  Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology
for a group $G$ with values in a module over the partial group algebra
$K_{\mathrm{par}}(G)$, which is different from the partial group cohomology
defined earlier by the first two named authors of the present paper. Given a
unital partial action $\alpha$ of $G$ on a (unital) algebra $\mathcal{A}$ we
consider $\mathcal{A}$ as a $K_{\mathrm{par}}(G)$-module in a natural way and
study the globalization problem for the cohomology in the sense of
Alvares-Alves-Redondo with values in $\mathcal{A}$. The problem is reduced to
an extendibility property of cocycles. Furthermore, assuming that $\mathcal{A}$
is a product of blocks, we prove that any cocycle is globalizable, and
globalizations of cohomologous cocycles are also cohomologous. As a consequence
we obtain that the Alvares-Alves-Redondo cohomology group
$H_{par}^n(G,\mathcal{A})$ is isomorphic to the usual cohomology group
$H^n(G,\mathcal{M}(\mathcal{B}))$, where $\mathcal{M}(\mathcal{B})$ is the
multiplier algebra of $\mathcal{B}$ and $\mathcal{B}$ is the algebra under the
enveloping action of $\alpha$.

Rings and Algebras;Group Theory