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时频分析与算子代数
Time-Frequency Analysis Meets Operator Algebras
Gabor分析中几个著名的基本定理（如对偶原理和稠密性定理）与群表示和算子代数理论密切相连.尽管时频分析与算子代数之间的某些联系是Jon von Neumann于1930年代建立的，可是它们在近期得到广泛研究，这主要应归于小波/Gabor理论或更一般的框架理论近二十年的发展.本文将讨论过去几年得到的一些主要结果，同时也给出一些新的结果、解释和问题.我们主要考虑来源于时频分析并能反映与群表示理论存在内在联系的那些结果.特别地，针对群表示的时频分析，将详细说明抽象的对偶原理及其与算子代数理论中几个公开问题的联系.
There are several well-known fundamental theorems in Gabor analysis that are naturally connected to group representation theory and theory of operator algebras.While some of these connections between time-frequency analysis and operator algebras were established by Jon von Neumann in 1930s, they have been extensively investigated more recently mainly due to the developments of wavelet/Gabor theory, or more generally, the theory of frames in the last two decades.In this article, we will discuss some of the main results we obtained in the last few years together with some new results, exposition and open problems.We will be mainly focused on the results that were originated from time-frequency analysis but reflect intrinsic connections with group representation theory.In particular, we give a detailed account on an abstract version of the duality principle in time-frequency analysis for group representations, and its connections with some open problems in the theory of operator algebras.
时频分析 / 框架 / Gabor表示 / 算子代数 / 群表示 {{custom_keyword}} /
time-frequency analysis / frames / Gabor representations / operator algebras / group representations {{custom_keyword}} /
[1] Arveson W., Dilation Theory Yesterday and Today, A Glimpse at Hilbert Space Operators, 99-123, Oper. Theory Adv. Appl., 207, Birkhäuser Verlag, Basel, 2010.
[2] Baggett L., Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math., 1990, 60/61:195-203.
[3] Bekka B., Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., 2004, 10:325-349.
[4] Casazza P., Consequences of the Marcus/Spielman/Stivastava Solution to the Kadison-Singer Problem, New Trends in Applied Harmonic Analysis, 2016:191-213.
[5] Casazza P., Fickus M., Tremain J., et al., The Kadison-Singer Problem in Mathematics and Engineering:A Detailed Account, Operator Theory, Operator Algebras, and Applications(Editors:Han D., D. Larson and P. Jorgensen), Contemporary Mathematics 414, Providence, RI:American Mathematical Society, 2006:299-335.
[6] Connes A., Classification of injective factors. Cases Ⅱ1, Ⅱ∞, Ⅲλ, λ≠1, Ann. of Math., 1976, 104:73-115.
[7] Dai X., Sun Q., The abc problem for Gabor systems, Memoirs of American Mathematical Society, 2016, 244(1152):1-99.
[8] Daubechies I., The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 1990, 39:961-1005.
[9] Daubechies I., Grossmann A., Frames in the Bargmann space of entire functions, Comm. Pure Appl. Math., 1988, 41:151-164.
[10] Daubechies I., Landau H., Landau Z., Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl., 1995, 1:437-478.
[11] Dykema K., Interpolated free group factors, Pacific J. Math., 1994, 163:123-135.
[12] Dutkay D., Han D., Larson D., A duality principle for groups, J. Funct. Anal., 2009, 257:1133-1143.
[13] Dutkay D., Han D., Picioroaga G., Frames for ICC groups, J. Funct. Analysis, 2009, 256:3071-3090.
[14] Duffin R. J., Schaeffer A. C., A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 1952, 72:341-366.
[15] Gabardo J. P., Han D., Subspace Weyl-Heisenberg frames, J. Fourier Anal. Appl., 2001, 7:419-433.
[16] Gabardo J. P., Han D., Frame representations for group-like unitary operator systems, J. Operator Theory, 2003, 49:223-244.
[17] Gabardo J. P., Han D., Li Y., Lattice titling and density conditions for subspace Gabor frames, J. Funct. Anal., 2013, 265:1170-1189.
[18] Gabardo J. P., Han D., Subspace Weyl-Heisenberg frames, J. Fourier Anal. Appl., 2001, 7:419-433.
[19] Gabardo J. P., Han D., Frame representations for group-like unitary operator systems, J. Operator Theory, 2003, 49:223-244.
[20] Gabardo J. P., Han D., The uniqueness of the dual of Weyl-Heisenberg subspace frames, Appl. Comput. Harmon. Anal., 2004, 17:226-240.
[21] Gabor D., Theory of communications, J. Inst. Elec. Eng. London, 1946, 93:429-457.
[22] Gu Q., Han D., When a characteristic function generates a Gabor frame, Appl. Comput. Harmonic Anal., 2008, 24:290-309.
[23] Gröchenig K., Han D., Heil C., et al., The Balian-Low theorem for symplectic lattices in higher dimensions, Appl. Comp. Harm. Anal., 2002, 13:169-176.
