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Let $A$ be a c-star algebra acting on a non separable Hilbert space. Can one always define a faithful normal state on it?

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No. That's the reason one considers weights.

For an easy example consider the von Neumann algebra $\ell^\infty(\mathbb R)$. Then, if $\{e_t\}$ denotes the canonical elements (that is, $e_t(r)=\delta_{r,t}$) you have the net of projections $$ p_t=\sum_{s\leq t}e_t. $$ This net converges strongly to the identity. If you had a faithful normal state $f$, we would have $f(p_t)\to f(I)=1$. This would imply that $f(e_t)=0$ for all $t$, a contradiction (you can check this by getting $f(p_{t+\varepsilon}-p_t)$ arbitrarily small, and putting a sequence in between).

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