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Abstract:

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

Registro:

Documento: Artículo
Título:Cyclic homology, tight crossed products, and small stabilizations
Autor:Cortiñas, G.
Filiación:Department of Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria Pab 1, Buenos Aires, 1428, Argentina
Palabras clave:Calkin's theorem; Crossed product; Cyclic homology; Karoubi's cone; Operator ideal
Año:2014
Volumen:8
Número:4
Página de inicio:1191
Página de fin:1223
DOI: http://dx.doi.org/10.4171/JNCG/184
Título revista:Journal of Noncommutative Geometry
Título revista abreviado:J. Noncommunitative Geom.
ISSN:16616952
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16616952_v8_n4_p1191_Cortinas

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Citas:

---------- APA ----------
(2014) . Cyclic homology, tight crossed products, and small stabilizations. Journal of Noncommutative Geometry, 8(4), 1191-1223.
http://dx.doi.org/10.4171/JNCG/184
---------- CHICAGO ----------
Cortiñas, G. "Cyclic homology, tight crossed products, and small stabilizations" . Journal of Noncommutative Geometry 8, no. 4 (2014) : 1191-1223.
http://dx.doi.org/10.4171/JNCG/184
---------- MLA ----------
Cortiñas, G. "Cyclic homology, tight crossed products, and small stabilizations" . Journal of Noncommutative Geometry, vol. 8, no. 4, 2014, pp. 1191-1223.
http://dx.doi.org/10.4171/JNCG/184
---------- VANCOUVER ----------
Cortiñas, G. Cyclic homology, tight crossed products, and small stabilizations. J. Noncommunitative Geom. 2014;8(4):1191-1223.
http://dx.doi.org/10.4171/JNCG/184