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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Autor principal:	Cortiñas, Guillermo Horacio
Publicado:	2014
Materias:	Calkin's theorem Crossed product Cyclic homology Karoubi's cone Operator ideal
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16616952_v8_n4_p1191_Cortinas http://hdl.handle.net/20.500.12110/paper_16616952_v8_n4_p1191_Cortinas
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_16616952_v8_n4_p1191_Cortinas
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spelling	paper:paper_16616952_v8_n4_p1191_Cortinas2023-06-08T16:25:43Z Cyclic homology, tight crossed products, and small stabilizations Cortiñas, Guillermo Horacio 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. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16616952_v8_n4_p1191_Cortinas 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, Guillermo Horacio 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
author	Cortiñas, Guillermo Horacio
author_facet	Cortiñas, Guillermo Horacio
author_sort	Cortiñas, Guillermo Horacio
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
publishDate	2014
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16616952_v8_n4_p1191_Cortinas http://hdl.handle.net/20.500.12110/paper_16616952_v8_n4_p1191_Cortinas
work_keys_str_mv	AT cortinasguillermohoracio cyclichomologytightcrossedproductsandsmallstabilizations
_version_	1768546692739629056

Cyclic homology, tight crossed products, and small stabilizations

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