In this paper, Property $\Gamma$ for a type II$_{1}$ von Neumann algebra is
introduced as a generalization of Murray and von Neumann's Property $\Gamma$
for a type II$_{1}$ factor. The main result of this paper is that if a type
II$_{1}$ von Neumann algebra $\mathcal{M}$ with separable predual has Property
$\Gamma$, then the continuous Hochschild cohomology group $H^{k}(\mathcal{M},
\mathcal{M})$ vanishes for every $k \geq 2$. This gives a generalization of an
earlier result due to E. Christensen, F. Pop, A.M. Sinclair and R.R. Smith.
