Norm‐controlled inversion in smooth Banach algebras, I

Every differential subalgebra of a unital $C^*$-algebra is spectrally invariant. We derive a quantitative version of this well-known fact and show that a minimal amount of smoothness, as given by a differential norm, already implies norm control. We obtain an explicit estimate for the differential norm of an invertible element $a$. This estimate depends only on the condition number of $a$ and the ratio of two norms.

[1]  K. Gröchenig,et al.  Wiener's lemma for twisted convolution and Gabor frames , 2003 .

[2]  M. Rieffel.  Leibniz seminorms for "Matrix algebras converge to the sphere" , 2007, 0707.3229.

[3]  Karlheinz Gröchenig,et al.  Norm‐controlled inversion in smooth Banach algebras, II , 2014 .

[4]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[5]  N. Nikolski.  The spectral localization property for diagonal operators and semigroups , 2010 .

[6]  Qiyu Sun,et al.  Wiener's lemma for infinite matrices with polynomial off-diagonal decay , 2005 .

[7]  R. Wheeden.  On hypersingular integrals and Lebesgue spaces of differentiable functions. II , 1968 .

[8]  Massimo Fornasier,et al.  Optimal adaptive computations in the Jaffard algebra and localized frames , 2010, J. Approx. Theory.

[9]  Andreas Klotz,et al.  Spectral invariance of Besov-Bessel subalgebras , 2010, J. Approx. Theory.

[10]  Andreas Klotz.  Inverse Closed Ultradifferential Subalgebras , 2012, 1201.2938.

[11]  Karlheinz Gröchenig,et al.  Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications , 2010 .

[12]  R. Balan,et al.  Density, overcompleteness, and localization of frames , 2006 .

[13]  Karlheinz Gröchenig,et al.  Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices , 2010 .

[14]  E. Kissin,et al.  Dense Q-subalgebras of Banach and C*-algebras and unbounded derivations of Banach and C*-algebras , 1993, Proceedings of the Edinburgh Mathematical Society.

[15]  Qiyu Sun,et al.  Frames in spaces with finite rate of innovation , 2008, Adv. Comput. Math..

[16]  A. Baskakov,et al.  Estimates for the entries of inverse matrices and the spectral analysis of linear operators , 1997 .

[17]  Karlheinz Gröchenig,et al.  Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices , 2009, 0904.0386.

[18]  A. Baskakov,et al.  Wiener's theorem and the asymptotic estimates of the elements of inverse matrices , 1990 .

[19]  J. Björk.  On the spectral radius formula in Banach algebras , 1972 .

[20]  O. Bratteli.  Derivations, Dissipations and Group Actions on C*-algebras , 1987 .

[21]  Karlheinz Gröchenig,et al.  Time-Frequency Analysis of Sj\"ostrand's Class , 2004 .

[22]  Karlheinz Gröchenig,et al.  Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices , 2006 .

[23]  O. El-Fallah,et al.  Majorations Uniformes de Normes D'Inverses Dans Les Algèbres de Beurling , 2002 .

[24]  Karlheinz Gröchenig,et al.  Time-Frequency Analysis of Sjöstrand's Class , 2004 .

[25]  Qiyu Sun,et al.  WIENER’S LEMMA FOR INFINITE MATRICES , 2007 .

[26]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[27]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[28]  Inverse closedness of approximation algebras , 2006 .

[29]  K. Gröchenig.  Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator , 2004 .

[30]  N. Nikolski.  In search of the invisible spectrum , 1999 .

[31]  J. Romero.  Explicit localization estimates for spline-type spaces , 2009, 0902.0557.

[32]  M. Abtahi,et al.  On the maximal ideal space of Dales–Davie algebras of infinitely differentiable functions , 2007 .

[33]  E. Kissin,et al.  Differential properties of some dense subalgebras of C*-algebras , 1994, Proceedings of the Edinburgh Mathematical Society.

[34]  J. Sjöstrand,et al.  Wiener type algebras of pseudodifferential operators , 1995 .

[35]  L. Brandenburg.  On identifying the maximal ideals in Banach algebras , 1975 .

[36]  A. Hulanicki.  On the spectrum of convolution operators on groups with polynomial growth , 1972 .

[37]  Marko Lindner,et al.  Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method , 2006 .

[38]  A. Davie,et al.  Quasianalytic Banach function algebras , 1973 .