Timeline for (Topological) K-theory for commutative $C^*$-algebras: operator and standard approaches
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 7, 2016 at 2:12 | comment | added | Denis Nardin | @truebaran If you have a vector bundle $V$ on $Y$ choose an embedding into a trivial vector bundle. Then the pullback of the embedding gives an embedding of the pullback into a trivial vector bundle. The orthogonal projections corresponding to these embeddings satisfy the naturality property you seek. | |
| May 6, 2016 at 23:16 | comment | added | truebaran | @Paul Siegel, I haven't found the answer in Blackadar's book (only exercise 9.4.3. at the end of chapter IV which does not answet my question: it only states that the Gelfand transform induces the isomorphism) | |
| May 6, 2016 at 23:16 | comment | added | truebaran | @Denis Nardin thank you for your comment, now I understand how to proceed to solve the fisrt problem. However could you provide some more details for the second one? | |
| May 6, 2016 at 2:36 | comment | added | Paul Siegel | Blackadar's "K-theory for Operator Algebras" provides textbook answers to both of these questions. | |
| May 6, 2016 at 0:01 | comment | added | Denis Nardin | In fact, even better: show that isomorphism classes of vector bundles of rank $k$ are the same things as equivalence classes of vector bundles of rank $k$ together with an embedding in $X\times \mathbb{R}^\infty$. Then you can send $(V,\eta:V\to X\times \mathbb{R}^\infty)$ to the function $X\to M_\infty(\mathbb{R})$ sending every point $x\in X$ to the orthogonal projection onto $V_x$. | |
| May 5, 2016 at 23:53 | comment | added | Denis Nardin | For problem 1 just put a metric on your vector bundle, embed the vector bundle isometrically in some trivial bundle and take the corresponding orthogonal projection. For problem 2, pullback the whole thing (metric, trivial bundle, isometrically embedding etc.). | |
| May 5, 2016 at 23:18 | history | asked | truebaran | CC BY-SA 3.0 |