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In the GNS construction, we have a $C^*$-algebra $A$ and a state $\phi$. Then we can construct a representation $\Pi: A\rightarrow B(H_\phi)$ for some GNS Hilbert space $H_\phi$.

I wonder if there's any useful general results regarding the relation between the norm in $A$ and that in $\Pi(A)$. I know if $\Pi$ is injective, it is isometric. In general, $\Pi$ is only contractive, i.e. $\|\Pi(A)\|\leq \|A\|$. Equality is always attained by unitary elements. Also clear is the lower bound $\|\Pi(A)\|^2\geq \phi(A^*A)$.

I'm interested in any results regarding this question: results for a specific class of elements or a specific class of algebras. If there isn't any, are things just behave randomly in between the two obvious bounds? Are there elements that attains the lower bound (besides identity)?

Alonso Delfín
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Singularity
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1 Answers1

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If $\phi$ is faithful, then $\pi$ is isometric. Otherwise, there is very little you can say.

The representation $\pi$ may fail to preserve the norm for "most" elements. For instance let $A=C[0,1]$, and $\phi(f)=f(0)$. Then $H_\phi=\mathbb C$, and $\pi(f)\lambda=f(0)\lambda$, $\|\pi(f)\|=|f(0)|$. So here $\pi$ will decrease norm for any $f$ that does not achieve its norm at $0$. And your lower bound is always achieved.

In general, little can be said, as a non-faithful state fails to "see" part of the algebra.

Martin Argerami
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