Noncommutative Mathematics for Quantum Systems
42 Noncommutative Mathematics for Quantum Systems over the algebraic probability space (L∞(Ω,F,P),E), where E denotes expectatio ...
Independence and L ́evy Processes in Quantum Probability 43 We will show that Levy processes on compact quantum groups ́ can be ...
44 Noncommutative Mathematics for Quantum Systems are satisfied, then the pair(A,∆)is called acompact quantum group (CQG). If th ...
Independence and L ́evy Processes in Quantum Probability 45 `≤ns}(wherensdenotes the dimension ofu(s)) is a linear basis of Pol( ...
46 Noncommutative Mathematics for Quantum Systems functional onA. Here,M(B)denotes the set of multipliers onBof a∗-algebraB, and ...
Independence and L ́evy Processes in Quantum Probability 47 (i) fz( 1 ) =1 forz∈C, (ii) the mappingC 3 z7→ fz(a) ∈Cis an entire ...
48 Noncommutative Mathematics for Quantum Systems is invertible, with inverse ( f 1 (u(jks)) ) jk ∈ Mns(C),cf. [Wor87a, Equation ...
Independence and L ́evy Processes in Quantum Probability 49 strongly continuous Markov semigroup on both its reduced and its uni ...
50 Noncommutative Mathematics for Quantum Systems Cu(G),cf. [Wor88, BMT01]. This fact can be deduced from the properties of the ...
Independence and L ́evy Processes in Quantum Probability 51 where φbt(x) := φt(b∗xb). Moreover, by the antipode property (1.6.2) ...
52 Noncommutative Mathematics for Quantum Systems completely positive,Ttis completely positive and contractive. Let us now check ...
Independence and L ́evy Processes in Quantum Probability 53 Proof Letb ∈ Nh ={a ∈A;h(a∗a) = 0 }. We can assume that T 6 =0, then ...
54 Noncommutative Mathematics for Quantum Systems Then (Tt)t≥ 0 is the quantum Markov semigroup of a (uniquely determined) L ́ev ...
Independence and L ́evy Processes in Quantum Probability 55 by Lindsay and Skalski (cf.[LS11, Proposition 3.2]). This covers, fo ...
56 Noncommutative Mathematics for Quantum Systems and itsreciprocal Cauchy–Stieltjes transform Fμby Fμ(z) = 1 Gμ(z) . Denote byF ...
Independence and L ́evy Processes in Quantum Probability 57 cf.[BB05] and the references therein. In the following, if X is an o ...
58 Noncommutative Mathematics for Quantum Systems then X+Y is essentially self-adjoint and the distribution w.r.t.Ωof its closur ...
Independence and L ́evy Processes in Quantum Probability 59 Note that here the free convolution of two discrete measures has a d ...
60 Noncommutative Mathematics for Quantum Systems alg{h(X),h(Y);h∈Cb(R)}Ω=H. Then the products √ XY √ X and √ YX √ Y are essenti ...
Independence and L ́evy Processes in Quantum Probability 61 Remark 1.7.10 (a) Note that this notion depends on the order, that i ...
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