Hi!


While studying C\*-algebras I found 2 different definitions for non degenerate representations (*-homomorphisms $\pi:\mathcal{A} \rightarrow B(\mathcal{h})$ where $\mathcal{A}$ is a C*-algebra and $B(\mathcal{h})$ is the space of bounded linear operators on some Hilbert space $\mathcal{h}$):


1) For every non-zero $\xi \in \mathcal{h}$ there exists $a \in \mathcal{A}$ such that $\pi(a)\xi \neq 0$;


2) The set $\{\pi(a)\xi \quad a \in \mathcal{A}, \xi \in \mathcal{h}\}$ is dense in $\mathcal{h}$.


Are they equivalent?


Thanks,
Alessandro



Non degenerate representations for C*-algebras
Yes they are. This is Proposition I.9.2 in Theory of Operator Algebras I by Takesaki.


Short proof:


2) => 1): suppose $\pi(a)\xi = 0$ for all $a$. Then $(\pi(a)\eta|\xi) = 0$ for all $\eta\in h$
and $a\in\mathcal{A}$ hence $\xi=0$.


1) => 2): Take $\xi \in h$ orthogonal to all $\pi(a) \eta$. Then from $(\xi| \pi(a^\* a) \xi)= 0$ for all $a$ it follows that $\xi =0$.


