Fréchet differentiability of the norm of 𝐿_{𝑝}-spaces associated with arbitrary von Neumann algebras

Potapov, D.; Sukochev, F.; Tomskova, A.; Zanin, D.

doi:10.1090/tran/7215

Fréchet differentiability of the norm of $L_p$-spaces associated with arbitrary von Neumann algebras
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by D. Potapov, F. Sukochev, A. Tomskova and D. Zanin PDF

Trans. Amer. Math. Soc. 371 (2019), 7493-7532 Request permission

Abstract:

Let $\mathcal M$ be a von Neumann algebra, and let $({\mathcal {L}}_p(\mathcal M),\|\cdot \|_p)$, $1\le p<\infty$ be the Haagerup $L_p$-space on $\mathcal M$. We prove that the differentiability properties of $\|\cdot \|_p$ are precisely the same as those of classical (commutative) $L_p$-spaces. Our main instruments are the theories of multiple operator integrals and singular traces.

References

A. B. Aleksandrov and V. V. Peller, Functions of operators under perturbations of class $\textbf {S}_p$, J. Funct. Anal. 258 (2010), no. 11, 3675–3724. MR 2606869, DOI 10.1016/j.jfa.2010.02.011
Huzihiro Araki and Tetsuya Masuda, Positive cones and $L_{p}$-spaces for von Neumann algebras, Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, 759–831 (339–411). MR 677270, DOI 10.2977/prims/1195183577
Jonathan Arazy and Yaakov Friedman, Contractive projections in $C_p$, Mem. Amer. Math. Soc. 95 (1992), no. 459, vi+109. MR 1086564, DOI 10.1090/memo/0459
N. A. Azamov, A. L. Carey, P. G. Dodds, and F. A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math. 61 (2009), no. 2, 241–263. MR 2504014, DOI 10.4153/CJM-2009-012-0
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
A. F. Ber, B. de Pagter, and F. A. Sukochev, Derivations in algebras of operator-valued functions, J. Operator Theory 66 (2011), no. 2, 261–300. MR 2844466
M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals, Probl. Math. Phys., No. I, Spectral Theory and Wave Processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 33–67 (Russian). MR 0209872
M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals. II, Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, pp. 26–60 (Russian). MR 0234304
M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals. III, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 27–53 (Russian). MR 0348494
Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
Vladimir I. Chilin, Andrei V. Krygin, and Pheodor A. Sukochev, Local uniform and uniform convexity of noncommutative symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 355–368. MR 1142755, DOI 10.1017/S0305004100075459
V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31 (1994), no. 1, 35–65. MR 1316983
A. van Daele, Continuous crossed products and type $\textrm {III}$ von Neumann algebras, London Mathematical Society Lecture Note Series, vol. 31, Cambridge University Press, Cambridge-New York, 1978. MR 500089
S. J. Dilworth, Special Banach lattices and their applications, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 497–532. MR 1863700, DOI 10.1016/S1874-5849(01)80014-0
Kenneth J. Dykema and Nigel J. Kalton, Sums of commutators in ideals and modules of type II factors, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 3, 931–971 (English, with English and French summaries). MR 2149407
Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
Daniele Guido and Tommaso Isola, Singular traces on semifinite von Neumann algebras, J. Funct. Anal. 134 (1995), no. 2, 451–485. MR 1363808, DOI 10.1006/jfan.1995.1153
Uffe Haagerup, $L^{p}$-spaces associated with an arbitrary von Neumann algebra, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977) Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 175–184 (English, with French summary). MR 560633
Marius Junge, Zhong-Jin Ruan, and Quanhua Xu, Rigid $\scr {OL}_p$ structures of non-commutative $L_p$-spaces associated with hyperfinite von Neumann algebras, Math. Scand. 96 (2005), no. 1, 63–95. MR 2142873, DOI 10.7146/math.scand.a-14945
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
N. J. Kalton, N. T. Peck, and James W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777, DOI 10.1017/CBO9780511662447
N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Hideki Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative $L^{p}$-spaces, J. Funct. Anal. 56 (1984), no. 1, 29–78. MR 735704, DOI 10.1016/0022-1236(84)90025-9
Hideki Kosaki, Applications of uniform convexity of noncommutative $L^{p}$-spaces, Trans. Amer. Math. Soc. 283 (1984), no. 1, 265–282. MR 735421, DOI 10.1090/S0002-9947-1984-0735421-6
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Steven Lord, Fedor Sukochev, and Dmitriy Zanin, Singular traces, De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013. Theory and applications. MR 3099777
L. A. Lusternik and V. J. Sobolev, Elements of functional analysis, Frederick Ungar Publishing Co., New York, 1961. Translated from the Russian by Anthony E. Labarre, Jr., Herbert Izbicki, H. Ward Crowley. MR 0141966
B. de Pagter and F. A. Sukochev, Differentiation of operator functions in non-commutative $L_p$-spaces, J. Funct. Anal. 212 (2004), no. 1, 28–75. MR 2065237, DOI 10.1016/j.jfa.2003.10.009
B. de Pagter, H. Witvliet, and F. A. Sukochev, Double operator integrals, J. Funct. Anal. 192 (2002), no. 1, 52–111. MR 1918492, DOI 10.1006/jfan.2001.3898
B. S. Pavlov, Multidimensional operator integrals, Problems of Math. Anal., No. 2: Linear Operators and Operator Equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1969, pp. 99–122 (Russian). MR 0415371
V. V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006), no. 2, 515–544. MR 2214586, DOI 10.1016/j.jfa.2005.09.003
Gilles Pisier and Quanhua Xu, Non-commutative $L^p$-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, DOI 10.1016/S1874-5849(03)80041-4
Denis Potapov, Anna Skripka, and Fedor Sukochev, Spectral shift function of higher order, Invent. Math. 193 (2013), no. 3, 501–538. MR 3091975, DOI 10.1007/s00222-012-0431-2
Denis Potapov and Fedor Sukochev, Fréchet differentiability of $\mathcal {S}^p$ norms, Adv. Math. 262 (2014), 436–475. MR 3228433, DOI 10.1016/j.aim.2014.05.011
Denis Potapov and Fyodor Sukochev, Lipschitz and commutator estimates in symmetric operator spaces, J. Operator Theory 59 (2008), no. 1, 211–234. MR 2404471
Kondagunta Sundaresan, Smooth Banach spaces, Math. Ann. 173 (1967), 191–199. MR 218876, DOI 10.1007/BF01361710
M. Terp, $L_p$ spaces associated with von Neumann algebras, notes, Math. Institute, Copenhagen Univ., 1981.
A. Thiago Bernardino, A simple natural approach to the uniform boundedness principle for multilinear mappings, Proyecciones 28 (2009), no. 3, 203–207. MR 2576334, DOI 10.4067/S0716-09172009000300001
N. Tomczak-Jaegermann, On the differentiability of the norm in trace classes $S_{p}$, Séminaire Maurey-Schwartz 1974–1975: Espaces $L^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. No. XXII, Centre Math., École Polytech., Paris, 1975, pp. 9. MR 0410443