Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we "complete" the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].
L. E. LABUSCHAGNE
,
W. A. MAJEWSKI
. DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES[J]. Acta mathematica scientia, Series B, 2020
, 40(5)
: 1249
-1270
.
DOI: 10.1007/s10473-020-0507-9
[1] Araki H. Relative entropy of states of von Neumann algebras. Publ RIMS, 1976, 11:809-833
[2] Araki H. Relative entropy of states of von Neumann algebras Ⅱ. Publ RIMS, 1977, 13:173-192
[3] Bennett C, Sharpley R. Interpolation of Operators. Academic Press Inc, 1988
[4] Cipriani F. Dirichlet forms on noncommutative spaces//Quantum Potential Theory. Lecture Notes in Mathematics 1954. Springer, 2008:161-276
[5] DiPerna R, Lions P L. On the Fokker-Planck-Boltzmann equation. Commun Math Phys, 1988, 120:1-23
[6] DiPerna R, Lions P L. On the Cauchy problem for Boltzmann equations:global existence and weak stability. Ann Math, 1989, 130:321-366
[7] Dodds P G, Dodds T K-Y, de Pagter B. Non-commutative Banach function spaces. Math Z, 1989, 201:583-597
[8] Dodds P G, Dodds T K-Y, de Pagter B. Fully symmetric operator spaces. Integral Equations and Operator Theory, 1992, 15:942-972
[9] Fack T, Kosaki H. Generalized s-numbers of τ-measurable operators. Pac J Math, 1986, 123:269-300
[10] Fagnola F, Umanita V. Generators of KMS symmetric Markov semigroups on B(H) Symmetry and Quantum Detailed Balance. Commun Math Phys, 2010, 298:523-547
[11] Glauber R J. Time-dependent statistics for the Ising model. J Math Physics, 1963, 4:294-307
[12] Goldstein S. Conditional expectation and stochastic integrals in non-commutative Lp spaces. Math Proc Camb Phil Soc, 1991, 110:365-383
[13] Goldstein S, Lindsay J M. KMS-symmetric Markov semigroups. Math Z, 1995, 219:591-608
[14] Goldstein S, Lindsay J M. Markov semigroups KMS-symmetric for a weight. Math Ann, 1999, 313:39-67
[15] Haagerup U. Operator-valued weights in von Neumann algebras, Ⅱ. J Funct Anal, 1979, 33:339-361
[16] Haagerup U, Junge M, Xu Q. A reduction method for noncommutative Lp-spaces and applications. Trans Amer Math Soc, 2010, 362:2125-2165
[17] Johnson W B, Lindenstrauss J, ed. Handbook of the Geometry of Banach Spaces, vol 2. Elsevier Science, 2003
[18] Kadison R V, Ringrose J R. Fundamentals of the Operator Theory, vol Ⅱ. Advanced Theory. Graduate Studies in Mathematics, vol 16. American Mathematical Society, 1997
[19] Kosaki H. Applications of complex interpolation method to a von Neumann algebra (Non-commutative Lp-spaces). J Funct Anal, 1984, 56:29-78
[20] Labuschagne L E. A crossed product approach to Orlicz spaces. Proc London Math Soc, 2013, 107:965-1003
[21] Labuschagne L E, Majewski W A. Maps on non-commutative Orlicz spaces. Illinois J Math, 2011, 55:1053-1081
[22] Labuschagne L E, Majewski W A. Quantum dynamics on Orlicz spaces. arXiv:1605.01210v1[math-ph]
[23] Labuschagne L E, Majewski W A. Integral and differential structures for quantum field theory. arXiv:1702.00665[math-ph]
[24] Majewski W A. The detailed balance condition in quantum statistical mechanics. J Math Phys, 1984, 25:614-616
[25] Majewski W A. Dynamical semigroups in the algebraic formulation of statistical mechanics. Fortsch Phys, 1984, 32:89
[26] Majewski W A, Labuschagne L E. On applications of Orlicz spaces to Statistical Physics. Ann H Poincare, (2014, 15:1197-1221
[27] Majewski W A, Labuschagne L E. Why are Orlicz spaces useful for Statistical Physics?//Daniel Alpay, et al, Eds. Noncommutative Analysis, Operator Theory and Applications. Linear Operators and Linear Systems Vol 252. Birkhauser-Basel, 2016
[28] Majewski W A, Labuschagne L E. On Entropy for general quantum systems. Adv Theor Math Phys, to appear (See also arXiv:1804.05579[math-ph])
[29] Majewski W A, Streater R F. The detailed balance condition and quantum dynamical maps. J Phys A:Math Gen, 1998, 31:7981-7995
[30] Nelson E. Notes on non-commutative integration. J Funct Anal, 1974, 15:103
[31] Paulsen V. Completely Bounded Maps and Operator Algebras. Cambridge University Press, 2002
[32] Pedersen G K, Takesaki M. The Radon-Nikodym theorem for von Neumann algebras. Acta Mathematica, 1973, 130:53-87
[33] Pistone G, Sempi C. An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. The Annals of Statistics, 1995, 23:1543-1561
[34] Rao M M, Ren Z D. Theory of Orlicz Spaces. Marcel Dekker, 1991
[35] Schmitt L. The Radon-Nikodym theorem for Lp-spaces of W* algebras. Publ Res Inst Math Sci Kyoto, 1986, 22:1025-1034
[36] Segal I E. A non-commutative extension of abstract integration. Ann Math, 1953, 57:401
[37] Takesaki M. Theory of Operator Algebras, vol I. Springer, 1979
[38] Takesaki M. Duality for crossed products and the structure of von Neumann algebras of type Ⅲ. Acta Math, 1973, 131:249-308
[39] Terp M. Lp spaces associated with von Neumann algebras. Københavs Universitet, Mathematisk Institut, Rapport No 3a, 1981
[40] Tolman R C. Foundation of Statistical Mechanics. Oxford, 1938
[41] Van Daele A. Continuous crossed products and type Ⅲ von Neumann algebras. Cambridge University Press, 1978
[42] Villani C. A review of mathematical topics in collisional kinetic theory//Handbook of Mathematical Fluid Dynamics, Vol I. Amsterdam:North-Holland, 2002:71-305
[43] Yeadon F J. Ergodic theorems for semifinite von Neumann algebras I. J London Math Soc, 1977, 16:326-332
[44] Zippin M. Interpolation of operators of weak type between rearrangement invariant Banach function spaces. J Funct Anal, 1971, 7:267-284
