Geodesic
Module and Hochschild Cohomology of Certain Semigroup Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 4, pp. 90-94.

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We study the relation between the module and Hochschild cohomology groups of Banach algebras. We show that, for every commutative Banach $\mathcal{A}$-$\mathfrak{A}$-bimodule $X$ and every $k\in\mathbb{N}$, the seminormed spaces $\mathcal{H}^{k}_{\mathfrak{A}}(\mathcal{A},X^*)$ and $\mathcal{H}^k(\mathcal{A}/J,X^*)$ are isomorphic, where $J$ is a specific closed ideal of $\mathcal{A}$. As an example, we show that, for an inverse semigroup $S$ with the set of idempotents $E$, where $\ell^1(E)$ acts on $\ell^1(S)$ by multiplication on the right and trivially on the left, the first module cohomology $\mathcal{H}^1_{\ell^1(E)}(\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is trivial for each $n\in\mathbb{N}$, where $G_S$ is the maximal group homomorphic image of $S$. Also, the second module cohomology $\mathcal{H}^2_{\ell^1(E)}(\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is a Banach space.
Keywords: module cohomology group, Hochschild cohomology group, inverse semigroup, semigroup algebra. 
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     title = {Module and {Hochschild} {Cohomology} of {Certain} {Semigroup} {Algebras}},
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A. Shirinkalam; A. Purabbas; M. Amini. Module and Hochschild Cohomology of Certain Semigroup Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 4, pp. 90-94. http://geodesic.mathdoc.fr/item/FAA_2015_49_4_a9/

[1] M. Amini, A. Bodaghi, D. Ebrahimi Bagha, Semigroup Forum, 80:2 (2010), 302–312 | DOI | MR | Zbl

[2] H. G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000 | MR | Zbl

[3] H. G. Dales, A. T.-M. Lau, D. Strauss, Dissertationes Math. (Rozprawy Mat.), 481 (2011), 1–121 | DOI | MR

[4] A. Ya. Khelemskii, Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo MGU, M., 1986 | MR

[5] E. Nasrabadi, A. Pourabbas, Semigroup Forum, 81:2 (2010), 269–276 | DOI | MR | Zbl

[6] E. Nasrabadi, A. Pourabbas, Bull. Iranian Math. Soc., 37:4 (2011), 157–168 | MR

[7] A. Pourabbas, Proc. Amer. Math. Soc., 132:2 (2004), 1403–1410 | DOI | MR | Zbl

[8] R. Rezavand, M. Amini, M. H. Sattari, and D. Ebrahimi Bagha, Semigroup Forum, 77:2 (2008), 300–305 | DOI | MR | Zbl

[9] V. Runde, Lectures on Amenability, Lecture Notes in Mathematics, 1774, Springer-Verlag, Berlin, 2002 | DOI | MR | Zbl

[10] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer-Verlag, London, 2002 | MR | Zbl