If $\pi$ is a zero representation of $C^*$ algebra $A$,there is no state$\tau$ on $A$ such that $\tau(a)=(\pi(a)\xi,\xi)$.
When we talk about GNS constuction,Should the zero representation be considered?
2026-03-26 08:31:18.1774513878
a question on gns construction
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The "GNS construction" starts with a given state $\tau$. With the C$^*$-algebra $A$ and the state $\tau$ you construct a Hilbert space $H$ and a representation $\pi:A\to B(H)$.
Any state is nonzero, so the zero representation cannot arise from the GNS construction.