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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
added pointer to:
added pointer to:
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.
added this pointer:
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.
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