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nLab > Latest Changes: state on a star-algebra

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2017
    • (edited Nov 30th 2017)

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2017
    • (edited Dec 3rd 2017)

    added a little bit more to state on a star-algebra, cross-linked with pure state

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2017

    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).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2020
    • (edited Jan 20th 2020)

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

    diff, v28, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2021

    added pointer to:

    diff, v32, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    added pointer to:

    diff, v33, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    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 ρ:𝒜→ℂ\rho \;\colon\; \mathcal{A} \to \mathbb{C} a state, with a non-null observable O∈𝒜O \in \mathcal{A}, ρ(O *O)≠0\rho(O^\ast O) \neq 0, then also

    ρ O:A↦1ρ(O *O)⋅ρ(O *⋅A⋅O) \rho_O \;\colon\; A \;\mapsto\; \tfrac{1}{ \rho(O^\ast O) } \cdot \rho\big( O^\ast \cdot A \cdot O \big)

    is a state.

    diff, v34, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2021

    added this pointer:

    diff, v36, current

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeSep 16th 2021
    Hi,
    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
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2021

    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.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2023

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

    diff, v42, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2023
    • (edited Jul 8th 2023)

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

    diff, v42, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2023
    • (edited Jul 9th 2023)

    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]

    diff, v45, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2024
    • (edited Jul 27th 2024)

    added pointer to:

    diff, v52, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2024

    added pointer to:

    diff, v53, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2024

    added (here) brief statement of the relation between multiplicative and pure states

    diff, v53, current

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