Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Sunder, V. S.

Abstract (Summary of the Work)

This thosin in devoted to the study of aome probloms related to the inclusion of a pair of (usually hyperfinite) Ili factors R. Specifically, the following and the relation between them are studied:(a) pairs of graphs which can occur as principal grapha for NC M;(b) construction of commuting squares starting with a pair of finite graphs;(c) computation of the higher relative commutants of sublactorn con- structed from specific commuting squares.The first chapter is introductory in nature and is included for the sake of comploteness and convenience of reference. It contains a rather perfunctory description of the basic construction for a pair of II,1 factors. The inclusion of finite dimensional C-algebras and the construction of a path-algebra on camp; tower of much algebras is described.We go on to describe Oceanu paragroup invariant for the inclusion of a pair of II1 factors, which includes a dencription of the principal graphs for NGM nud Oencnu's biunitarily conditlon for the existonco uf a com- muting square. An iterative procedure for constructing a pair of subfactors is describod and a complete proof of Ocnoanu Compactness Theorem is given.In the second chapter the properties of a pair of graphs which arise as principal graphs for NCM are studied. Based on this, a property called weak duality, for a pair of finite, bipartite, connected graphs is defined. The Loclinical result proved hore is that a graph g with at most triple points, no multiple bonds and not containing two speciflic subgraphs can be weakly dual only to itself. Combining this with Ocneanu triple point obstruction, It is shown that a tree, with trivial contragredient map, can occur as a principal graph only if it contains a copy of E.b(1)In the third chapter, starting with two spocific pairs of finite, bipar- tite, connected graphs, explicit constructions of commuting squares, cach with these graphs as inclusions, is given. The first example is taken from the theory of hyporgroups and the BOcond occurs as a principal graph of Rx HCR a G, whore II in a particular subgroup of a specilic group G.In the last chapter, a special class of commuting squares, called vertex models, is studied. Two classes of such commuting squares are considered and the principal grapha of subfactora, constructed from these following the standard itorative procodure described in chapter 1, aro computed. It is shown that one of these is related to the group dual of a suitable (clased) subgroup of U(N), and the other to the Cayley graph of a (non-closed) group, modulo scalars, generated by N claments of U(N).Further, the vertax models in tho case N= 2 are classified and the possible resulting principal graphs are identified as A-n1 sns 00. A brief comment is mado on nomo resulta which have been obtained in the case when N- 3 and the gonerating bilunitary matrix is a permutation matrix. The most interesting result in the occurrence of infinite graphs among the possible principal graphs arising froin such commuting squares.


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