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.)
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$\begingroup$ 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 YuanCommented May 20, 2014 at 20:44
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$\begingroup$ I don't necessarily expect composition to be compatible - just about any bimodule would be interesting. $\endgroup$– Edwin BeggsCommented May 21, 2014 at 5:48
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$\begingroup$ 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 HenryCommented May 21, 2014 at 8:14
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1 Answer
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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!
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$\begingroup$ 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. $\endgroup$ Commented May 21, 2014 at 8:16
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3$\begingroup$ Lance's book, Hilbert C*-modules, Chapter 5 is a book reference. $\endgroup$ Commented May 21, 2014 at 13:52