Characterization of CPH-Maps - What Might They Be Good For?

If $\tau$ is a linear map from a $C^*$-algebra $B$ to a $C^*$-algebra $C$, a {\bf $\tau$-map} is a map $T$ from a Hilbert $B$-module $E$ to a Hilbert $C$-module $F$ such that $$ \langle T(x),T(x')\rangle ~=~ \tau(\langle x,x'\rangle). $$ Such maps are well-known in the case when $\tau$ is a homomorphism. The next step is to look a the case when $\tau$ is a CP-map. (This was proposed by Asadi and examined in a very special version with $\tau$ unital,$C=B(G)$, $F=B(G,H)$, and yet another strong strong extra condition.) Since then, such maps have drawn some attention. Some people started calling them CP-maps between Hilbert modules. But still nobody seem to have a clue what they actually might be good for. In this talk we characterize $\tau$-maps in full generality --- and without knowing $\tau$! We give a number of equivalent conditions. Two of them are intrinsic and allow to say if a map $T$ is a $\tau$-map for some CP-map $\tau$, by just looking at $T$ and checking some simple conditions. A third condition in terms of extension of $T$ to a map between the linking algebras of $E$ and of $F$ shows that CP-map is {\bf not} the right name for such maps $T$. We shall explain why we are going to call them {\bf CPH-maps}. Finally, we explain that CPH-maps occur in a context that generalizes the situation of (weak) dilation of CP-maps, so-called {\bf CPH-dilations}. We are not sure, if this, finally, can answer the question what CPH-maps might be good for. But, at least, we believe that there are some obvious connections with CPD-kernels and with Morita equivalence that, finally, might {\bf justify} the effort to analyze CPH-maps.