# Graded $C^*$-algebras can be faithfully represented on a graded Hilbert space

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

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What is a graded $C^{*}$-algebra? And what are its odd elements? –  Benjamin Hayes May 7 '11 at 14:01
A $\mathbb{Z}/2$ graded C*-algebra is a C*-algebra $A$ with a *-automorphism $\alpha$ st $\alpha^2=1_A$. the even elements is the elements $a$ st $\alpha(a)=a$ and the odd elements is the elements $a$ st $\alpha(a)=-a$. –  m07kl May 7 '11 at 15:49
An equivalent definition of a graded $C^*$−algebra is as a $C^*$−algebra with a $\mathbb Z/2$-action. –  Rasmus Bentmann May 7 '11 at 20:13
Hence you can ask more generally, whether every $G$−$C^*$-algebra has a faithful equivariant representation on a $G$-Hilbert space. –  Rasmus Bentmann May 7 '11 at 20:16
I try to show that the graded representations of $A\widehat{\otimes}B$ on a graded Hilbert space are in natural one-one correspondence with the pairs of graded -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

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.
I know this, but Hilbert space $H$ should be graded and faithful represenation should respect the gradings on $A$ and $H$. –  m07kl May 8 '11 at 9:46
As another comment: I think you are confusing faithfulness of the GNS representation with existence of states that separate points. The separation property you need is that if $x\neq y$, then there exists a state $\phi$ and elements $z,w$ so that $\phi(wxz)\neq \phi(wyz)$ which is much weaker than $\phi(x)\neq \phi(y)$. –  Dima Shlyakhtenko May 8 '11 at 16:37