ABSTRAC T

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

B(B(L2(G)))

of the operator algebra

B(L2(G))

of the Hilbert space

L2(G).

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

regular representation.

Received by the editor June 5, 1991; and in revised form December 23, 1994.