INTRODUCTION In 1980, E. St0rmer, [82], among other things, proved the existence of an isometric isomorphism \I from the convolution measure algebra M(G) of a locally compact abelian group G into the operator algebra B(B(L2(G))) of the operator algebra B(L2(G)) of the Hilbert space L2(G). He also showed that when \I is restricted to the Schatten class ci of Hilbert-Schmidt operators on L2(G) then it is a *-isomorphism. Independently, F. Ghahramani [34] proved the first result for non-commutative G by a different method. An attempt to investigate the situation for the analogous map 3/ for a hypergroup K with a left Haar measure, i.e., a convo as studied by R. I. Jewett [47] with a left Haar measure or a compact P*-hypergroup as studied by C. F. Dunkl [24] or a commutative hypergroup as studied by R. Spector [77] revealed the following. I. (Theorem 4.7.7) ^ is not a homomorphism for a commutative hermitian dis- crete hypergroup K which is not a group or for any of the compact hypergroups Ha studied by C. F. Dunkl and D. E. Ramirez [26]. II. (i) (Proposition 1.6.3 and Remark 4.3.3) For a compact hypergroup K and ii, a scalar multiple of a non-negative measure on K, ||\P(/i)|| = ||JU||. (ii) (Theorem 4.7.8) For infinitely many two-point hypergroups is not an isometry. In fact, for a measure \i on a two-point discrete space K, which is not a scalar multiple of a non-negative measure on K, ||\I/(//)|| ^ ||//|| for infinitely many hypergroup structures on K. It is well known that dilations of *-homomorphisms are positive definite, com- pletely positive and completely bounded, cf., M. A. Naimark [57], B. Sz. Nagy [84], W. F. Stinespring [79] and W. B. Arveson [2]. III. For a non-negative measure /x, \£(/x) is completely positive when B(L2(K)) is given the usual matrix orders and also ^ is completely bounded when M(if) is given the natural matrix norm and B(L2(K)) is given the usual matrix norm. General matrix ordered spaces were introduced and studied by M. D. Choi and E. G. Effros [13] and general matricially normed spaces by E. G. Effros and Z.-J. Ruan, [30], [70], [71]. Different notions of complete positivity and complete boundedness have emerged in general set-ups, cf., P. Masani [52], T. Itoh and M. Nagisa [45] and A. S. Holevo [42], [43]. We give a general idea of such developments in Chapter 2 and carry out a detailed study in Chapter 3 when the measure algebra M(K) with the Bernoulli topology is given the matrix orders coming from the set of bounded continuous positive definite functions on K. We define presentations and actions on the lines of representations and actions of W. B. Arveson [3] in Chapter 1 and study the properties of complete positivity, complete boundedness, etc. for them and xi
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