Noncommutative Mathematics for Quantum Systems
22 Noncommutative Mathematics for Quantum Systems In the GNS representation of this state (see the lecture ‘Quantum dynamical sy ...
Independence and L ́evy Processes in Quantum Probability 23 1.4 Infinite Divisibility in Classical Probability Before we study q ...
24 Noncommutative Mathematics for Quantum Systems μ = μn?···?μn ︸ ︷︷ ︸ . ntimes Definition 1.4.2A family (μt)t≥ 0 of probability ...
Independence and L ́evy Processes in Quantum Probability 25 Proposition 1.4.5 If(Xst)is a L ́evy process with values in a topolo ...
26 Noncommutative Mathematics for Quantum Systems Recall that a non-empty subsetK in a real or complex vector space is called a ...
Independence and L ́evy Processes in Quantum Probability 27 1.4.7 The Markov semigroup of a Levy process ́ Markov processes are ...
28 Noncommutative Mathematics for Quantum Systems Fix a basis(Xj, 1≤j≤n)ofgand define the dense subspace C 2 L(G)by C 2 L(G) = { ...
Independence and L ́evy Processes in Quantum Probability 29 See also [App14, Theorem 5.3.3] for more details and an outline of t ...
30 Noncommutative Mathematics for Quantum Systems random variables (jt)t∈I. For a quantum random variable j:B →Awe will callφj=Φ ...
Independence and L ́evy Processes in Quantum Probability 31 Definition 1.5.1 Let(A,Φ)be a quantum probability space andB a∗-alge ...
32 Noncommutative Mathematics for Quantum Systems ∆f(g 1 ,g 2 ) =f(g 1 g 2 ), forg 1 ,g 2 ∈Gandε(f) =f(e). Definition 1.5.3 LetB ...
Independence and L ́evy Processes in Quantum Probability 33 In particular, it follows that the antipodeSof an involutive Hopf al ...
34 Noncommutative Mathematics for Quantum Systems The continuity is an immediate consequence of the last condition in Definition ...
Independence and L ́evy Processes in Quantum Probability 35 (b) Let(φt)t≥ 0 be a convolution semigroup on some coalgebraC. Then ...
36 Noncommutative Mathematics for Quantum Systems Letb∈ B,ε(b) =0. If allφtare states, then we haveφt(b∗b)≥ 0 for allt≥0 and the ...
Independence and L ́evy Processes in Quantum Probability 37 We will call a Schurmann triple ̈ surjective, if the cocycleη:B →D i ...
38 Noncommutative Mathematics for Quantum Systems Let nowLbe a generating functional. The sesqui-linear form 〈·,·〉L:B×B →Cdefine ...
Independence and L ́evy Processes in Quantum Probability 39 holds for allk∈Nand all 0≤s≤t. Construct the Schurmann triple for Br ...
40 Noncommutative Mathematics for Quantum Systems If a generating functionalLsatisfies one of these conditions, then we call it ...
Independence and L ́evy Processes in Quantum Probability 41 1.5.4 Examples L ́evy processes on the circleTand on the real lineR ...
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