Kinds of derivations on Hilbert $C^*$-modules and their operator algebras
Abstract: Let M be a Hilbert C-module. A linear mapping d W M ! M is called a derivation if d.< x; y >´/ D< dx; y >´C < x; dy >´C < x; y > d´for all x; y;´2 M. We give some results for derivations and automatic continuity of them on M. Also, we will characterize generalized derivations and strong higher derivations on the algebra of compact operators and adjointable operators of Hilbert C-modules, respectively.
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 3 publications
References 7 publications
Suppose A is a pro-C*-algebra. Let L A (E) be the pro-C*-algebra of adjointable operators on a Hilbert A-module E and let K A (E) be the closed two sided * -ideal of all compact operators on E. We prove that if E be a full Hilbert A-module, the innerness of derivations on K A (E) implies the innerness of derivations on L A (E). We show that if A is a commutative pro-C*-algebra and E is a Hilbert A-bimodule then every derivation on K A (E) is zero. Moreover, if A is a commutative σ-C*-algebra and E is a Hilbert A-bimodule then every derivation on L A (E) is zero, too.2010 Mathematics Subject Classification. Primary 46L08; Secondary 46L05, 46H25, 47B47.
Suppose A is a pro-C*-algebra. Let L A (E) be the pro-C*-algebra of adjointable operators on a Hilbert A-module E and let K A (E) be the closed two sided * -ideal of all compact operators on E. We prove that if E be a full Hilbert A-module, the innerness of derivations on K A (E) implies the innerness of derivations on L A (E). We show that if A is a commutative pro-C*-algebra and E is a Hilbert A-bimodule then every derivation on K A (E) is zero. Moreover, if A is a commutative σ-C*-algebra and E is a Hilbert A-bimodule then every derivation on L A (E) is zero, too.2010 Mathematics Subject Classification. Primary 46L08; Secondary 46L05, 46H25, 47B47.
Let ℳ {\mathcal{M}} be a Hilbert C * {\mathrm{C}^{*}} -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * {\mathrm{C}^{*}} -module higher derivations { φ n : ℳ → ℳ } n = 0 ∞ {\{\varphi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=0}^{\infty}} with φ 0 = I {\varphi_{0}=I} satisfying φ n ( 〈 x , y 〉 z ) = ∑ i + j + k = n 〈 φ i ( x ) , φ j ( y ) 〉 φ k ( z ) ( x , y , z ∈ ℳ , n ∈ ℕ ∪ { 0 } ) \varphi_{n}(\langle x,y\rangle z)=\sum_{i+j+k=n}\langle\varphi_{i}(x),\varphi_% {j}(y)\rangle\varphi_{k}(z)\quad(x,y,z\in\mathcal{M},\,n\in\mathbb{N}\cup\{0\}) and all Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n : ℳ → ℳ } n = 1 ∞ {\{\psi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=1}^{\infty}} satisfying ψ n ( 〈 x , y 〉 z ) = 〈 ψ n ( x ) , y 〉 z + 〈 x , ψ n ( y ) 〉 z + 〈 x , y 〉 ψ n ( z ) ( x , y , z ∈ ℳ , n ∈ ℕ ) , \psi_{n}(\langle x,y\rangle z)=\langle\psi_{n}(x),y\rangle z+\langle x,\psi_{n% }(y)\rangle z+\langle x,y\rangle\psi_{n}(z)\quad(x,y,z\in\mathcal{M},\,n\in% \mathbb{N}), and we show that for every Hilbert C * {\mathrm{C}^{*}} -module higher derivation { φ n } n = 0 ∞ {\{\varphi_{n}\}_{n=0}^{\infty}} on ℳ {\mathcal{M}} , there exists a unique sequence of Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n } n = 1 ∞ {\{\psi_{n}\}_{n=1}^{\infty}} on ℳ {\mathcal{M}} such that ψ n = ∑ k = 1 n ( ∑ ∑ j = 1 k r j = n ( - 1 ) k - 1 r 1 φ r 1 φ r 2 … φ r k ) \psi_{n}=\sum_{k=1}^{n}\biggl{(}\sum_{\sum_{j=1}^{k}r_{j}=n}(-1)^{k-1}~{}r_{1}% \varphi_{r_{1}}\varphi_{r_{2}}\dots\varphi_{r_{k}}\biggr{)} for all positive integers n, where the inner summation is taken over all positive integers r j {r_{j}} with ∑ j = 1 k r j = n {\sum_{j=1}^{k}r_{j}=n} .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
