Completely bounded multilinear maps and \(C^ *\)-algebraic cohomology (Q1103824)

The category of completely bounded mappings is considered. The first order case concerns completely bounded derivations from a given \(C^*\)- algebra A into B(H) which are known to be inner. The higher dimensional cohomology problem is solved in the affirmative: It is shown that the cohomology groups \(H^ n_{cb}(A,B(H))\) always vanish. (The subscript refers to the completely bounded category.) Two cases are considered when the bounded cohomology groups are also shown to vanish. The conditions on the given \(C^*\)-algebra A are: (i) The weak closure \(\bar A\) is properly infinite, or alternatively, (ii) \(\bar A\simeq R{\bar \otimes}N\) with N the hyperfinite \(II_ 1\)- factor.