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I have expanded the Idea section at state on a star-algebra and added a bunch of references.
The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.
added a little bit more to state on a star-algebra, cross-linked with pure state
Started an Examples-section (here) with making explicit the two archetypical examples (classical probability measure as state on measurable functions and element on Hilbert space as state on bounded operators).
added a sentence at the very beginning, connecting back to quantum probability theory and AQFT
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Under “Properties – Closure properties” I added mentioning of convex combinations of states
and then I added (here) the “operator-state correspondence” (one way) saying that for $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$ a state, with a non-null observable $O \in \mathcal{A}$, $\rho(O^\ast O) \neq 0$, then also
$\rho_O \;\colon\; A \;\mapsto\; \tfrac{1}{ \rho(O^\ast O) } \cdot \rho\big( O^\ast \cdot A \cdot O \big)$is a state.
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Thanks for catching this, it was of course not stated correctly. I have now adjusted the wording (here, adding the previously missing condition that functions vanish at infinity) and have added a pointer to a textbook reference with more details.
This could certainly be expanded on further, but I leave it as is for the moment. If you feel like improving on it, please be invited to edit.
added some more references, such as the useful
and the original reference for the characterization of states on group algebras:
И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943) 301–316 [mathnet pdf]
Israel Gelfand, Dmitri Raikov, Irreducible unitary representations of locally bicompact groups, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [mathnet:eng/sm6181]
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