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)|| =
(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
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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