Estimates for spectral density functions of matrices over [ d ]  [ Estimation de fonctions de densité spectrale de matrices de [ d ] ]
Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 73-88.

Nous donnons une estimation polynomiale pour la fonction de densité spectrale d’une matrice sur l’algèbre complexe du groupe d . Ce résultat donne une borne inférieure explicite à l’invariant de Novikov-Shubin associé à la matrice, montrant en particulier que l’invariant de Novikov-Shubin est strictement positif.

We give a polynomial bound on the spectral density function of a matrix over the complex group ring of d . It yields an explicit lower bound on the Novikov-Shubin invariant associated to this matrix showing in particular that the Novikov-Shubin invariant is larger than zero.

DOI : https://doi.org/10.5802/ambp.346
Classification : 46L99,  58J50
Mots clés : Invariants de Novikov-Shubin, fonction de densité spectrale
@article{AMBP_2015__22_1_73_0,
     author = {L\"uck, Wolfgang},
     title = {Estimates for spectral density functions of matrices over $\mathbb{C}[\mathbb{Z}^d]$},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {73--88},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {22},
     number = {1},
     year = {2015},
     doi = {10.5802/ambp.346},
     language = {en},
     url = {www.numdam.org/item/AMBP_2015__22_1_73_0/}
}
Lück, Wolfgang. Estimates for spectral density functions of matrices over $\mathbb{C}[\mathbb{Z}^d]$. Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 73-88. doi : 10.5802/ambp.346. http://www.numdam.org/item/AMBP_2015__22_1_73_0/

[1] Grabowski, L. Group ring elements with large spectral density (2014) (http://arxiv.org/abs/1409.3212)

[2] Grabowski, L.; Virág, B. Random Walks on Lamplighters via random Schrödinger operators (2013) (Preprint)

[3] Lawton, Wayne M. A problem of Boyd concerning geometric means of polynomials, J. Number Theory, Volume 16 (1983) no. 3, pp. 356-362 | Article | MR 707608 | Zbl 0516.12018

[4] Lott, John Heat kernels on covering spaces and topological invariants, J. Differential Geom., Volume 35 (1992) no. 2, pp. 471-510 | MR 1158345 | Zbl 0770.58040

[5] Lott, John Delocalized L 2 -invariants, J. Funct. Anal., Volume 169 (1999) no. 1, pp. 1-31 | Article | MR 1726745 | Zbl 0958.58027

[6] Lott, John; Lück, Wolfgang L 2 -Topological invariants of 3-manifolds, Invent. Math., Volume 120 (1995) no. 1, pp. 15-60 | Article | MR 1323981 | Zbl 0876.57050

[7] Lück, Wolfgang L 2 -Invariants: Theory and Applications to Geometry and K -Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Volume 44, Springer-Verlag, Berlin, 2002, pp. xvi+595 | MR 1926649 | Zbl 1009.55001

[8] Lück, Wolfgang Twisting L 2 -invariants with finite-dimensional representations (2015) (in preparation)

[9] Lück, Wolfgang; Rørdam, Mikael Algebraic K-theory of von Neumann algebras, K-Theory, Volume 7 (1993) no. 6, pp. 517-536 | Article | MR 1268591 | Zbl 0802.19001

[10] Novikov, S. P.; Shubin, M. A. Morse inequalities and von Neumann II 1 -factors, Dokl. Akad. Nauk SSSR, Volume 289 (1986) no. 2, pp. 289-292 | MR 856461 | Zbl 0647.46049

[11] Novikov, S. P.; Shubin, M. A. Morse inequalities and von Neumann invariants of non-simply connected manifolds, Uspekhi. Matem. Nauk, Volume 41 (1986) no. 5, p. 222-223 (in Russian)

[12] Sauer, Roman Power series over the group ring of a free group and applications to Novikov-Shubin invariants, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 449-468 | MR 2048733 | Zbl 1051.16013