Given a positive (or completely positive map) $\phi:A\to B$ between C\* algebras, is there a way to construct an $A-B$ bimodule? This would more or less generalise the following construction: If $\phi$ was an algebra map, we could have ${}\_\phi B$, which is $B$ as a vector space, with $B$ product as the right $B$ action, and left action $a. b=\phi(a) b$. If there was a Hilbert C\* module, so much the better. (This is part of the idea that bimodules generalise algebra maps, and I want to look at the differentiable properties of bimodules.)



positive maps and bimodules
At least for unital C\*-algebras, the answer is yes. A proof plus explanatory comments are provided by Paul Skoufranis, <http://www.math.ucla.edu/~pskoufra/OANotes-HilbertC-Bimodules.pdf> - look for the theorem on page 11 (here the algebras are assumed to be unital, though I didn't check whether one can extend this to the non-unital case). If you look at the theorem on page 16, you'll see that the map you started with needs to be completely positive. 


edit: The paper of Jürgen Schweizer in <http://www.ams.org/mathscinet-getitem?mr=1796906> may be of interest to you. Look for paragraph 1.5. The relation between a Hilbert bimodule and the notion of a correspondence used there is probably a bit subtle. A warning to end with: I have seen other notions of a correspondence, e.g. the one used in Katsura's papers *A class of C\*-algebras generalizing...*, so be aware! 


