Globalization of group cohomology in the sense of Alvares-Alves-Redondo

Abstract

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 Kpar(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 α of G on a (unital) algebra A we consider A as a Kpar(G)-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that 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 n par (G, A) is isomorphic to the usual cohomology group H n (G, M(B)), where M(B) is the multiplier algebra of B and B is the algebra under the enveloping action of α.

Group cohomology based on partial representations

“…• (4) ⇒ (5). As s = ss −1 t then s −1 = t −1 ss −1 and s −1 s = t −1 ss −1 t, hence s = ss −1 (tt −1 t) = t(t −1 ss −1 t) = ts −1 s.…”

“…In this section we are going to use Theorem 3.15 to give another characterization of the 1-st cohomology group H 1 par (G, M ) using maps d : G → M satisfying a certain property. This section correspond to the study of [5].…”

“…Partial actions, partial representations, the corresponding crossed product and the interaction between them were introduced by R. Exel in [17,18,19] as methods of study C * -algebras. Those works started the development of the theory of partial projective group representations in [10,11,12,13], and the study of group cohomology based on partial actions in [2], [5] and [14]. For an overview of publications on partial actions, related concepts and more details see [16] and [9].…”

“…After that we relate the partial group cohomology with the vector space of partial derivations Der par (G, M ) and the partial augmentation ideal. In Sections 3.2 and 3.3 we study a part of the theory developed in [5]. Using the results obtained in the previous section to give another characterization of the 1-st cohomology group H 1 par (G, M ) in term of classes of maps d : G → M of maps that satisfy certain conditions, which form a vector space denoted D(G, M ), showing an isomorphism between the vector space of partial derivations Der par (G, M ) and D(G, M ).…”

“…In Section 5.1 we study the final section of [5]. We work with an algebra A over a commutative ring K and a unital partial action α of G on A.…”