Let $(\A, \A_o)$ be a topological \qxa, which means in particular that $\A_o$ is a topological \xa, dense in \A. Let $\pi^o$ be a *-representation of $\A_o$ in some pre-Hilbert space $\D \subset \H$. Then we present several ways of extending $\pi^o$, by closure, to some larger \qxa\ contained in \A, either by Hilbert space operators, or by sesquilinear forms on \D. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.
Antoine, J., Bagarello, F., Trapani, C. (1998). Extension of representations in quasi *-algebras. ANNALES DE L'INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE, 69(6), 241-264.
Extension of representations in quasi *-algebras
BAGARELLO, Fabio;TRAPANI, Camillo
1998-01-01
Abstract
Let $(\A, \A_o)$ be a topological \qxa, which means in particular that $\A_o$ is a topological \xa, dense in \A. Let $\pi^o$ be a *-representation of $\A_o$ in some pre-Hilbert space $\D \subset \H$. Then we present several ways of extending $\pi^o$, by closure, to some larger \qxa\ contained in \A, either by Hilbert space operators, or by sesquilinear forms on \D. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.
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