Vol.33 (2) 2011 — 4 articles found.

On coarse spectral geometry in even dimension

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 57–64
Vol.33 (2) 2011
Robert Yuncken Details
(Received: 2010-10-25 )
(Received: 2010-10-25 )

Robert Yuncken, Laboratoire de Mathematiques, Universite Blaise Pascal, Clermont-Ferrand II, CampusUniversitaire des Cezeaux, 63177 Aubiere cedex

Abstract/Résumé:

Let \(\sigma\) be the involution of the Roe algebra \(C^*|\mathbf{R}|\) which is induced from the reflection \(\mathbf{R}\colon \mathbf{R}\); \(x\mapsto -x\). A graded Fredholm module over a separable \(C^*\)-algebra \(A\) gives rise to a homomorphism \(\tilde{\rho} \colon A\colon C^*|\mathbf{R}|^\sigma\) to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the \(K\)-theory of this symmetric Roe algebra is \(K_0 (C^*|\mathbf{R}|^\sigma) \cong \mathbf{Z}\), \(K_1(C^*|\mathbf{R}|^\sigma) = 0\), and that the induced map \(\tilde{\rho}_* \colon K_0(A) \colon \mathbf{Z}\) on \(K\)-theory gives the index pairing of \(K\)-homology with \(K\)-theory.

Soit \(\sigma\) l’involution de l’algèbre de Roe \(C^*|\mathbf{R}|\) induite par la réflexion \(\mathbf{R}\colon \mathbf{R}\); \(x\mapsto -x\). Un module de Fredholm gradué sur une \(C^*\)-algèbre séparable \(A\) donne lieu à un homomorphisme \(\tilde{\rho} \colon A\colon C^*|\mathbf{R}|^\sigma\) à valeurs dans la sous-algèbre des éléments invariants. En utilisant cette observation, nous montrons un analogue en dimension paire d’un résultat de Roe. Plus précisément, nous montrons que la \(K\)-théorie de cette algèbre de Roe symétrique est \(K_0 (C^*|\mathbf{R}|^\sigma) \cong \mathbf{Z}\), \(K_1(C^*|\mathbf{R}|^\sigma) = 0\) et que l’application induite \(\tilde{\rho}_* \colon K_0(A) \to \mathbf{Z}\) coïncide avec l’accouplement entre \(K\)-homologie et \(K\)-théorie.

Keywords:

AMS Subject Classification: 58G12

PDF(click to download): On coarse spectral geometry in even dimension

Unimodal sequences show that Lambert $W$ is Bernstein

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 50–56
Vol.33 (2) 2011
G.A. Kalugin; D.J. Jeffrey Details
(Received: 2010-11-28 )
(Received: 2010-11-28 )

G.A. Kalugin, Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada, N6A 5B7; gkalugin@uwo.ca

D.J. Jeffrey, Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada, N6A 5B7; djeffrey@uwo.ca

Abstract/Résumé:

We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert \(W\) function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property implies that the positive real branch of the Lambert \(W\) function is a Bernstein function.

Nous considérons une séquence de polynômes que l’on retrouve dans l’expression des dérivées de la fonction Lambert \(W\). Nous montrons que les coefficients de chaque polynôme forment une séquence positive qui est log-concave et unimodale. Cette propriété implique que la branche réelle positive de la fonction Lambert \(W\) est une fonction de Bernstein.

Keywords: Bernstein function, completely monotonic function, unimodal sequence

AMS Subject Classification: Special sequences and polynomials 11B83

PDF(click to download): Unimodal sequences show that Lambert $W$ is Bernstein

Quasitraces are traces: A short proof of the finite-nuclear-dimension case

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 44–49
Vol.33 (2) 2011
Nathanial P. Brown; Wilhelm Winter Details
(Received: 2010-05-12 , Revised: 2010-10-21 )
(Received: 2010-05-12 , Revised: 2010-10-21 )

Nathanial P. Brown, Department of Mathematics, Penn State University, State College, PA 16802, USA; e-mail: nbrown@math.psu.edu

Wilhelm Winter, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK; e-mail: wilhelm.winter@nottingham.ac.uk

Abstract/Résumé:

Uffe Haagerup proved that quasitraces on unital exact \(C^*\)-algebras are traces. We give a short proof under the stronger hypothesis of locally finite nuclear dimension; our result generalizes to the case of lower semicontinuous extended quasitraces on nonunital \(C^*\)-algebras.

Uffe Haagerup a démontré qu’une quasi-trace sur une \(C^*\)-algèbre exacte à élément unité est une trace. Nous donnons une courte démonstration sous l’hypothèse plus forte de dimension nucléaire localement finie; ce résultat se généralise jusqu’au cas d’une quasi-trace étendue semicontinue inférieurement sur une \(C^*\)-algèbre sans élément unité.

Keywords:

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): Quasitraces are traces: A short proof of the finite-nuclear-dimension case

Higher derivations and Hochschild homology

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 33–43
Vol.33 (2) 2011
A. Banerjee Details
(Received: 2010-10-25 , Revised: 2010-11-29 )
(Received: 2010-10-25 , Revised: 2010-11-29 )

A. Banerjee, Dept. of Mathematics, Ohio State University, 231 W 18th Ave, Columbus, Ohio 43210, USA; e-mail: abhishekbanerjee1313@gmail.com

Abstract/Résumé:

Let \(K\) be a commutative ring and let \(A\) be a \(K\)-algebra. A \(K\)-linear derivation \(D\) on \(A\) induces a morphism \(L_D\) on the Hochschild and cyclic homologies of \(A\), which is the analogue of the Lie derivative in noncommutative geometry. In this paper, we extend this to higher (or Hasse–Schmidt) derivations on \(A\).

Soit \(K\) un anneau commutatif et soit \(A\) une \(K\)-algèbre. Une dérivation \(K\)-linéaire \(D\) de \(A\) induit un endomorphisme \(L_D\) de l’homologie de Hochschild et de l’homologie cyclique de \(A\), qui est l’analogue en géométrie non-commutative de la dérivée de Lie. Dans cet article, nous généralisons cette construction aux dérivations d’ordre supérieur (ou dérivations de Hasse–Schmidt) de \(A\).

Keywords:

AMS Subject Classification: Derivations; actions of Lie algebras 16W25

PDF(click to download): Higher derivations and Hochschild homology

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