Studia Scientiarum Mathematicarum Hungarica 57 (1), 54–67 (2020) DOI: 10.1556/012.2020.57.1.1455 THE FIRST COHOMOLOGY GROUP OF SOME OPERATOR ALGEBRAS ON HILBERT C ∗ -MODULES HOGER GHAHRAMANI 1,2 and SAMAN SATTARI 1,3 1 Department of Mathematics, University of Kurdistan, P. O. Box 416, Sanandaj, Iran 2 e-mail: h.ghahramani@uok.ac.ir; hoger.ghahramani@yahoo.com 3 e-mail: saman.sattari92@gmail.com Communicated by A. Kro´o (Received January 26, 2019; accepted February 10, 2020) Abstract Let X be a Hilbert C ∗ -module over a C ∗ -algebra B. In this paper we introduce two classes of operator algebras on the Hilbert C ∗ -module X called operator algebras with property K and operator algebras with property Z, and we study the first (continuous) cohomology group of them with coefficients in various Banach bimodules under several conditions on B and X . Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras. 1. Introduction and preliminaries Let A be a Banach algebra and let E be a Banach A-bimodule. A deriva- tion d : A→E is a linear map satisfying d(ab)= d(a)b + ad(b) (a, b ∈A). A derivation d : A→E is called inner derivation, if there is an element x ∈E such that d(a)= xa − ax (a ∈A). 2010 Mathematics Subject Classification. Primary 46L08, 47L10, 47B47, 16E40. Key words and phrases. Hilbert C ∗ -module, operator algebra, derivation, first (contin- uous) cohomology group. 0081–6906 c  2020 Akad´ emiai Kiad´ o, Budapest Unauthenticated | Downloaded 05/09/20 06:55 AM UTC