Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Norm of a derivation on a von Neumann algebra


Author: P. Gajendragadkar
Journal: Trans. Amer. Math. Soc. 170 (1972), 165-170
MSC: Primary 46L10
MathSciNet review: 0305090
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A derivation on an algebra $ \mathfrak{A}$ is a linear function $ \mathcal{D}:\mathfrak{A} \to \mathfrak{A}$ satisfying $ \mathcal{D}(ab) = \mathcal{D}(a)b + a\mathcal{D}(b)$ for all $ a,b$ in $ \mathfrak{A}$. If there exists an $ a$ in $ \mathfrak{A}$ such that $ \mathcal{D}(b) = ab - ba$ for $ b$ in $ \mathfrak{A}$, then $ \mathcal{D}$ is called the inner derivation induced by $ a$. If $ \mathfrak{A}$ is a von Neumann algebra, then by a theorem of Sakai [7], every derivation on $ \mathfrak{A}$ is inner. In this paper we compute the norm of a derivation on a von Neumann algebra. Specifically we prove that if $ \mathfrak{A}$ is a von Neumann algebra acting on a separable Hilbert space $ \mathcal{H},T$ is in $ \mathfrak{A}$, and $ {\mathcal{D}_T}$ is the derivation induced by $ T$, then $ \vert\vert{\mathcal{D}_T}\vert\mathfrak{A}\vert\vert = 2\inf \{ \vert\vert T - Z\vert\vert,Z\;{\text{in}}\;{\text{centre}}\;\mathfrak{A}\} $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L10

Retrieve articles in all journals with MSC: 46L10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0305090-X
PII: S 0002-9947(1972)0305090-X
Keywords: Centre of a bounded operator, decomposition of a Hilbert space into a direct integral, norm of a derivation, separable, von Neumann algebra
Article copyright: © Copyright 1972 American Mathematical Society