I think that there is no tracial state on non-unital simple $C^*$ algebras.Is my thought correct? I 'll appreciate it if anyone can supply me a counterexample.
2026-03-26 04:31:19.1774499479
tracial state on a non-unital simple $C^*$ algebra
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No this is not correct. There are simple non-unital C*-algebras that have tracial states.
Maybe not the easiest example, but the so-called Jacelon-Razak algebra is such an example. It is obtained as an inductive limit of certain non-unital C*-algebras, such that the limit algebra has a unique tracial state and is simple.