1st Edition

Algebras, Graphs and their Applications

By Ilwoo Cho Copyright 2014

    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.

    Algebra on Graphs
    Groupoids under Operations
    Operations on Graphs

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

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

    Fractal on Graph Groupoids
    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 on Finite Graphs
    Entropy of Finite Fractal Graphs

    Jones Index Theory on Graph Groupoids
    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

    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
    Projections in AG
    Projections in M∞ю(AG)
    K0-Groups of AG


    Ilwoo Cho