Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 9 Issue 2 (2018), Pages 37-47. SECOND ORDER COHOMOLOGY OF MATRIX BANACH ALGEBRAS WITH RESPECT TO CHARACTERS FEREIDOUN HABIBIAN * , NAJMEH MIRZASANI Abstract. In this paper, we investigate the second order cohomology of Ba- nach algebras Ep(I ) (1 ≤ p< ∞) with coefficients corresponding to characters. Consequently, the necessary and sufficient conditions are established for con- tractibility of these Banach algebras. As an application, we investigate weak character amenability of A(G) and contractibility of L 2 (G), when in either case G is a compact group. 1. Introduction The Banach algebras E p (I ), where p ∈ [1, ∞] ∪{0}, have been introduced and extensively studied in Section 28 of [8]. Amenability, weak amenability and ap- proximate weak amenability of these Banach algebras have been studied by H. Samea in [15](see also [11]). The concept of ϕ-amenability was also first introduced by Kaniuth, Lau, and Pym (see [9], [10]), and simultaneously by Monfared (see [12])and weak ϕ-amenability was recently studied for a Banach algebra A by R. Nasr-Isfahani, S. Shahmoradi and S. Soltani Renani([13]). In this paper, we in- vestigate weak character amenability for the subalgebras of E(I ) for which their character space is non-empty, together with their applications to a number of con- volution Banach algebras on compact groups. The paper concluded by studying contractibility of Banach algebras E p (I ) (1 ≤ p< ∞). Let I be an arbitrary index set. Suppose that {H i ; i ∈ I } be a family of Hilbert spaces H i of dimension d i (i ∈ I ) and consider B(H i ) is the space of all bounded linear operators on H i . Clearly we can identify B(H i ) with M di (C) (the space of all d i × d i -matrices on C). The set E(I ) :=  i∈I B(H i ) is a ∗-algebra, the scaler multiplication, addition, multiplication, and the adjoint of an element, are defined coordinatewise. In E(I ) one introduces various norms: for E := (E i ) i∈I ∈ E(I ) and (a i ) i∈I with a i ≥ 1 for all i ∈ I ; ‖E‖ p :=  ( ∑ i∈I a i ‖E i ‖ p ϕp ) 1 p if 1 ≤ p< ∞; sup i∈I ‖E i ‖ ϕ∞ if p = ∞, 2010 Mathematics Subject Classification. Primary 46H25, 16E40; Secondary 46H20, 22D99. Key words and phrases. Banach algebra; Cohomology; Character amenable; Contractible; Banach module; Locally compact group. c 2018 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted December 23, 2017. Revised February 17, 2018. Accepted .... Communicated by E.G.Madjid. * Corresponding author. 37