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



Graded $C^*$-algebras can be faithfully represented on a graded Hilbert space
In your case, if your algebra is separable, you can always produce a $\mathbb{Z}/2\mathbb{Z}$-invariant faithful state by averaging over the group: if $\phi$ is any faithful state, then $\psi(x) = \frac{1}{2}(\phi(x)+\phi(\alpha(x)))$ is $\mathbb{Z}/2\mathbb{Z}$-invariant (and, obviously, faithful). 


Much more generally, if $G$ is any locally compact group acting on a $C^\ast$-algebra $A$, then there exists a faithful representation $\pi$ of $A$ on a Hilbert space and a unitary representation $\rho$ of $G$ so that $\rho(g) \pi(x) \rho(g)^\ast = \pi(g\cdot x)$. This is standard in the theory of crossed products (indeed, this is how one shows that crossed product $C^\ast$-algebras exist, since an equivariant representation of this kind is exactly a representation of the universal $C^\ast$-crossed product) -- I think one place to look would be G. Pedersen's book on $C^\ast$-algebras and their automorphism groups.


