Elementary operators on the algebra of adjointable operators on a Hilbert module

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:475

Article

Magiatis, Charalampos¹ 

[1] Univ Aegean, Dept Math, Mitilini, Greece

关键词: Multiplication operator;    Elementary operator;    Prime C*-algebra;    Hilbert C*-module;   

DOI  :  10.1016/j.jmaa.2019.02.059

来源: Elsevier

【 摘 要 】

Let A be a prime unital C*-algebra, X a countably generated Hillbert A-module, B(X) the C*-algebra of adjointable operators on X and K(X) the C*-algebra of (generalised) compact operators on X. We characterise multiplication operators and elementary operators on B(X) in terms of the size of their images. To obtain these characterisations we introduce the concept of a uniformly approximable subset of a C*-algebra. We show that M-A,M-B(B(X)) subset of K(X) if and only if at least one of A or B belongs to K(X). We show that the set M-A,M-B (B(X)(1)) is a uniformly approximable subset of K(X), (B(X)(1) is the unit ball of B(X)), if and only if A, B is an element of K(X). If Phi is an elementary operator on B(X), we show that Phi(B(X)) subset of K(X) (resp. is a uniformly approximable subset of K(X)) if and only if there exist {A(i)}(k)i=1, {B-i}(k)i=1 subset of B(X) such that at least one of A(i) or B-i (resp. both) belong to K(X) for i = 1,...,k and Phi = Sigma(k)(i=1) M-AiBi. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free