Déterminant associé à une trace sur une algèbre de Banach
Annales de l'Institut Fourier, Volume 34 (1984) no. 1, p. 241-260

Let A be a complex Banach algebra, let GL (A) be the stable general linear group of A and let GL 0 (A) be its connected component for the norm topology. We show that, for any non zero trace r:AC, one may define a homomorphism Δ r from GL 0 (A) onto the quotient of the additive group C by the image r ̲(K 0 (A)) of the Grothendieck group of A. If A=M n (C) (respectively if A is a finite continuous factor) with the usual trace, than exp (i2πΔ r ) is the usual determinant (resp. exp (Re(i2πΔ r )) is that of Fuglede and Kadison). In general, the determinants Δ r make it possible to study the derived groups of GL 1 0 (A) and GL 0 (A), et thus also the relationship between the Whitehead groups K 1 (A) and K 1 top (A).

Soient A une algèbre de Banach complexe, GL (A) le groupe général linéaire stable de A et GL 0 (A) sa composante connexe pour la topologie normique. Nous montrons que toute trace non nulle r:AC permet de définir un homomorphisme Δ r de GL 0 (A) sur le quotient du groupe additif C par l’image r ̲(K 0 (A)) du groupe de Grothendieck de A. Si A=M n (C) (respectivement si A est un facteur fini continu) avec la trace usuelle, alors exp (i2πΔ r ) est le déterminant usuel (resp. exp ( Re (i2πΔ r )) est celui de Fuglede et Kadison). Dans le cas général, les déterminants Δ r permettent d’étudier les groupes dérivés de GL 1 0 (A) et GL 0 (A), donc aussi la relation entre les groupes de Whitehead K 1 (A) et K 1 top (A).

@article{AIF_1984__34_1_241_0,
     author = {Harpe, Pierre De La and Skandalis, Georges},
     title = {D\'eterminant associ\'e \`a une trace sur une alg\`ebre de Banach},
     journal = {Annales de l'Institut Fourier},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {1},
     year = {1984},
     pages = {241-260},
     doi = {10.5802/aif.958},
     zbl = {0521.46037},
     mrnumber = {87i:46146a},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1984__34_1_241_0}
}
Harpe, Pierre De La; Skandalis, Georges. Déterminant associé à une trace sur une algèbre de Banach. Annales de l'Institut Fourier, Volume 34 (1984) no. 1, pp. 241-260. doi : 10.5802/aif.958. http://www.numdam.org/item/AIF_1984__34_1_241_0/

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