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

Abstract

Abstract. We introduced a family of infinite graphs directly associated with a class of von Neumann automaton model A G . These are finite state models used in symbolic dynamics: stimuli models and in control theory. In the context of groupoid von Neumann algebras, and an associated fractal group, we prove a classification theorem for representations of automata.

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

“…Recall that, in this case, the graph groupoid G a , induced by the unitary a (on H a ) is groupoid-isomorphic to the infinite abelian cyclic group Z = a , without containing the empty word (also see [9,10,[12][13][14] and [11]). So, for y ∈ A q , α w (y) = wyw * = a n ya n * = a n ya −n = a n−1 (aya −1 )a −(n−1) = u n−1 (α a (y))u n−1 = · · · = α n u (y),…”

Directed Graphs, von Neumann Algebras, and Index

“…Our analysis depends on a number of technical points in the theory of discrete analysis and operator algebras. For this, the reader may find the following helpful: [3,4,8,[13][14][15].…”

Fractal Property in B(H) Induced by Partial Isometries

“…The groupoid W * -algebras generated by graph groupoids have been recently studied in [9], through [2,3], and [4]. Also, the groupoid C * -algebras have been studied in [5][6][7]11,15,16], and [17]: precisely, the groupoid C * -algebras, generated by graph groupoids, are studied in [6,7,15], and [16].…”

“…In the previous works, we have studied the fractality of algebraic structures induced by directed graphs, and then create operator algebras generated by the structures, under suitable representations (e.g., [1][2][3][4], and [5]). Reversely, in this paper, we will fix an operator algebra B(H ), and then characterize the "embedded" sub-structures of B(H ) induced by the fractality.…”