LEADER 05362nam 2200625 450 001 996466600003316 005 20220909003020.0 010 $a3-662-21558-6 024 7 $a10.1007/978-3-662-21558-6 035 $a(CKB)3390000000043620 035 $a(SSID)ssj0001058626 035 $a(PQKBManifestationID)11602544 035 $a(PQKBTitleCode)TC0001058626 035 $a(PQKBWorkID)11071910 035 $a(PQKB)11701766 035 $a(DE-He213)978-3-662-21558-6 035 $a(MiAaPQ)EBC5591411 035 $a(Au-PeEL)EBL5591411 035 $a(OCoLC)1066191961 035 $a(MiAaPQ)EBC6841964 035 $a(Au-PeEL)EBL6841964 035 $a(OCoLC)1292359664 035 $a(PPN)238017923 035 $a(EXLCZ)993390000000043620 100 $a20220909d1993 uy 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aQuantum probability for probabilists /$fPaul-Andre? Meyer 205 $a1st ed. 1993. 210 1$aBerlin, Germany ;$aNew York, New York :$cSpringer-Verlag,$d[1993] 210 4$dİ1993 215 $a1 online resource (X, 293 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1538 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-56476-4 327 $aI: Non Commutative Probability -- II: Spin -- III: The Harmonic Oscillator -- IV: Fock Space (1) -- V. Multiple Fock Spaces -- VI. Stochastic Calculus on Fock Space -- VII. Independence -- Appendix 1: Functional Analysis -- Hilbert-Schmidt operators (1) -- Trace class operators (2) -- Duality properties (3) Weak convergence properties (4) -- Weak topologies for operators (5) -- Tensor products of Hilbert spaces (6?7) -- Appendix 2: Conditioning and Kernels -- Conditioning: discrete case (1) -- Conditioning: continuous case (2) -- Example of the canonical pair (3) -- Multiplicity theory (4) -- Classical kernels (5) -- Non commutative kernels, first form (6) -- second form (7) -- Completely positive maps (8) -- Some difficulties (9) -- Appendix 3: Two Events -- 1. Elementary theory -- Application of spectral theory (2) -- Some elementary properties (3) -- Positive elements (4) -- Symbolic calculus for s.a. elements (5) -- Applications (6) -- Characterization of positive elements (7) -- A few inequalities (8) -- Existence of many states (1) -- Representations and the GNS theorem (2?3) -- Examples from toy Fock space theory (4) -- Quotient algebras and approximate units (5) -- 3. Von Neumann algebras -- Weak topologies and normal states (1) -- Von Neumann?s bicommutant theorem (2?3) -- Kaplanski?s density theorem (4) -- The predual (5) -- Normality and order continuity (6) -- About integration theory (7) -- Measures with bounded density (8) -- The linear Radon-Nikodym theorem (9) -- The KMS condition (10) -- Entire vectors (11) -- 4. The Tomita-Takesaki theory -- Elementary geometric properties (1) -- The main operators (2?3) -- Interpretation of the adjoint (4) -- The modular property (5) -- Using the linear RN theorem (6) -- The main computation (7) -- The three main theorems (8) -- Additional results (9) -- Examples (10) -- Appendix 5: Local Times and Fock Space -- 1. Dynkin?s formula -- Symmetric Markov semigroups and processes (1) -- Dynkin?s formula (2) -- Sketch of the Marcus-Rosen approach to the continuity of local times (3) -- 2. Le Jan?s ?supersymmetric? approach -- Notations of complex Brownian motion (1) -- Computing the Wiener product (2) -- Stratonovich integral and trace (4) -- Expectation of the exponential of an element of the second chaos (5) -- Exponential formula in the antisymmetric case (7) -- Supersymmetric Fock space: the Wick and Wiener products (8) -- Properties of the Wiener product (9) -- Applications to local times (sketch) (10) -- References -- Index of Notation. 330 $aThese notes contain all the material accumulated over six years in Strasbourg to teach "Quantum Probability" to myself and to an audience of commutative probabilists. The text, a first version of which appeared in successive volumes of the Seminaire de Probabilite8, has been augmented and carefully rewritten, and translated into international English. Still, it remains true "Lecture Notes" material, and I have resisted suggestions to publish it as a monograph. Being a non-specialist, it is important for me to keep the moderate right to error one has in lectures. The origin of the text also explains the addition "for probabilists" in the title : though much of the material is accessible to the general public, I did not care to redefine Brownian motion or the Ito integral. More precisely than "Quantum Probability" , the main topic is "Quantum Stochastic Calculus" , a field which has recently got official recognition as 81825 in the Math. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1538 606 $aDistribution (Probability theory)$vTables 606 $aQuantum theory 615 0$aDistribution (Probability theory) 615 0$aQuantum theory. 676 $a519.2 686 $a81S25$2msc 700 $aMeyer$b Paul Andre?$0350920 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466600003316 996 $aQuantum probability for probabilists$978689 997 $aUNISA