Does there exist a non-unital $C^*$ algebra which have uncountable tracial states?
2026-03-26 04:32:32.1774499552
tracial state on a non-unital $C^*$ algebra
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Yes. Let $X$ be an uncountable, non-compact, locally compact Hausdorff space (for example, $X=\mathbb R$), and consider the algebra $C_0(X)$. This is non-unital, and the collection of all evaluation functionals $f\mapsto f(x)$ form an uncountable family of tracial states on $C_0(X)$.