Abstract:
Let $A$ be a complex $*$-algebra with involution $a\to a^+$. Let $\mathcal X$ be a Hilbert $\frak A$-module for a $C^*$-algebra $\frak A$ and $\mathcal D$ a $\frak B$-submodule of $\mathcal X$ for some $*$-subalgebra $\frak B$ of $\frak A$. A $*$-representation of $A$ on $\mathcal D$ is an algebra homomorphism $\pi$ of $A$ into the algebra of $\frak B$-linear operators of $\mathcal D$ such that $\langle \pi(a)x, y\rangle_{\mathcal X} = \langle x, \pi(a^+)y\rangle_{\mathcal X}$ for $a\in A$, $x, y \in\mathcal X$. An important special case is when $\frak B = \frak A$. Any Hilbert space $*$-representation of the $C^*$-algebra $\frak A$ induces a $*$-representation of $A$ on some dense domain of the Hilbert space. This induction procedure is developed in detail. Various examples (Hermitean quantum plane, enveloping algebras) are discussed.
