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

Abstract

In this paper, we characterize a C * -subalgebra C * (x) of B(H ), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C * (x) is * -isomorphic to the groupoid crossed product algebra A q × α G a , where A q = C * (q) and G a is the graph groupoid induced by a partial isometry part a of x.

Directed Graphs, von Neumann Algebras, and Index

“…In [1] through [5], we showed that every directed graph G induces the corresponding algebraic structure G, called the graph groupoid of G (See Section 2 below). The graph groupoid G is indeed a categorial groupoid.…”

Free probability induced by electric resistance networks on energy Hilbert spaces

“…By considering groupoid elements as multiplication operators on certain Hilbert spaces, they become natural groupoid actions on Hilbert spaces. Such groupoid actions induce groupoid dynamical systems (e.g., [3,4,4,6], and [5]). …”

“…The radial operators in the "canonical" graph-groupoidal settings have been studied in [3,4], and [6].…”

“…, n} mean the partitions without crossings. For example, π = {(1, 2, 5), (3,4), (6, 7)} is a noncrossing partition in N C (7), however θ = {(1, 2, 5), (3,4,6), (7)} is a "crossing" partition, because the blocks (1, 2, 5) and (3, 4, 6) of the partition θ are crossing from each other. Let N C(n) be the collection of all noncrossing partitions over {1, .…”

“…Then we define the size |π| of π by the cardinality of the set of blocks in π. For example, if π = {(1, 3, 6, 7), (2), (4, 5)}, with blocks (1, 3, 6, 7), (2), and (4,5), in N C(7), then |π| = 3.…”