# positive maps and bimodules

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.)

• 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. – Qiaochu Yuan May 20 '14 at 20:44
• I don't necessarily expect composition to be compatible - just about any bimodule would be interesting. – Edwin Beggs May 21 '14 at 5:48
• This is actually provided by (the non commutative version of) the GNS construction. It works if and only if the map is completely positive. – Simon Henry May 21 '14 at 8:14