'Twisted duality' in the ${\rm C^*}$ Clifford algebra

TL;DR

This paper establishes a duality relationship in the ${ m C}^*$ Clifford algebra, showing that the supercommutant of a subalgebra corresponds to the Clifford algebra of the orthogonal complement.

Contribution

It proves a new duality property in ${ m C}^*$ Clifford algebras relating subalgebras and their supercommutants.

Findings

Supercommutant of $C[Z]$ equals $C[Z^{ot}]$ in $C[V]$

Provides a duality framework for subalgebras in Clifford algebras

Enhances understanding of algebraic structures in quantum physics

Abstract

Let $V$ be a real inner product space and $C[V]$ its ${\rm C}^*$ Clifford algebra. We prove that if $Z$ is a subspace of $V$ then $C[Z^{\perp}]$ coincides with the supercommutant of $C[Z]$ in $C[V]$.