Algebras, graphs and their applications. Edited by Palle E. T. Jorgensen (Q2849085)

Quoted from the preface: ``This book aims to introduce the study of algebras induced by combinatorial objects, directed graphs. They serve as tools in an analysis of graph-theoretic problems. Also conversely, some analytic problems can be characterized and resolved''.NEWLINENEWLINENEWLINEThe book starts with a chapter that introduces graph groupoids. These are algebraic structures associated to directed graphs. Towards constructing new graphs from existing ones, operations on graphs are considered such as union, gluing, product, quotient, pull-back, complement.NEWLINENEWLINEChapter 2 brings an operator algebraic flavor to the analysis by introducing topological algebras determined by representations of graph groupoids. For a countable discrete graph \(G\), the author constructs first a canonical representation of the associated graph groupoid \(\mathbb{G}\) on the Hilbert space \(l^2(\mathbb{G})\). Next a groupoid \(W^*\)-dynamical system is introduced, and the crossed product yields the graph von Neumann algebra. One interesting connection here is with free probability. It is shown that the groupoid crossed product \(W^*\)-algebra can be characterized by an amalgamated free product \(W^*\)-algebra over a certain diagonal. Other connections with free probability are also explored. The chapter also contains an extended study of \(C^*\)-subalgebras generated by one partial isometry.NEWLINENEWLINENEWLINEChapter three contains an operator-theoretic analysis, and addresses characterization of properties such as self-adjointness, unitarity, normality and hyponormality of graph operators.NEWLINENEWLINEChapter four is devoted to an analysis of fractals on graph groupoids. The objects of interest here are graph fractaloids. The principle is that algebraic fractality can be used, via representations of graph groupoids, for studying fractality of analytic structures.NEWLINENEWLINENEWLINEEntropy on graphs is the subject of Chapter five, and here, in particular, the entropy of a finite fractal graph is computed.NEWLINENEWLINENEWLINEChapter six is devoted to developing a Jones index theory on graph groupoids. The author introduces a notion of index for finite directed graphs, from which a basic construction for graphs emerges. These are used as the basis for a comprehensive study of index theory of graph von Neumann algebras.NEWLINENEWLINENEWLINEChapter 7 discusses some aspects on applying graph-groupoid theory to network theory.NEWLINENEWLINEThe final chapter concludes with some \(K\)-theory computations for \(C^*\)-algebras generated by graph groupoids of connected directed graphs.NEWLINENEWLINEThe book seems generally well-written, and the contents are well-motivated. There is an abundance of examples, making this book accessible to graduate and postgraduate students who want to learn about topics in operator theory and operator algebras from a combinatorial point of departure.