×

A non-commutative Wiener-Wintner theorem. (English) Zbl 1325.47022

Summary: For a von Neumann algebra \(\mathcal{M}\) with a faithful normal tracial state \(\tau\) and a positive ergodic homomorphism \(\alpha :\mathcal{L}^{1}(\mathcal{M},\tau)\to\mathcal{L}^{1}(\mathcal{M},\tau)\) such that \(\tau\circ\alpha =\tau\) and \(\alpha \) does not increase the norm in \(\mathcal{M}\), we establish a non-commutative counterpart of the classical Wiener-Wintner ergodic theorem.

MSC:

47A35 Ergodic theory of linear operators
46L51 Noncommutative measure and integration
PDFBibTeX XMLCite
Full Text: arXiv Euclid

References:

[1] \beginbbook \bauthor\binitsI. \bsnmAssani, \bbtitleWiener Wintner ergodic theorems, \bpublisherWorld Scientific, \blocationRiver Edge, \byear2003. \endbbook \OrigBibText I. Assani, Wiener Wintner ergodic theorems , World Scientific (2003) \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1041.37004 · doi:10.1142/4538
[2] \beginbarticle \bauthor\binitsV. \bsnmChilin and \bauthor\binitsS. \bsnmLitvinov, \batitleUniform equicontinuity for sequences of homomorphisms into the ring of measurable operators, \bjtitleMethods Funct. Anal. Topology \bvolume12 (\byear2006), no. \bissue2, page 124-\blpage130. \endbarticle \OrigBibText V. Chilin, S. Litvinov, Uniform equicontinuity for sequences of homomorphisms into the ring of measurable operators , Methods of Funct. Anal. Top., 12 (2) (2006), 124-130 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1124.46047
[3] \beginbotherref \oauthor\binitsV. \bsnmChilin and \oauthor\binitsS. \bsnmLitvinov, Ergodic theorems in fully symmetric spaces of \(\tau\)-measurable operators , 2015; available at \arxivurl Ergodic theorems in fully symmetric spaces of \(\tau\)-measurable operators , arXiv:1410.1451v2 (2015) \endOrigBibText \bptokstructpyb \endbibitem arXiv:
[4] \beginbarticle \bauthor\binitsV. \bsnmChilin, \bauthor\binitsS. \bsnmLitvinov and \bauthor\binitsA. \bsnmSkalski, \batitleA few remarks in non-commutative ergodic theory, \bjtitleJ. Operator Theory \bvolume53 (\byear2005), no. \bissue2, page 331-\blpage350. \endbarticle \OrigBibText V. Chilin, S. Litvinov, A. Skalski, A few remarks in non-commutative ergodic theory , J. Operator Theory, 53 (2) (2005), 331-350 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1119.46314
[5] \beginbarticle \bauthor\binitsM. \bsnmJunge and \bauthor\binitsQ. \bsnmXu, \batitleNoncommutative maximal ergodic theorems, \bjtitleJ. Amer. Math. Soc. \bvolume20 (\byear2007), no. \bissue2, page 385-\blpage439. \endbarticle \OrigBibText M. Junge, Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. , 20 (2)(2007), 385-439 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1116.46053 · doi:10.1090/S0894-0347-06-00533-9
[6] \beginbbook \bauthor\binitsY. \bsnmKatznelson, \bbtitleAn introduction to harmonic analysis, \bpublisherDover, \blocationNew York, \byear1976. \endbbook \OrigBibText Y. Katznelson, An introduction to harmonic analysis , Dover Publications (1976) \endOrigBibText \bptokstructpyb \endbibitem
[7] \beginbarticle \bauthor\binitsS. \bsnmLitvinov, \batitleUniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems, \bjtitleProc. Amer. Math. Soc. \bvolume140 (\byear2012), page 2401-\blpage2409. \endbarticle \OrigBibText S. Litvinov, Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems , Proc. of Amer. Math. Soc., 140 (2012), 2401-2409 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1279.46048 · doi:10.1090/S0002-9939-2011-11483-7
[8] \beginbotherref \oauthor\binitsS. \bsnmLitvinov, Weighted ergodic theorems , Ph.D. thesis, North Dakota State University, 1999. \endbotherref \OrigBibText S. Litvinov, Weighted ergodic theorems , Doctoral Dissertation, North Dakota State University (1999) \endOrigBibText \bptokstructpyb \endbibitem
[9] \beginbarticle \bauthor\binitsE. \bsnmNelson, \batitleNotes on non-commutative integration, \bjtitleJ. Funct. Anal. \bvolume15 (\byear1974), page 103-\blpage116. \endbarticle \OrigBibText E. Nelson, Notes on non-commutative integration , J. Funct. Anal., 15 (1974), 103-116 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7
[10] \beginbarticle \bauthor\binitsC. P. \bsnmNiculescu, \bauthor\binitsA. and \bauthor\binitsL. , \batitleNoncommutative extensions of classical and multiple recurrence theorems, \bjtitleJ. Operator Theory \bvolume50 (\byear2005), page 3-\blpage52. \endbarticle \OrigBibText C. P. Niculescu, A. Ströh, L. Zsidó, Noncommutative extensions of classical and multiple recurrence theorems , J. Operator Theory, 50 (2005), 3-52 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 1036.46053
[11] \beginbarticle \bauthor\binitsI. \bsnmSegal, \batitleA non-commutative extension of abstract integration, \bjtitleAnn. of Math. (2) \bvolume57 (\byear1953), page 401-\blpage457. \endbarticle \OrigBibText I. Segal, A non-commutative extension of abstract integration , Ann. of Math., 57 (1953), 401-457 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0051.34201 · doi:10.2307/1969729
[12] \beginbarticle \bauthor\binitsN. \bsnmWiener and \bauthor\binitsA. \bsnmWintner, \batitleHarmonic analysis and ergodic theory, \bjtitleAmer. J. Math. \bvolume63 (\byear1941), no. \bissue2, page 415-\blpage426. \endbarticle \OrigBibText N. Wiener, A. Wintner, Harmonic Analysis and Ergodic Theory , Amer. J. of Math., 63 (2) (1941), 415-426 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0025.06504 · doi:10.2307/2371534
[13] \beginbarticle \bauthor\binitsF. J. \bsnmYeadon, \batitleErgodic theorems for semifinite von Neumann algebras-I, \bjtitleJ. Lond. Math. Soc. (2) \bvolume16 (\byear1977), no. \bissue2, page 326-\blpage332. \endbarticle \OrigBibText F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras-I , J. London Math. Soc., 16 (2) (1977), 326-332 \endOrigBibText \bptokstructpyb \endbibitem · Zbl 0369.46061 · doi:10.1112/jlms/s2-16.2.326
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.