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In very rough terms, let $A$ be a complex unital $C^*$-algebra. Assume it nuclear for convenience, but it doesn't matter much. Consider the 'Fock'-type $C^*$-algebra (don't know a better name for it) $...

Let $T_+$ be the set of a positive trace-class operators over some separable Hilbert space and $A: T_+ \to \mathbb{R}\cup \{\infty\}$ some linear functional. In general, $A$ will not be continuous. ...