#### Date of Submission

2-28-1975

#### Date of Award

2-28-1976

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Quantitative Economics

#### Department

Research and Training School (RTS)

#### Supervisor

Nandkarni, M. G. (RTS-Kolkata; ISI)

#### Abstract (Summary of the Work)

Mackeys theorem on inducud representation gives all the systems of imprimitivity for a locally compact second countable group G, acting in a separable Hilbert space and based on a transitive G-space X. These systems of imprimitivity are obtained from the unitary representationg of the closed subgroup H of G defining the transitive G-space X. The main tool for this study is a class of functions called cocycles. MaÃ§key showed that the (G, x) systems of imprimitivity are connected in a one-one way to unitary operator valued cocycles on GXX: This cocycle in turn gives a representation of the closed subgroup H, defining x. Systems of imprimitivity on non-transitive actions are not as well studied. n this thesis we study the systeme of imprimitivity on some important cases of essentially non-transitive actions, and connect them to systems of imprimitivity on simpler spaces.Using the notion of cocycles, Gamelin [7] showed that the systems of imprimitivity on the pair (R, B) where R is the real line ad B is the Bohr group are related in a one-one way to systems of imprimi-tivity on the pair (N, K) is the integer where grÉ”up and K is the annihilator of a cyclic subgroup of the dual group of Ð’. More precisely, he showed that every (N, K) cocycle extends to an (IR, B) cocycle and in the cohomology (an equivalence relation defined in the set of all cocycles) class of every (R, B) cocycle, there is a cocycle which is extended from an (N, K) cocycle. Gamelin proved the above result for scalar valued cocycles and it was extended to the vector valued case by Muhly [20] and Bagchi [3]. The method Gamelin used is that of a flow built under a functiÃ³n. He views the action of R on B as a flow built under the const ant function with base space K. In chapter II of this thesis we show that Gamelins method of extending cocycles can be used for a general flow built under a function.To study the systems of imprimitivity on strictly ergodic actions, Mackey [19] introduced the notion of virtual subgroups. He showed, among other thinge, that using this notion one can generalise the notion of a flow built. under a function. We show that Gamelins method of extending cocycles is applicable to a OF o py of o - 3 - generalized flow built under a constant function The action of a locally compact second countable abelian subgroup H acting continuously on a locally compact second countable abelian group G by translation, can be viewed as a generalized flow built under a constant function.In chapter III we consider in detail, this particular case of group actions. The situation is similar to that of the pair (IR, B).This thesis is divided into three chapters. The first chapter is mainly introductory. We introduce the notion of a unitary eperator valued cocycle and a system of imprimitivity. The definitions and results are taken from Varadarajan [24]. Strict cocycles are easier to handle. and we mention some cases where we need to consider only strict cocycles. We show that we can take a unitary operator valued cocycle to satisfy a much stringent condition. This is obtained by gener- alizing a method of Doobs in obtaining a measurable stochastic process from a continuous in probability stochastic process. This definition of a cocycle is needed for chapters II and III.

#### Control Number

ISILib-TH24

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Mathew, Joseph Dr., "System of Imprimitivity for Ergodic Actions of Locally Compact Abelian Groups." (1976). *Doctoral Theses*. 183.

https://digitalcommons.isical.ac.in/doctoral-theses/183

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842960