show/hide this revision's text 2 corrected a typo, not that anyone seems to have read my answer

This is in response to Dmitri's remark. The reason I didn't bring up Sinclair & Smith's book (which is where I first started trying to learn Hochschild cohomology) is that it deals with continuous cochains with coefficients in the algebra. I understood DP's original question as being about purely algebraic cyclic cohomology, which involves not-necessarily continuous cochains taking values in the dual of the algebra (not the algebra itself). I hope this addresses your "surprise". FWIW, I'm more interested in the Sinclair & Smith setting myself, but I don't think that's what DP was asking about - though I may have misunderstood.

And yes, there is still no example of a von Neumann algebra M for which H^n(M,M)=0 H^n(M,M) is nonzero for all some n \geq > 1; while the vanishing or otherwise of H^2(L(F_2),L(F_2)) is still unknown...
show/hide this revision's text 1

This is in response to Dmitri's remark. The reason I didn't bring up Sinclair & Smith's book (which is where I first started trying to learn Hochschild cohomology) is that it deals with continuous cochains with coefficients in the algebra. I understood DP's original question as being about purely algebraic cyclic cohomology, which involves not-necessarily continuous cochains taking values in the dual of the algebra (not the algebra itself). I hope this addresses your "surprise". FWIW, I'm more interested in the Sinclair & Smith setting myself, but I don't think that's what DP was asking about - though I may have misunderstood.

And yes, there is still no example of a von Neumann algebra M for which H^n(M,M)=0 for all n \geq 1; while the vanishing or otherwise of H^2(L(F_2),L(F_2)) is still unknown...