The main purpose of this monograph is to show the existence of distortions of histories, and analyze the distorted histories in operator-algebraic points of view. We provide two types of distortions occured by some finite number of objects. In dynamical systems, we define a history by a pair, consisting of a type I factor, and an E_0-group with some additional conditions. We distort the given history by a finite number of partial isometries. Two types of distortions can happen: inner distortions, and outer distortions. The fundamental properties of distorted histories are characterized. We...

In this monograph, we consider the framing process on given graphs. For a fixed graph, we put a frame on it. Such a frame is determined to be an independent mathematical structure (possibly other than graphs). In particular, we frame a graph with measure spaces and groups. This framing technique can be extendable to frame other mathematical structures; for instance, rings, fields, (topological vector) spaces, operator algebra, or operator spaces, etc. Here, we restrict our interests to the cases where the frames are either measure spaces or groups. In Part 1 through Part 3, we...

In this monograph, we study the relation between graph-index and Watatanis extended Jones index of certain von Neumann algebras. This research provides not only interesting examples of Jones index theory but also the connection between combinatorics (graph theory), algebra (grouopoid theory), operator algebra, and noncommutative dynamical systems. Moreover, the study of graph-index, itself, is an interesting topic in graph theory because these quantities give invariants for quotient structures induced by graphs. We first define the indexes (or the index numbers) of graph...