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
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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}\} $.

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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