Cyclic homology, tight crossed products, and small stabilizations
In [1] we associated an algebra (Formula presented.) to every bornological algebra (Formula presented.) and an ideal (Formula presented.) to every symmetric ideal (Formula presented.). We showed that (Formula presented.) has K-theoretical properties which are similar to those of the usual stabilizat...
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todo:paper_16616952_v8_n4_p1191_Cortinas2023-10-03T16:28:41Z Cyclic homology, tight crossed products, and small stabilizations Cortiñas, G. Calkin's theorem Crossed product Cyclic homology Karoubi's cone Operator ideal In [1] we associated an algebra (Formula presented.) to every bornological algebra (Formula presented.) and an ideal (Formula presented.) to every symmetric ideal (Formula presented.). We showed that (Formula presented.) has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal (Formula presented) of the algebra β of bounded operators in Hilbert space which corresponds to S under Calkin's correspondence. In the current article we compute the relative cyclic homology (Formula presented.). Using these calculations, and the results of loc. cit., we prove that if (Formula presented) is a C∗-algebra and c0 the symmetric ideal of sequences vanishing at infinity, then (Formula presented.) is homotopy invariant, and that if ∗≥ 0, it contains (Formula presented.) as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem ([20]) that says that for the ideal Κ = Jc0 of compact operators and the C∗-algebra tensor product (Formula presented.), we have (Formula presented.). Similarly, we prove that if (Formula presented.) is a unital Banach algebra and (Formula presented.), then (Formula presented.) is invariant under Hölder continuous homotopies, and that for ∗≥ 0 it contains (Formula presented.) as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups (Formula presented.) in terms of (Formula presented.) for general (Formula presented.) and S. For (Formula presented.) = ℂ and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map (Formula presented.) is an isomorphism in many cases. © European Mathematical Society Fil:Cortiñas, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_16616952_v8_n4_p1191_Cortinas |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Calkin's theorem Crossed product Cyclic homology Karoubi's cone Operator ideal |
| spellingShingle |
Calkin's theorem Crossed product Cyclic homology Karoubi's cone Operator ideal Cortiñas, G. Cyclic homology, tight crossed products, and small stabilizations |
| topic_facet |
Calkin's theorem Crossed product Cyclic homology Karoubi's cone Operator ideal |
| description |
In [1] we associated an algebra (Formula presented.) to every bornological algebra (Formula presented.) and an ideal (Formula presented.) to every symmetric ideal (Formula presented.). We showed that (Formula presented.) has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal (Formula presented) of the algebra β of bounded operators in Hilbert space which corresponds to S under Calkin's correspondence. In the current article we compute the relative cyclic homology (Formula presented.). Using these calculations, and the results of loc. cit., we prove that if (Formula presented) is a C∗-algebra and c0 the symmetric ideal of sequences vanishing at infinity, then (Formula presented.) is homotopy invariant, and that if ∗≥ 0, it contains (Formula presented.) as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem ([20]) that says that for the ideal Κ = Jc0 of compact operators and the C∗-algebra tensor product (Formula presented.), we have (Formula presented.). Similarly, we prove that if (Formula presented.) is a unital Banach algebra and (Formula presented.), then (Formula presented.) is invariant under Hölder continuous homotopies, and that for ∗≥ 0 it contains (Formula presented.) as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups (Formula presented.) in terms of (Formula presented.) for general (Formula presented.) and S. For (Formula presented.) = ℂ and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map (Formula presented.) is an isomorphism in many cases. © European Mathematical Society |
| format |
JOUR |
| author |
Cortiñas, G. |
| author_facet |
Cortiñas, G. |
| author_sort |
Cortiñas, G. |
| title |
Cyclic homology, tight crossed products, and small stabilizations |
| title_short |
Cyclic homology, tight crossed products, and small stabilizations |
| title_full |
Cyclic homology, tight crossed products, and small stabilizations |
| title_fullStr |
Cyclic homology, tight crossed products, and small stabilizations |
| title_full_unstemmed |
Cyclic homology, tight crossed products, and small stabilizations |
| title_sort |
cyclic homology, tight crossed products, and small stabilizations |
| url |
http://hdl.handle.net/20.500.12110/paper_16616952_v8_n4_p1191_Cortinas |
| work_keys_str_mv |
AT cortinasg cyclichomologytightcrossedproductsandsmallstabilizations |
| _version_ |
1807316509426450432 |