added some more references, such as the useful

- James D. Steward,
*Positive definite functions and generalizations, an historical survey*, The Rocky Mountain Journal of Mathematics**6**3 (1976) 409-434 [jstor:44236118]

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]

added (here) brief statement of the example/theorem asserting that quantum states on group algebras are equivalently unitary representations with a cyclic vector.

]]>added (here) at least brief mentioning of “positive linear functionals”.

]]>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.

]]>I am confused about Proposition 3.1 where you say that L^1(Omega) is an algebra under pointwise operations, where $Omega$ is a probability space,

since the product of two integrable functions is not necessarily integrable (for example, the reciprocal of the square root of x, where Omega is the interval (0,1), multiplied by itself, is not integrable). I cannot see where I am wrong.

Thanks for your attention.

Fausto di Biase

fausto.dibiase@unich.it ]]>

added this pointer:

- Anatoly Vershik, Def. 1 in:
*Gel’fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis*, pp. 619 in: Pavel Etingof, Vladimir S. Retakh, Isadore Singer (eds.)*The Unity of Mathematics*In Honor of the Ninetieth Birthday of I. M. Gelfand, (arXiv:math/0503140)

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 pointer to:

- Paolo Facchi, Giovanni Gramegna, Arturo Konderak, around (6) in:
*Entropy of quantum states*(arXiv:2104.12611, spire:1860877)

added pointer to:

- Paul-André Meyer, Section I.1.1 in:
*Quantum Probability for Probabilists*, Lecture Notes in Mathematics**1538**, Springer 1995 (doi:10.1007/BFb0084701)

added a sentence at the very beginning, connecting back to quantum probability theory and AQFT

]]>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 little bit more to *state on a star-algebra*, cross-linked with *pure state*

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.

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