Timeline for An extension of $K$-theory to topological $^*$-algebras
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2015 at 19:45 | answer | added | Kolya Ivankov | timeline score: 5 | |
| Aug 29, 2015 at 15:03 | comment | added | Yemon Choi | One person who seems to have looked properly at how algebraic and topological K-theory for operator algebras relate to each other is Jonathan Rosenberg. I recommend math.umd.edu/~jmr/algtopK.pdf | |
| Aug 29, 2015 at 15:00 | comment | added | Yemon Choi | @K.J.Moi It is far from obvious to me that your recipe would give the "usual" K-theory for Banach or C*-algebras. Do you have any evidence to substantiate this intuition? | |
| Aug 27, 2015 at 15:05 | comment | added | K.J. Moi | You could for instance take a look at chapter IV of the K-book: math.rutgers.edu/~weibel/Kbook.html | |
| Aug 27, 2015 at 14:51 | comment | added | Jonathan Gleason | "The algebraic K-groups of a ring A are the homotopy groups of a certain space (or spectrum) K(A) which is defined in terms of the category of finitely generated projective A-modules" --- Do you have a reference for this so I can study it in more detail? And also possibly a reference for the extension of algebraic $K$-theory to schemes? | |
| Aug 27, 2015 at 14:50 | comment | added | K.J. Moi | However, I'm not sure that this would capture the topology in a good way if your ring were say the $p$-adic numbers with their usual topology. Maybe someone else can comment on that. | |
| Aug 27, 2015 at 14:46 | comment | added | K.J. Moi | First of all I think you can drop the $\ast$. The algebraic K-groups of a ring A are the homotopy groups of a certain space (or spectrum) K(A) which is defined in terms of the category of finitely generated projective A-modules. If A has a topology you can incorporate this into the construction in a way that (should) give the usual thing for Banach/C*-algebras and definitely gives the right thing for discrete rings. | |
| Aug 27, 2015 at 13:19 | comment | added | Jonathan Gleason | @K.J.Moi The immediate interest is more in having a conceptual unification than because I am interested in the $K$-theory of topological $^*$-algebras that are neither purely algebraic nor $C^*$. Not only do I find it aesthetically pleasing, but I also usually find unifications like this to aid in understanding of the special cases. | |
| Aug 27, 2015 at 11:38 | comment | added | K.J. Moi | If you take Segal's $\Gamma$-space construction or Waldhausen's $S.$-construction of $K$-theory and topologize your hom-sets using the topology of your algebra I think you get something that satisfies your criteria, but I don't know if it behaves the way you want on topological $\ast$-algebras that are neither discrete nor $C^{\ast}$. Which kinds of algebras do you have in mind? | |
| Aug 26, 2015 at 18:03 | history | edited | Jonathan Gleason | CC BY-SA 3.0 |
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| Aug 26, 2015 at 17:38 | comment | added | Tomasz Kania | Might be relevant: research.lancs.ac.uk/portal/en/publications/… | |
| Aug 26, 2015 at 17:20 | comment | added | Yemon Choi | If you want something a bit closer to K-theory for Banach algebras than (notoriously uncomputable) algebraic K-theory, how about the variants explored by Cuntz and others? See springer.com/us/book/9783764383985 for some blurb and perhaps some pointers to the literature -- I'm afraid I don't really know enough about said literature, though | |
| Aug 26, 2015 at 17:17 | comment | added | Yemon Choi | @BranimirĆaćić However, on general Banach $*$-algebras the theories can differ, IIRC | |
| Aug 26, 2015 at 16:21 | answer | added | Simon Henry | timeline score: 5 | |
| Aug 26, 2015 at 16:20 | comment | added | Branimir Ćaćić | You can forget about involutions entirely, since the $K$-theory of a $C^\ast$-algebra as a $C^\ast$-algebra (i.e., in terms of projections and unitaries) is identical to its $K$-theory as a Banach algebra (i.e., in terms of idempotents and invertible matrices). | |
| Aug 26, 2015 at 16:12 | history | edited | Jonathan Gleason | CC BY-SA 3.0 |
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| Aug 26, 2015 at 15:37 | history | asked | Jonathan Gleason | CC BY-SA 3.0 |