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.

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