Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài:Một đặc tính cohomological hữu hạn chiều C *- đại số. | OPERATOR THEORY Ị4 W35ỉ 239-247 Copyright by INCREST 1985 A COHOMOLOGICAL CHARACTERIZATION OF FINITE--DIMENSIONAL C -ALGEBRAS A. J. LAZAR . TSUI s. WRIGHT 1. INTRODUCTION Let A be a c -algebra B a c -algebra containing A and let Ỗ A - B be a derivation . a linear map for which ỗ ab aỗ b ỗ a b for all a b e A. In 12 and 13 the authors have investigated c -algebras B with the property that for every c -subalgebra A of B each such derivation Ờ of A into B extends to an inner derivation of B . there exists b e B with 0 a ba ab a A. Such c 5-algebras are interesting for a number of reasons some of which are documented in the introduction to 12 . A question which naturally arises in this situation occurs when one reverses the quantifiers what c -algebras A have the property that for every c -algebra B which contains A each derivation of A into B is inner in Bl It is easy to see that all finite-dimensional c -algebras A have this latter property and in Problem of 8 George Elliott motivated by questions about derivations of matroid c -algebras asked if the converse holds this is not completely accurate see the remarks at the end of next section . We show in this paper that the converse does indeed hold. In order to state our results precisely and concisely we introduce some concepts and notation from the Hochschild cohomology of algebras 11 Let A be a linear associative algebra over the complex numbers c M a two-sided X-module. We denote by Z A M the set of all derivations of A into M . linear maps of A into M which satisfy ỗ ab aỏ b ỗ a b for all a b e A and we denote by B A M the set of all inner derivations of A into M . all maps of the form a ma am a e A m ranging over the elements of M. Z A M becomes an abelian group under vector-space addition B A M is a subgroup of Z A M and H1 M the first Hochschild cohomology group of A with coefficients in M is defined to be the quotient group Z A M B A M . Thus LPỘl M 0 if and only if every derivation of A .
