Operator-valued free Fisher information and modular frames

Meng, Bin; Guo, Maozheng; Cao, Xiaohong

doi:10.1090/S0002-9939-05-08111-6

Operator-valued free Fisher information and modular frames
HTML articles powered by AMS MathViewer

by Bin Meng, Maozheng Guo and Xiaohong Cao PDF

Proc. Amer. Math. Soc. 133 (2005), 3087-3096 Request permission

Abstract:

We introduce the operator-valued free Fisher information for a random variable in an operator-valued noncommutative probability space and point out its relations to the amalgamated freeness. Using M. Frank and D. Larson’s modular frame notion we can construct the conjugate variable for an operator-valued semicircle variable with conditional expectation covariance. Then we obtain its free Fisher information and show it is equal to the index of the conditional expectation. At last the conjugate variable with respect to a modular frame operator for a semicircle variable is also constructed.

References

Michel Baillet, Yves Denizeau, and Jean-François Havet, Indice d’une espérance conditionnelle, Compositio Math. 66 (1988), no. 2, 199–236 (French). MR 945550
Thomas M. Cover and Joy A. Thomas, Elements of information theory, Wiley Series in Telecommunications, John Wiley & Sons, Inc., New York, 1991. A Wiley-Interscience Publication. MR 1122806, DOI 10.1002/0471200611
Michael Frank and Eberhard Kirchberg, On conditional expectations of finite index, J. Operator Theory 40 (1998), no. 1, 87–111. MR 1642530
Michael Frank and David R. Larson, A module frame concept for Hilbert $C^\ast$-modules, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 207–233. MR 1738091, DOI 10.1090/conm/247/03803
Michael Frank and David R. Larson, Frames in Hilbert $C^\ast$-modules and $C^\ast$-algebras, J. Operator Theory 48 (2002), no. 2, 273–314. MR 1938798
Liming Ge, Applications of free entropy to finite von Neumann algebras. II, Ann. of Math. (2) 147 (1998), no. 1, 143–157. MR 1609522, DOI 10.2307/120985
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
Alexandru Nica, Dimitri Shlyakhtenko, and Roland Speicher, Operator-valued distributions. I. Characterizations of freeness, Int. Math. Res. Not. 29 (2002), 1509–1538. MR 1907203, DOI 10.1155/S1073792802201038
Alexandru Nica, Dimitri Shlyakhtenko, and Roland Speicher, Some minimization problems for the free analogue of the Fisher information, Adv. Math. 141 (1999), no. 2, 282–321. MR 1671766, DOI 10.1006/aima.1998.1780
Sorin Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), no. 2, 375–405. MR 1198815, DOI 10.1007/BF01231293
Sorin Popa, Classification of subfactors and their endomorphisms, CBMS Regional Conference Series in Mathematics, vol. 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR 1339767, DOI 10.1090/cbms/086
Roland Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898, DOI 10.1090/memo/0627
Dimitri Shlyakhtenko, Free entropy with respect to a completely positive map, Amer. J. Math. 122 (2000), no. 1, 45–81. MR 1737257, DOI 10.1353/ajm.2000.0005
V. S. Sunder, An invitation to von Neumann algebras, Universitext, Springer-Verlag, New York, 1987. MR 866671, DOI 10.1007/978-1-4613-8669-8
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424
D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
Dan Voiculescu, Operations on certain non-commutative operator-valued random variables, Astérisque 232 (1995), 243–275. Recent advances in operator algebras (Orléans, 1992). MR 1372537
D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), no. 1, 172–199. MR 1371236, DOI 10.1007/BF02246772
Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information, Adv. Math. 146 (1999), no. 2, 101–166. MR 1711843, DOI 10.1006/aima.1998.1819