Derivations on the Algebra of Operators in Hilbert C~*-Modules  

作　　者：Peng Tong LI De Guang HAN Wai Shing TANG 

机构地区：[1]Department of Mathematics,Nanjing University of Aeronautics and Astronautics [2]Department of Mathematics,University of Central Florida [3]Department of Mathematics,National University of Singapore

出　　处：《Acta Mathematica Sinica,English Series》2012年第8期1615-1622,共8页数学学报（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171151);Natural Science Foundation of Jiangsu Province of China(Grant No.BK2011720);supported by Singapore Ministry of Education Academic Research Fund Tier1(Grant No.R-146-000-136-112)

摘　　要：Let M be a full Hilbert C＊-module over a C＊-algebra A, and let End^（.A4） be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End＊A（M） is an inner derivation, and that if A is a-unital and commutative, then innerness of derivations on ＂compact＂ operators completely decides innerness of derivations on EndA（M）. If .4 is unital （no commutativity is assumed） such that every derivation of A is inner, then it is proved that every derivation of EndA（Ln（A）） is also inner, where Ln（A） denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist xo,yo ∈M such that 〈xo,yo〉 = 1, we characterize the linear A-module homomorphisms on EndA（M） which behave like derivations when acting on zero products.Let M be a full Hilbert C＊-module over a C＊-algebra A, and let End^（.A4） be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End＊A（M） is an inner derivation, and that if A is a-unital and commutative, then innerness of derivations on ＂compact＂ operators completely decides innerness of derivations on EndA（M）. If .4 is unital （no commutativity is assumed） such that every derivation of A is inner, then it is proved that every derivation of EndA（Ln（A）） is also inner, where Ln（A） denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist xo,yo ∈M such that 〈xo,yo〉 = 1, we characterize the linear A-module homomorphisms on EndA（M） which behave like derivations when acting on zero products.

关 键 词：DERIVATIONS inner derivations C＊-algebras Hilbert C＊-modules 

分 类 号：O177.5[理学—数学]