Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Norm of a derivation on a von Neumann algebraHTML articles powered by AMS MathViewer

Trans. Amer. Math. Soc. 170 (1972), 165-170 Request permission

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 $||{\mathcal {D}_T}|\mathfrak {A}|| = 2\inf \{ ||T - Z||,Z\;{\text {in}}\;{\text {centre}}\;\mathfrak {A}\}$.
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