C *-algebras generated by partial isometries

Abstract

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 co-isometry; 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 symetric operator with dense domain on the Hilbert space H. Our classification parameters for the "Wold decomposed" set G W of 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 W of partial isometries from the Cayley transform theory and Hermitian extensions of unbounded Hermitian operators with dense domain.

C *-Subalgebras Generated by a Single Operator in B(H)

“…In particular, by the groupoidal properties, if w is a reduced finite path, then θ(w) is a partial isometry on H 0 (and hence on H ), and if w is a vertex, then θ(w) is a projection on H 0 (and hence on H ). This connection of graph groupoidal elements and operators on H 0 are considered in [8][9][10], and [12]. Under the measure theoretical setting, similar observation was done in [7].…”

“…In [7,9,10] and [12], we showed the connection from directed graphs to Hilbert space operators. In particular, we can match each edge and each vertex of a given directed graph to a partial isometry and a projection, respectively.…”

“…By using graph-groupoid technique, we construct other types of graph-depending operator algebras: each edge of a directed graph induces a partial isometry on a certain Hilbert space under the canonical matricial representation (see [12]). (In fact, the similar connection of graphs and partial isometries are found, whenever we study any kinds of graph-depending operator algebras.)…”

Markov measures and extended zeta functions

“…To make our paper more accessible, we offer below a few pointers to the relevant literature. Readers familiar with one of these areas, but perhaps not the others, may wish to check the following references covering aspects of these areas used below: operator algebras, Cuntz algebras and their representations ( [13], [4], [10], [8], [16], [31], [22], [24], [25], [26], [40]) ; multiresolutions and their diverse uses ( [1], [12], [3], [23], [2], [14] , [27] ) ; Zeta functions ( [6], [39], [20], [38], [37], [36]); and Markov measures ( [32], [33], [21], [5], [51], [52], [7], [9]). We further use results from harmonic analysis, such as ( [19], [29], [15], [17], [18], [49]).…”

Applications of automata and graphs: Labeling operators in Hilbert space. II.

“…In [10], [11], [13], [14] and [15], we introduced graph groupoids induced by countable directed graphs. A graph groupoid is a categorial groupoid having as a base the set of all vertices of the given graph.…”