Applications of uniform convexity of noncommutative 𝐿^{𝑝}-spaces

Kosaki, Hideki

doi:10.1090/S0002-9947-1984-0735421-6

Applications of uniform convexity of noncommutative $L^{p}$-spaces
HTML articles powered by AMS MathViewer

by Hideki Kosaki

Trans. Amer. Math. Soc. 283 (1984), 265-282

DOI: https://doi.org/10.1090/S0002-9947-1984-0735421-6

PDF | Request permission

Abstract:

We consider noncommutative ${L^p}$-spaces, $1 < p < \infty$, associated with a von Neumann algebra, which is not necessarily semifinite, and obtain some consequences of their uniform convexity. Among other results, we obtain (i) the norm continuity of the "absolute value part" map from each ${L^p}$-space onto its positive part; (ii) a certain continuity result on Radon-Nikodym derivatives in the context of positive cones introduced by H. Araki; and (iii) the necessary and sufficient condition for certain ${L^p}$-norm inequalities to become equalities. Some dominated convergence theorems for a probability gage are also considered.

References

Huzihiro Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309–354. MR 350437, DOI 10.2140/pjm.1974.50.309
Huzihiro Araki, Relative entropy for states of von Neumann algebras. II, Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 1, 173–192. MR 0454656, DOI 10.2977/prims/1195190105
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
Huzihiro Araki and Shigeru Yamagami, An inequality for Hilbert-Schmidt norm, Comm. Math. Phys. 81 (1981), no. 1, 89–96. MR 630332, DOI 10.1007/BF01941801
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115, DOI 10.24033/asens.1247
A. Connes, On the spatial theory of von Neumann algebras, J. Functional Analysis 35 (1980), no. 2, 153–164. MR 561983, DOI 10.1016/0022-1236(80)90002-6
Uffe Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), no. 2, 271–283. MR 407615, DOI 10.7146/math.scand.a-11606
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
Frank Hansen, An operator inequality, Math. Ann. 246 (1979/80), no. 3, 249–250. MR 563403, DOI 10.1007/BF01371046

Les espaces

d’une algébre de von Neumann

40

Hideki Kosaki, Positive cones associated with a von Neumann algebra, Math. Scand. 47 (1980), no. 2, 295–307. MR 612702, DOI 10.7146/math.scand.a-11891
Hideki Kosaki, Positive cones and $L^{p}$-spaces associated with a von Neumann algebra, J. Operator Theory 6 (1981), no. 1, 13–23. MR 636997

Applications of the complex interpolation method to a von Neumann algebra (Non-commutative

spaces)

Hideki Kosaki, Remarks on positive cones associated with a von Neumann algebra, Tohoku Math. J. (2) 33 (1981), no. 4, 587–591. MR 643238, DOI 10.2748/tmj/1178229358

Topological vector spaces

R. A. Kunze, $L_{p}$ Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519–540. MR 100235, DOI 10.1090/S0002-9947-1958-0100235-1
Charles A. McCarthy, $c_{p}$, Israel J. Math. 5 (1967), 249–271. MR 225140, DOI 10.1007/BF02771613
Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116. MR 0355628, DOI 10.1016/0022-1236(74)90014-7
A. R. Padmanabhan, Probabilistic aspects of von Neumann algebras, J. Functional Analysis 31 (1979), no. 2, 139–149. MR 525948, DOI 10.1016/0022-1236(79)90058-2

Methods of modern mathematical physics

I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401–457. MR 54864, DOI 10.2307/1969729
Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149, DOI 10.1007/BFb0064579
Christian F. Skau, Positive selfadjoint extensions of operators affiliated with a von Neumann algebra, Math. Scand. 44 (1979), no. 1, 171–195. MR 544585, DOI 10.7146/math.scand.a-11801
W. Forrest Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15–56. MR 102761, DOI 10.1090/S0002-9947-1959-0102761-9
M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168, DOI 10.1007/BFb0065832
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
P. K. Tam, Isometries of $L_{p}$-spaces associated with semifinite von Neumann algebras, Trans. Amer. Math. Soc. 254 (1979), 339–354. MR 539922, DOI 10.1090/S0002-9947-1979-0539922-3

spaces associated with von Neumann algebras

F. J. Yeadon, Convergence of measurable operators, Proc. Cambridge Philos. Soc. 74 (1973), 257–268. MR 326411, DOI 10.1017/s0305004100048052