Completely Positive Hypergroup Actions

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 $B(B(L^2(G)))$ of the operator algebra $B(L^2(G))$ of the Hilbert space $L^2(G)$. In this memoir, the author studies the situation in hypergroups and finds 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.