Our result is stated in terms of the C*-algebra generated by G. The result is new even in the case of a single partial isometry which is not an isometry or a coisometry, and in this case, it extends the Wold decomposition for isometries. We give applications to groupoid C*-algebras generated by graph groupoids and to partial isometries which have finite defect indices and which parametrize the extensions of a fixed Hermitian symmetric operator with a dense domain on the Hilbert space H. Our classification parameters for our finite set G of partial isometries involve infinite and explicit Cartesian product sets, and they are computationally attractive. Moreover, our classification labels generalize the notion of defect indices in the special case of the family G of partial isometries from the Cayley transform theory and Hermitian extensions of unbounded Hermitian operators with dense domain.