Hi everyone I try to use GNS-construction to show every graded C*-algebras can be faithfully represented on a graded Hilbert space. If $A$ is a graded C*-algebra with grading automorphism $\alpha$ of order 2, $\phi$ is a state on $A$. I think I need $\alpha (\phi(a))=\phi(a)$ for all $a\in A$ to get a grading on the Hilbert space coming from $\phi$. Since automorphism is given, we need to restrict to homogeneous states (i.e. states that vanish on odd elements in $A$). But I can not see that homogeneous states separete points of $A$, hence the universal representation is not injective. Is there someone that can help me to prove this fact?

Thanks

`$C^{*}$`

-algebra? And what are its odd elements? – Benjamin Hayes May 7 '11 at 14:01-homomorphisms $\phi:A \rightarrow C$, $\psi:A\rightarrow C$ with graded commuting ranges, where $C$ is some graded C-algebra. So one direction I need that a graded C∗-algebras can be faithfully represented on a graded Hilbert space. The another direction I need the fact that a graded rep of $A\widehat{\otimes}B$ on a graded Hilbert space $H$ can be restrict to a pair of graded reps on $H$. – m07kl May 7 '11 at 22:16