Pardon me if this problem is obvious. I am trying to figure out if given a $C^*$-algebra $A$, whether every closed subspace of $A^*$ has non-trivial positive functionals in it.
Thanks!
2026-03-30 17:15:51.1774890951
Does a closed subspace of the dual of a C*-algebra always have positive elements?
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What about the linear span of a single element of $A^*$ that is neither positive nor negative?