Homotopy invariance through small stabilizations
| Autor: | Beatriz Abadie, Guillermo Cortiñas |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Homotopy invariance
Matemáticas Cyclic homology Calkin's theorem Matemática Pura Combinatorics Crossed product Operator ideal Mathematics::K-Theory and Homology FOS: Mathematics Inverse semigroup crossed products Operator Algebras (math.OA) Mathematics::Representation Theory Mathematics Algebra and Number Theory Mathematics::Commutative Algebra Homotopy Mathematics - Operator Algebras K-Theory and Homology (math.KT) Mathematics - Rings and Algebras Invariant (physics) Operator ideals Number theory Rings and Algebras (math.RA) Bounded function Mathematics - K-Theory and Homology Geometry and Topology Complex number CIENCIAS NATURALES Y EXACTAS |
| DOI: | 10.1007/s40062-013-0069-9 |
| Popis: | We associate an algebra $\Gami(\fA)$ to each bornological algebra $\fA$. The algebra $\Gami(\fA)$ contains a two-sided ideal $I_{S(\fA)}$ for each symmetric ideal $S\triqui\elli$ of bounded sequences of complex numbers. In the case of $\Gami=\Gami(\C)$, these are all the two-sided ideals, and $I_S\mapsto J_S=\cB I_S\cB$ gives a bijection between the two-sided ideals of $\Gami$ and those of $\cB=\cB(\ell^2)$. We prove that Weibel's $K$-theory groups $KH_*(I_{S(\fA)})$ are homotopy invariant for certain ideals $S$ including $c_0$ and $\ell^p$. Moreover, if either $S=c_0$ and $\fA$ is a local $C^*$-algebra or $S=\ell^p,\ell^{p\pm}$ and $\fA$ is a local Banach algebra, then $KH_*(I_{S(\fA)})$ contains $K_*^{\top}(\fA)$ as a direct summand. Furthermore, we prove that for $S\in\{c_0,\ell^p,\ell^{p\pm}\}$ the map $K_*(\Gamma^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)})$ fits into a long exact sequence with the relative cyclic homology groups $HC_*(\Gamma^\infty(\fA):I_{S(\fA)})$. Thus the latter groups measure the failure of the former map to be an isomorphism. Comment: 32 pages. The original paper has been split into two parts, of which this is the first part. The second part is now arXiv:1304.3508 |
| Databáze: | OpenAIRE |
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