Given a positive (or completely positive map) $\phi:A\to B$ between C* algebras, is there a way to construct an $AB$ 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.)

$\begingroup$ Presumably you also want composition of positive maps to be compatible with composition of bimodules. I would be surprised if you could do this. Positivity doesn't seem to play an important role (that is, I would be surprised if you could only do this under a positivity assumption), so one can ask an analogous question about linear maps between rings and it might be easier to prove a negative result there. $\endgroup$– Qiaochu YuanMay 20, 2014 at 20:44

$\begingroup$ I don't necessarily expect composition to be compatible  just about any bimodule would be interesting. $\endgroup$– Edwin BeggsMay 21, 2014 at 5:48

$\begingroup$ This is actually provided by (the non commutative version of) the GNS construction. It works if and only if the map is completely positive. $\endgroup$– Simon HenryMay 21, 2014 at 8:14
1 Answer
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/OANotesHilbertCBimodules.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 nonunital 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/mathscinetgetitem?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!

$\begingroup$ Thanks! I was hoping that there was a result somewhere, and that looks just what I wanted. It would be very good to have a journal or book reference for it. $\endgroup$ May 21, 2014 at 8:16

3$\begingroup$ Lance's book, Hilbert C*modules, Chapter 5 is a book reference. $\endgroup$ May 21, 2014 at 13:52