[Merged by Bors] - feat: the Fuglede–Putnam–Rosenblum theorem for C⋆-algebras#37569
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[Merged by Bors] - feat: the Fuglede–Putnam–Rosenblum theorem for C⋆-algebras#37569j-loreaux wants to merge 10 commits intoleanprover-community:masterfrom
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PR summary 35186be320Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Thanks for doing this, it's a cool result and I'm happy we are in a position to add it to mathlib!
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Co-authored-by: David Loeffler <[email protected]>
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This is looking great now, just a few more pieces of unrestrained pedantry from me
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Co-authored-by: David Loeffler <[email protected]>
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Thanks, both to author and reviewer!
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Let `A` be a C⋆-algebra, and let `a b x : A`. The Fuglede–Putnam–Rosenblum theorem states that if `a` and `b` are normal and `x` intertwines `a` and `b` (i.e., `SemiconjBy x a b`). Then `x` also intertwines `star a` and `star b`. Fuglede's original result was for `a = b` (i.e., if `x` commutes with `a`, then `x` also commutes with `star a`), and Putnam extended it to intertwining elements. Rosenblum later gave the elementary proof formalized here using Liouville's theorem. A version of the Fuglede–Putnam theorem also holds for unbounded operators, but it necessitates a different proof technique and holds in different generality than the one given here.
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…r-community#37569) Let `A` be a C⋆-algebra, and let `a b x : A`. The Fuglede–Putnam–Rosenblum theorem states that if `a` and `b` are normal and `x` intertwines `a` and `b` (i.e., `SemiconjBy x a b`). Then `x` also intertwines `star a` and `star b`. Fuglede's original result was for `a = b` (i.e., if `x` commutes with `a`, then `x` also commutes with `star a`), and Putnam extended it to intertwining elements. Rosenblum later gave the elementary proof formalized here using Liouville's theorem. A version of the Fuglede–Putnam theorem also holds for unbounded operators, but it necessitates a different proof technique and holds in different generality than the one given here.
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Let
Abe a C⋆-algebra, and leta b x : A. The Fuglede–Putnam–Rosenblum theorem states that ifaandbare normal andxintertwinesaandb(i.e.,SemiconjBy x a b). Thenxalso intertwinesstar aandstar b. Fuglede's original result was fora = b(i.e., ifxcommutes witha, thenxalso commutes withstar a), and Putnam extended it to intertwining elements. Rosenblum later gave the elementary proof formalized here using Liouville's theorem.A version of the Fuglede–Putnam theorem also holds for unbounded operators, but it necessitates a different proof technique and holds in different generality than the one given here.