The First and Second Hochschild Cohomology Groups of Banach Algebras with Coefficients in Special Symmetric Bimodules
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Complex Analysis and Operator Theory
The First and Second Hochschild Cohomology Groups of Banach Algebras with Coefficients in Special Symmetric Bimodules E. Feizi1
· H. Ghahramani2 · V. Khodakarami2
Received: 9 December 2019 / Accepted: 25 August 2020 © Springer Nature Switzerland AG 2020
Abstract Let A be a Banach algebra and φ be a character on A. In this paper we consider the class S MφA of Banach A-bimodules X for which the module actions of A on X is given by a · x = x · a = φ(a)x (a ∈ A, x ∈ X ) and we study the first and second continuous Hochschild cohomology groups of A with coefficients in X ∈ S MφA . We obtain some sufficient conditions under which H 1 (A, X ) = {0} and H 2 (A, X ) is Hausdorff, where X ∈ S MφA . We also consider the property that H 1 (A, X ) = {0} for every X ∈ S MφA and get some conclusions about this property. Finally, we apply our results to some Banach algebras related to locally compact groups. Keywords First cohomology group · Second cohomology group · Banach algebra · Symmetric bimodule · Character Mathematics Subject Classification 16E49 · 46M20 · 46H99 · 43A95
Communicated by Serap Oztop.
B
E. Feizi [email protected] H. Ghahramani [email protected]; [email protected] V. Khodakarami [email protected]; [email protected]
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Department of Mathematics, Bu-Ali Sina University, Hamadan 65174-4161, Iran
2
Department of Mathematics, University of Kurdistan, P. O. Box 416, Sanandaj, Iran 0123456789().: V,-vol
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1 Introduction Let A be a Banach algebra, X be a Banach A-bimodule, and H n (A, X ) (n ≥ 1) be the nth continuous Hochschild cohomology group of A with coefficients in X . It is interesting to study the structure of H n (A, X ) (n ≥ 1), where coefficients are in different Banach A-bimodules X , and its vanishing conditions. The Banach Abimodule X is called symmetric if a · x = x · a, for all a ∈ A and x ∈ X . We use symbols M A and S M A , for the class of all Banach A-bimodules and symmetric Banach A-bimodules, respectively. Johnson studied cohomology of Banach algebras in [9] and defined the concept of amenable Banach algebra which was based on the amenability of locally compact groups. A Banach algebra A is amenable if H 1 (A, X ∗ ) = {0}, for every X ∈ M A . It is proved in [9, Proposition 8.2] that if A is an amenable commutative Banach algebra, then H 1 (A, X ) = H 2 (A, X ) = {0}, for every X ∈ S M A . With this in mind, the notion of weak amenability for commutative Banach algebras was introduced in [1]. A commutative Banach algebra A is weakly amenable if H 1 (A, X ) = {0}, where X ∈ S M A , which is equivalent to H 1 (A, A∗ ) = {0}. According to this result, Johnson in [10], extended the weak amenability notion to all Banach algebras, by H 1 (A, A∗ ) = {0}. Note that, C is a Banach L 1 (G)-module with the following module actions: f (x)d x ( f ∈ L 1 (G), z ∈ C). f ·z =z· f =z G
If define ϕ : L 1 (G) → C by ϕ( f ) = G f (x)d x, it is easy to check that ϕ is a character on L 1 (G), that is
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