nLab
relative commutant

Contents

Definition

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

𝒜 c():={B:BA=AB,A𝒜} \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()=𝒜 \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)