The twisted partial group algebra and (co)homology of partial crossed products

Given a group $G$ and a partial factor set $\sigma$ of $G,$ we introduce the twisted partial group algebra ${\kappa }_{par}^{\sigma }G,$ which governs the partial projective $\sigma$-representations of $G$ into algebras over a filed $\kappa \mathrm{.}$ Using the relation between partial projective representations and twisted partial actions we endow ${\kappa }_{par}^{\sigma }G$ with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of ${\kappa }_{par}^{\sigma }G\mathrm{.}$ Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product $A{\ast }_{\mathrm{\Theta }}G,$ involving the Hochschild homology of $A$ and the partial homology of $G,$ where $\mathrm{\Theta }$ is a unital twisted partial action of $G$ on a $\kappa$-algebra $A$ with a $\kappa$-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.