It is now well known that the measure algebra M(G) of a locally compact
group can be regarded as a subalgebra of the operator algebra
of the operator algebra
of the Hilbert space
We study the
situation in hypergroups and find that, in general, the analogous map for them
is neither an isometry nor a homomorphism. However, it is completely positive
and completely bounded in certain ways. This work presents the related general
theory and special examples.
Key words and phrases, presentations, opresentations, actions, opactions, completely posi-
tive maps, completely bounded maps, hypergroups, matrix orders on the hypergroup measure
algebra, completely positive hypergroup actions, actions and opactions associated with the left
Received by the editor June 5, 1991; and in revised form December 23, 1994.