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

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