Abstract
We point out that the quasicentral modulus is a noncommutative analogue of a nonlinear rearrangement invariant Sobolev condenser capacity. In the case of the shifts by the generators of a finitely generated group, the quasicentral modulus coincides with a corresponding nonlinear condenser capacity on the Cayley graph of the group. Some other capacities related to the quasicentral modulus are also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer Verlag, 1999.
V. Anandam, Harmonic Functions and Potentials on Finite and Infinite Networks, Lecture Notes of Unione Matematica Italiana 12, Springer Verlag, 2011.
B. Blackadar and J. Cuntz, Differential Banach algebra norms and smooth subalgebras of C*-algebras, J. Operator Theory, 26 (1991), 255–282.
A. Cianchi and L. Pick, Sobolev embeddings into BMO,VMO and L∞, Ark. Mat., 36 (1998), 317–340.
F. Cipriani, Noncommutative potential theory: A survey, J. Geometry and Physics, 105 (2016), 25–59.
S. Costea, Scaling invariant Sobolev-Lorentz capacity on \(\mathbb{R}^n\), Indiana Univ. Math. J., 56 (2007), 2641–2669.
S. Costea and V. G. Maz’ya, Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities, Sobolev spaces in mathematics II, Applications in Analysis and Partial Differential Equations, Springer, 2009, p. 103–121.
G. David and D. Voiculescu, s-numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimension, J. Funct. Anal., 94 (1990), 14–26.
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Florida, 1992.
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, RI, 1969.
J. Heinonen, T. Kipelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, 1993.
J. Kauhanen, P. Koskela and J. Maly, On functions with derivatives in a Lorentz space, Manuscripta Math., 100 (1999), 87–101.
V. G. Maz’ya, Sobolev Spaces, Springer Verlag, 1985.
M. A. Rieffel, Standard deviation is a strongly Leibniz seminorm, New York J. Math., 20 (2014), 35–56.
B. Simon, Trace Ideals and Their Applications, 2nd Ed., Mathematical Surveys and Monographs, Vol. 120, Amer. Math. Soc., Providence, RI, 2005.
P. M. Soardi, Potential Theory on Infinite Networks, Lecture Notes in Mathematics 1590, Springer Verlag, Berlin – Heidelberg, 1994.
E. M. Stein, Editor’s note: The differentiability of functions in Rn, Annals of Math., 113 (1981), 383–385.
D. V. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators I, J. Operator Theory, 2 (1979), 3–37.
D. V. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators II, J. Operator Theory, 5 (1981), 77–100.
D. V. Voiculescu, On the existence of quasicentral approximate units relative to normed ideals I., J. Funct. Anal., 91 (1990), 1–36.
D. V. Voiculescu, Perturbations of operators, connections with singular integrals, hyperbolicity and entropy, Harmonic Analysis and Discrete Potential Theory (Frascati, 1991), Plenum Press, New York, 1992, 181–191.
D. V. Voiculescu, Almost normal operators mod Hilbert–Schmidt and the K-theory of the algebras \(E\Lambda(\Omega)\), J. Noncommut. Geom., 8 (2014), 1123–1145.
D. V. Voiculescu, Commutants mod normed ideals, Advances in Noncommutative Geometry on the Occasion of Alain Connes’ 70th Birthday, Springer Verlag, 2020, 585–606.
D. V, Voiculescu, The formula for the quasicentral modulus in the case of spectral measures on fractals, arXiv (2020), arXiv:2006.14456 .
D. V. Voiculescu, The condenser quasicentral modulus, arXiv (2021), arXiv: 2109.07633 preprint of preliminary version.
M. Yamasaki, Parabolic and hyperbolic networks, Hiroshima Math. J., 7(1977), 135–146.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Voiculescu, DV. Capacity and the quasicentral modulus. Acta Sci. Math. (Szeged) 88, 515–525 (2022). https://doi.org/10.1007/s44146-022-00030-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44146-022-00030-1