Contents

Definition

Let $𝒜\subset ℬ$ be an inclusion of $*-$ algebras. The relative commutant ${𝒜}^{c}\left(ℬ\right)$ is defined by

${𝒜}^{c}\left(ℬ\right):=\left\{B\in ℬ:BA=AB,\phantom{\rule{thickmathspace}{0ex}}A\in 𝒜\right\}$\mathcal{A}^c(\mathcal{B}) := \{ B \in \mathcal{B} : B A = A B, \; A \in \mathcal{A} \}

If the algebras are operator algebras defined on a Hilbert space, then

${𝒜}^{c}\left(ℬ\right)=𝒜\prime \bigcap ℬ$\mathcal{A}^c(\mathcal{B}) = \mathcal{A}' \bigcap \mathcal{B}

Revised on May 4, 2011 09:37:36 by Urs Schreiber (87.212.203.135)