The Structure and Represenation of a ${\mathrm{C}}^{\ast }$-Algebra Associated to the Super-Poincaré Group

Abstract

The representation theory of a class of algebras associated to certain graded Lie groups is investigated. To a group whose even part is central is associated a natural involutive algebra all of whose ${}^{\ast }$-representations factor through a quotient algebra of continuous Clifford algebra-valued fields. The irreducible representations of crossed products of the algebra by a Lie algebra, such as the super-Poincaré group, are then constructed by Takesaki's method. It is then shown that they may also be constructed by Rieffel's ${\mathrm{C}}^{\ast }$-algebraic induction. Tensor product decompositions are briefly discussed.