Index Theory on Graphs: Graph-Index and Extended Jones Index of von Neumann Algebras Induced by Graphs - Softcover

CHO, ILWOO

Softcover

ISBN 10: 3639332717 ISBN 13: 9783639332711

Publisher: VDM Verlag Dr. Müller, 2011

Synopsis

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.

About this title

About the Author

Master Degree (Math): Sungkyunkwan Univ., and Univ. of Iowa / PhD (Math) : Univ. of Iowa / Current Position : Assistant Professor in St. Ambrose Univ., Dep. of Math (Davenport, Iowa, U. S. A.)

Edition details

• Publisher: VDM Verlag Dr. Müller
• Publication date: 2011
• Language: English
• ISBN 10: 3639332717
• ISBN 13: 9783639332711
• Binding: Paperback
• Number of pages: 136