Norm of a derivation on a von Neumann algebra

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