Symmetry in tensor algebras over Hilbert space

Abstract

This paper deals with three issues: (1) Unitary representations $U$ of a scale of (finite and infinite dimensional) non-compact Lie groups $G(H)$ built on a fixed complex Hilbert space $H$; and their covariant systems. Our computations for these representations make use of the associated Lie algebras. (2) The covariant representations involve the $C^{*}$-algebras going by the names, the Toeplitz algebras, and the Cuntz algebras. (3) An essential result which also is used throughout is our computation of the commutant of the unitary representation $U$ of $G(H)$ mentioned in (1). For a fixed Hilbert space $H$, we apportion the commutant as a specific projective limit-algebra of operators.