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nLab > Latest Changes: state on a star-algebra

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  1.  
    • 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

1 to 8 of 8

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