Ralf Meyer,Mathematisches Institut, Georg-August Universitaet Goettingen, Bunsenstrasse 3-5, 37073 Goettingen, Germany; email: rmeyer2@uni-goettingen.de

Abstract/Résumé:

The index map in topological K-theory for real Banach algebra extensions is a natural transformation from the first K-theory of the quotient to the zeroth K-theory of the ideal. We show that any such natural transformation is an integer multiple of the index map.

L’application index dans la K-théorie des extensions d’algèbres de Banach réelles est une transformation naturelle entre le premier K-groupe de l’algèbre quotient et le zéroième K-groupe de l’idéal. On démontre qu’une telle transformation naturelle doit être un multiple intégral de l’application index.

Keywords: Index map, real K-theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Real K-theory

George A. Elliott,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in the theory of real Banach algebras is unique as a natural transformation, up to an integral multiple, and modulo a (unique) two-torsion “ghost” map arising from the order-two K\(_1\)-group of the Banach algebra \({\mathbb R}\) (of real numbers). (In the earlier paper this was shown for complex Banach algebras, of course without the “ghost” map, but in way—using Bott periodicity to pass to the opposite parity—that is not available for real Banach algebras. The present approach yields a new proof in the complex case.)

On démontre que l’application index dans la K-théorie des algèbres de Banach réelles (ou complexes) est essentiellment unique.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-theory, II

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in Banach algebra K-theory, as a natural map from the K\(_1\)-group of a quotient of a Banach algebra to the K\(_0\)-group of the corresponding ideal, is unique (up to an integral multiple).

Il est démontré que l’application index dans la K-théorie des algèbres de Banach est unique, dans un sens très naturel.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory 19K56

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-Theory