[24] Gröchenig K., Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
[25] Haagerup U., Solutions of the similarity problem for cyclic representations of C*-algebras, Ann. Math., 1983, 118:215-240.
[26] Han D., Frame Representations and Parseval Duals with Applications to Gabor Frames, Trans. Amer. Math. Soc., 2008, 360:3307-3326.
[27] Han D., The existence of tight Gabor duals for Gabor frames and subspace Gabor frames, J. Funct. Anal., 2009, 256:129-148.
[28] Han D., Dilations and completions for Gabor systems, J. Fourier Anal. Appl., 2009, 15:201-217.
[29] Han D., A note on the duality theorem for frame representations, Proc. Amer. Math. Sco., 2016, to appear.
[30] Han D., Larson D., Frames, bases and group representations, Memoirs Amer. Math. Soc., 2000, 697:1-94.
[31] Han D., Larson D., Frame duality properties for projective unitary representations, Bull. London Math. Soc., 2008, 40:685-695.
[32] Han D., Larson D. R., Liu B., et al., Operator-valued measures, dilations, and the theory of frames, Mem. Amer. Math. Soc., 2014, 229(1075):1-84.
[33] Han D., Larson D. R., Liu B., et al., Dilations of frames, operator valued measures and bounded linear maps, Contemp. Math., Amer. Math. Soc., 2014, 626:33-54.
[34] Han D., Larson D. R., Liu B., et al., Dilations for systems of imprmimitivity acting on Banach spaces, J. Funct. Anal., 2014, 266:6914-6937.
[35] Han D., Wang Y., Lattice tiling and the Weyl-Heisenberg frames, Geom. Funct. Anal., 2001, 11:742-758.
[36] Han D., Wang Y., The existence of Gabor bases, Contemp. Math., 2004, 345:183-192.
[37] Heil C., History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl., 2007, 13:113-166.
[38] Heil C., The Density Theorem and the Homogeneous Approximation Property for Gabor Frames, in:"Representations, Wavelets, and Frames:A Celebration of the Mathematical Work of Lawrence Baggett, " P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, eds., Birkhäuser, Boston, 2008, 71-102.
[39] Janssen A., Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl., 1995, 1:403-436.
[40] Janssen A., Representations of Gabor Frame Operators, in:Twentieth Century Harmonic Analysis-A Celebration NATO Sci. Ser. Ⅱ, Math. Phys. Chem., vol. 33, Kluwer Academic, Dordrecht, 2001:73-101.
[41] Kadison R. V., On the orthogonalization of operator representations, Amer. J. Math., 1955, 77:600-620.
[42] Kadison R., Singer I, Extensions of pure states, Amer. J. Math., 1959, 81:383-400.
[43] Lyubarskii Yu. I., Frames in the Bargmann Space of Entire Functions, in Entire and subharmonic Functions, Amer. Math. Soc., Providence, RI, 1992:167-180.
[44] Marcus A., Spielman D., Srivastava N., Interlacing families Ⅱ:Mixed characteristic polynomials and the Kadison-Singer problem, Ann. Math., 2015, 182:327-350.
[45] Murray F. J., von Neumann J., On rings of operators. IV, Ann. Math., 1943, 44:716-808.
[46] von Neumann J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955.
[47] Paulsen V., Completely Bounded Maps and Operator Algebras, Cambridge University Press, Cambridge, 2002.
[48] Pisier G., Simliarity Problems and Completely Bounded Maps, 2nd, Expanded Edition, Springer-Verlag Lecture Notes, Vol. 1618, 2001.
[49] Popa S., On the fundamental group of type Ⅱ1 factors, Proc. Natl. Acad. Sci., 2004, 101:723-726.
[50] Popa S., On a class of type Ⅱ1 factors with Betti numbers invariants, Ann. of Math., 2006, 163:809-899.
[51] Radulescu F., The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R+\{0}, J. Amer. Math. Soc., 1992, 5:517-532.
[52] Radulescu F., Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math., 1994, 115:347-389.
[53] Rieffel M., Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann., 1981, 257:403-418.
[54] Ron A., Shen Z., Weyl-Heisenberg frames and Riesz bases in L2(Rd), Duke Math. J., 1997, 89:237-282.
[55] Seip K., Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math., 1992, 429:91-106.
[56] Seip K., Wallstén R., Sampling and interpolation in the Bargmann-Fock space Ⅱ, J. Reine Angew. Math., 1992, 429:107-113.
[57] Stinespring W. F., Positive Functions on C*-algebras, Proc. Amer. Math. Soc., 1955, 6:211-216.
[58] Varadarajan V. S., Geometry of Quantum Theory, Second Edition, Springer-Verlag, New York, Berlin, 1985.
[59] Voiculescu D. V., Dykema K. J., Nica A., Free Random Variables, CRM Monograph Series, A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, American Mathematical Society, Providence, RI, 1992, vi+70.
NSF of USA(DMS-1106934, DMS-1403400)
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