Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas.

Features Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas.

This book deals with two important branches of mathematics, namely, logic and set theory. Logic and set theory are closely related and play very crucial roles in the foundation of mathematics, and together produce several results in all of mathematics. The topics of logic and set theory are required in many areas of physical sciences, engineering, and technology. The book offers solved examples and exercises, and provides reasonable details to each topic discussed, for easy understanding. The book is designed for readers from various disciplines where mathematical logic and set theory play a crucial role. The book will be of interested to students and instructors in engineering, mathematics, computer science, and technology.

This book introduces the study of algebra induced by combinatorial objects called directed graphs. These graphs are used as tools in the analysis of graph-theoretic problems and in the characterization and solution of analytic problems. The book presents recent research in operator algebra theory connected with discrete and combinatorial mathematical objects. It also covers tools and methods from a variety of mathematical areas, including algebra, operator theory, and combinatorics, and offers numerous applications of fractal theory, entropy theory, K-theory, and index theory.

In Spectral Properties of Certain Operators on a Free Hilbert Space and the Semicircular Law, the authors consider the so-called free Hilbert spaces, which are the Hilbert spaces induced by the usual l2 Hilbert spaces and operators acting on them. The construction of these operators itself is interesting and provides new types of Hilbert-space operators. Also, by considering spectral-theoretic properties of these operators, the authors illustrate how “free-Hilbert-space” Operator Theory is different from the classical Operator Theory. More interestingly, the authors demonstrate how such operators affect the semicircular law induced by the ONB-vectors of a fixed free Hilbert space. Different from the usual approaches, this book shows how “inside” actions of operator algebra deform the free-probabilistic information—in particular, the semicircular law.

In this monograph, we study graph-index, Watatani's extended Jones index, and the relations between them. From this investigation, we realize that graph-index of graphs and Jones index of von Neumann algebras generated by "graph groupoids" are equivalent in a certain sense, whenver given graphs are finite trees. We provides the classification of graph-groupoid von Neumann algebras induced by finite trees with respect to these equivalent indexes. By understanding the indexings as morphisms, we establish actions of morphisms and construct the corresponding algebraic structure, called the "index semigroup." We study the fundamental properties of them. As application, we consider the connection between our index semigroup and K-groups of finite-tree-groupoid von Neumann algebras.

In this monograph, we study the relation between graph-index and Watatani's 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 inclusions, which are determined by corresponding subgroupoid inclusions. We call them graph-indexes (of graph inclusions). If we take a special graph inclusion, a vertex-full subgraph inclusion, then our graph-index has close connection with Watatani's extended Jones index. We show that Jones indexes of von-Neumann algebra- inclusions are characterized by our graph-indexes, and vice versa, whenever von Neumann algebras are generated by groupoids of graphs. This characterization will be extended to the case where we have graph-groupoid dynamical systems.

In this monograph, we consider natural operations on directed graphs. And we find the connections between our operations on graphs and the groupoid-perations on "graph" groupoids. Remark that we cannot guarantee the (algebraic or categorial) groupoids generated by the grouopoid-operations; sum, product, quotient or complement; of graph groupoids are again graph groupoids. By defining suitable operations on graphs, we can conclude the groupoids generated by the groupoid-operations of graph groupoids are again graph groupoids; for example, the product groupoid of two graph groupoids is groupoid-isomorphic to the graph groupoid of the product graph, etc. This provides another bridge connceting combinatorics and algebra. Recently, the von Neumann algebras generated by graph groupoids, called graph von Neumann algebras, have been studied. By using the fundamental techniques from graph von Neumann algebra theory, we can characterize the properties of groupoid von Neumann algebras, generated by groupoids obtained from the groupoid-operations, as certain graph von Neumann algebras.

In this monograph, we consider the connection between graphs and Hilbert space operators. In particular, we are interested in the algebraic structures, called graph groupoids, embedded in operator algebras. In Part 1, we consider the connection from graphs to partial isometries. Every element in graph groupoids assigns an operator, which is either a partial isometry or a projection, under suitable representations. The von Neumann algebras induced by the dynamical systems of graph groupoids are characterized. In Part 2, we observe the connection from partial isometries to graphs. We show that a finite family of partial isometries on a fixed Hilbert space H creates the corresponding graph, and the graph groupoid of it is an embedded groupoid inside B(H). Moreover, the C*-subalgebra generated by the family is *-isomorphic to the groupoid algebra generated by the graph groupoid of the corresponding graph. As application, we consider the C*-subalagebras generated by a single operator.

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 also study the operator algebras generated by distorted histories. Also, by comparing operator algebras generated by histories and those generated by distorted histories, we investigate the differences between them. As application, we observe the fractally distorted histories, which are one type of the patterned'' distorted histories.

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 investigate the measure-space framing: Part 1 shows the fundamental properties of the measure- space framing, and Part 2 provides the biggest application for the measure-space framing; the free stochastic integration. Part 3 is the extension of Part 1 to the groupoid dynamical systems. In Part 4, we concentrate on the group-framing on graphs. The basic properties are studied. As application, we briefly mention about free-group-framing, and distorted histories.

Recently, graphs have been studied and applied in various math and science fileds. In this monograph, we consider graphs with fractal property. Starting with graphs (combinatorial objects), we construct the corresponding groupoids (algebraic objects). The fractal property of graphs and groupoids is detected by the automata labelings (automata-theoretic objects). The groupoids with fractal property will be called graph fractaloids. By defining suitable representations of groupoids, we establish von Neumann algebras (operator-algebraic objects). As elements of the von Neumann algebras, we define the labeling operators (operator-theoretic objects) of graph fractaloids. In Part 1, by computing the free moments (free-probabilistic data) of the operators, we verify how the graph fractaloids act in the von Neumann algebras. Also, based on such computations, we can classify the graph fractaloids, in Part 2. Our classification shows the richness of graph fractaloids which are not fractal groups, in general. In Part 3, we show that, for any finite graph, there always exists a finite fractal graph containing it as its part.