Comparison between algebraic and topological Ktheory of locally convex algebras
This paper is concerned with the algebraic Ktheory of locally convex Calgebras stabilized by operator ideals, and its comparison with topological Ktheory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological Ktheory agree on the compl...
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2008


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paper:paper_00018708_v218_n1_p266_Cortinas20230608T14:21:43Z Comparison between algebraic and topological Ktheory of locally convex algebras Algebraic and topological Ktheory Karoubi's conjecture Locally convex algebra Operator ideal This paper is concerned with the algebraic Ktheory of locally convex Calgebras stabilized by operator ideals, and its comparison with topological Ktheory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological Ktheory agree on the completed projective tensor product L over(⊗, ̂) J, and that the obstruction for the comparison map K (L over(⊗, ̂) J) → Ktop (L over(⊗, ̂) J) to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1). {A figure is presented}. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of submultiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic Ktheory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485496]. We also consider stabilization with respect to a wider class of operator ideals, called subharmonic. Every Fréchet ideal is subharmonic, but not conversely; for example the Schatten ideal Lp is subharmonic for all p > 0 but is Fréchet only if p ≥ 1. We prove a variant of the exact sequence above which essentially says that if A is a Calgebra and J is subharmonic, then the obstruction for the periodicity of K* (A ⊗C J) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for Hunital algebras announced in [M. Wodzicki, Algebraic Ktheory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485496]. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal Ktheory, due to the first author. © 2007 Elsevier Inc. All rights reserved. 2008 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v218_n1_p266_Cortinas http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p266_Cortinas 
institution 
Universidad de Buenos Aires 
institution_str 
I28 
repository_str 
R134 
collection 
Biblioteca Digital  Facultad de Ciencias Exactas y Naturales (UBA) 
topic 
Algebraic and topological Ktheory Karoubi's conjecture Locally convex algebra Operator ideal 
spellingShingle 
Algebraic and topological Ktheory Karoubi's conjecture Locally convex algebra Operator ideal Comparison between algebraic and topological Ktheory of locally convex algebras 
topic_facet 
Algebraic and topological Ktheory Karoubi's conjecture Locally convex algebra Operator ideal 
description 
This paper is concerned with the algebraic Ktheory of locally convex Calgebras stabilized by operator ideals, and its comparison with topological Ktheory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological Ktheory agree on the completed projective tensor product L over(⊗, ̂) J, and that the obstruction for the comparison map K (L over(⊗, ̂) J) → Ktop (L over(⊗, ̂) J) to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1). {A figure is presented}. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of submultiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic Ktheory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485496]. We also consider stabilization with respect to a wider class of operator ideals, called subharmonic. Every Fréchet ideal is subharmonic, but not conversely; for example the Schatten ideal Lp is subharmonic for all p > 0 but is Fréchet only if p ≥ 1. We prove a variant of the exact sequence above which essentially says that if A is a Calgebra and J is subharmonic, then the obstruction for the periodicity of K* (A ⊗C J) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for Hunital algebras announced in [M. Wodzicki, Algebraic Ktheory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485496]. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal Ktheory, due to the first author. © 2007 Elsevier Inc. All rights reserved. 
title 
Comparison between algebraic and topological Ktheory of locally convex algebras 
title_short 
Comparison between algebraic and topological Ktheory of locally convex algebras 
title_full 
Comparison between algebraic and topological Ktheory of locally convex algebras 
title_fullStr 
Comparison between algebraic and topological Ktheory of locally convex algebras 
title_full_unstemmed 
Comparison between algebraic and topological Ktheory of locally convex algebras 
title_sort 
comparison between algebraic and topological ktheory of locally convex algebras 
publishDate 
2008 
url 
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v218_n1_p266_Cortinas http://hdl.handle.net/20.500.12110/paper_00018708_v218_n1_p266_Cortinas 
_version_ 
1768543061949808640 