1st Edition

Algebras, Graphs and their Applications




ISBN 9781466590199
Published September 11, 2013 by CRC Press
444 Pages

USD $145.00

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Book Description

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.

Table of Contents

Algebra on Graphs
Introduction
Preliminaries
Groupoids under Operations
Operations on Graphs

Representations and Operator Algebras of Graph Groupoids
Introduction
Partial Isometries
Graph von Neumann Algebras
M-Diagonal Graph W*-Probability Spaces
C*-Subalgebras Generated by Partial Isometries

Operator Theory on Graphs
Introduction
Self-Adjointness and Unitarity
Normality of Graph Operators
Operators in Free Group Factors
Graph Operators Induced by Regular Trees

Fractal on Graph Groupoids
Introduction
Labeled Graph Groupoids and Graph Automata
Graph Fractaloids
Labeling Operators of Graph Fractaloids
Graph-Theoretical Characterization
Fractal Graphs Constructed by Fractal Graphs
Fractal Pairs of Graph Fractaloids
Equivalence Classes of Graph Fractaloids
Completely Finite Fractalization
Fractalized-Graph von Neumann Algebras
Fractalized Labeling Operators

Entropy Theory on Graphs
Entropy
Entropy on Finite Graphs
Entropy of Finite Fractal Graphs

Jones Index Theory on Graph Groupoids
Introduction
Quotient Graphs and Graph Index
Watatani's Extended Jones Index Theory
Index Theory on Graph von Neumann Algebras
Basic Construction Induced by Full-Vertex Subgraph Inclusion
Ladders and Nets on Graph von Neumann Algebras
Index-Morphisms

Network Theory on Graphs
Electric Resistance Network Theory
Representations of ERNs
ERN-Actions on Energy Hilbert Spaces
Free Structures Induced by ERN-Groupoids

K-Theory on Graphs
Introduction
K-Theory
Projections in AG
Projections in M∞ю(AG)
K0-Groups of AG

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