xii INTRODUCTION
the associated opresentations and opactions for general spaces in Chapter 2 and
for hypergroups in Chapter 3. Other notions such as spectral subspaces [3] and
completely positive instruments as in M. Ozawa [59] and A. S. Holevo [42], [43]
are also studied with the help of spectral synthesis in hypergroups as developed
by the author and K. A. Ross [12]. Chapter 4 gives some concrete examples of
presentations and actions of hypergroups.
I learned hypergroups from Kenneth A. Ross while on a visiting appointment
at the University of Oregon, completely positive maps from K. R. Parthasarathy
of Indian Statistical Institute, Delhi while at University of Delhi. I was able to
make these two notions interact successfully while on a visiting appointment at
the Center of Advanced Study in Mathematics of Panjab University, Chandigarh.
I derived encouragement mainly from its first Professor and Head, Late Hansraj
Gupta and his mathematical family there and visitors like Chandler H. Davis. I
take this opportunity to thank them all and dedicate this paper with reverence
to Professor Hansraj Gupta.
I thank the referee for useful comments and suggestions which have consider-
ably improved the presentation of the paper. I also thank Kenneth A. Ross for
useful discussions and his kind hospitality at University of Oregon, Eugene dur-
ing my recent visits. My thanks are due to Clark Atlanta University and Georgia
Institute of Technology in Atlanta, Georgia, USA and University of Manitoba,
Winnipeg, Canada, for making their facilities available to me while the paper
was being revised and typed. I thank Ms. Annette Rohrs for her very efficient
typing under unusually tight time restraints and for cheerfully deciphering my
difficult handwriting.
This work was presented at an invited talk at the Joint AMS-IMS-SIAM Summer Re-
search Conference on 'Applications of hypergroups and related measure algebras' held at the
University of Washington, Seattle, USA from August 1 to August 6, 1993.
A preliminary report was given in an invited talk at the International Conference on Con-
temporary Analysis in the memory of the Late Professor U. N. Singh held at the University of
Delhi in August 1990.
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