Graph von Neumann Algebras

Abstract

In this paper, we will construct a graph von Neumann algebra M G = M × α G over a fixed von Neumann algebra M, as a crossed product algebra of M and a graph groupoid G induced by a given countable directed graph G, and we will observe 96 I. Cho the amalgamated free probabilistic properties of M G . The construction of such crossed product algebras is motivated by the Exel's crossed product construction. Here, the pair (α w , α −1 w ) is an intertwining analogue of an Exel's interaction, for all w ∈ G. Recently, graph algebras are studied by various operator algebraists and indeed they are very important topics in Operator Algebra because they provides the way to understand operator algebraic objects (for instance, partial isometries and projections) visually. The construction of graph algebras is based on the construction of Cuntz-Kieger algebras. After fixing a Cuntz-Kieger family, we can construct the (topological) operator algebra generated by the family. So, it is natural to restrict our interests to the case when we have "good" graphs satisfying the CuntzKieger relation. The construction of graph von Neumann algebras is free from this restriction. In our construction, we can use any arbitrary countable directed graphs since the construction of graph groupoids is quite canonical. Moreover, by using the (groupoid) crossed product technique, we can study not only a graph algebrawhere v N α (X 1 , X 2 ) means a von Neumann algebra generated by X 1 and X 2 satisfying the relation depending on an action (a partial representation) α of G. Throughout this paper, we concentrate on observing the amalgamated free probabilistic properties of a graph von Neumann algebra M G . This means that we will consider each element of M G as an operator determined by a certain (reduced) word depending on a new algebraic structure G induced by G.

C *-algebras generated by partial isometries

“…The various equivalent characterizations of partial isometries are well-known: the operator a is a partial isometry if and only if a = aa * a, if and only if its adjoint a * is a partial isometry, too, in B(H [5], [6], [9], [10], [17], [19], [24], [35] and [36]). Not only are they connected with certain noncommutative structures but they also let us visualize such structures.…”

Free probability induced by electric resistance networks on energy Hilbert spaces

“…As groupoids, our graph groupoids can have their groupoid actions. Canonical actions induced by graphs are introduced in [1,2], and [3]. By considering groupoid elements as multiplication operators on certain Hilbert spaces, they become natural groupoid actions on Hilbert spaces.…”

“…In this section, we re-define electric resistance networks to apply our graph-groupoidal research (e.g., [1] through [5]). i.e., we construct graph groupoids induced by electric resistance networks and study certain operator algebras generated by electric resistance networks.…”

“…In [1] and [2], we showed that if two graphs G 1 and G 2 have graph-isomorphic shadowed graphs G 1 and G 2 , then the corresponding graph groupoids G 1 and G 2 are groupoid-isomorphic.…”

“…Independently, the first named author initiated an approach to graph groupoids based on free probability theory and representations (e.g., see [1] through [5]). …”