Noncommutative Mathematics for Quantum Systems
82 Noncommutative Mathematics for Quantum Systems Multiplicative boolean convolution onM 1 (R+) Bercovici defined a boolean conv ...
Independence and L ́evy Processes in Quantum Probability 83 It is easy to deduce from Subsection 1.7.1 that the multiplicative b ...
84 Noncommutative Mathematics for Quantum Systems processes with tensor-independent increments. In order to define and study Lev ...
Independence and L ́evy Processes in Quantum Probability 85 ‰k∈IAk their free product, with canonical inclusions {ik:Ak→‰k∈IAk}k ...
86 Noncommutative Mathematics for Quantum Systems fora 1 ⊗···⊗an∈ Ae,b 1 ⊗···⊗bm∈ Aδ. Note that in the case en=δ 1 the productan ...
Independence and L ́evy Processes in Quantum Probability 87 How is this functor defined on the morphisms? Show that the followin ...
88 Noncommutative Mathematics for Quantum Systems NuAlgProb, ∗-AlgProb, and ∗-NuAlgProb by restricting to commutative algebras. ...
Independence and L ́evy Processes in Quantum Probability 89 and with values in two possibly distinct measurable spaces (E 1 ,E 1 ...
90 Noncommutative Mathematics for Quantum Systems 1.8.3 Products of algebraic probability spaces We will now define several prod ...
Independence and L ́evy Processes in Quantum Probability 91 (i) The images of j 1 and j 2 commute, that is, [ j 1 (a 1 ),j 2 (a ...
92 Noncommutative Mathematics for Quantum Systems Voiculescu’s[VDN92]free productφ 1 ∗φ 2 :A 1 ‰A 2 →Cof two unital linear funct ...
Independence and L ́evy Processes in Quantum Probability 93 φ 1 φ 2 (a 1 a 2 ···am) = m ∏ k= 1 φek(ak), φ 1 .φ 2 (a 1 a 2 ···am ...
94 Noncommutative Mathematics for Quantum Systems This leads again to the formulae φ 1 φ 2 (a 1 a 2 ···an) = n ∏ i= 1 φei(ai), ...
Independence and L ́evy Processes in Quantum Probability 95 Associativity: ( (φ 1 ·φ 2 )·φ 3 ) ◦αA 1 ,A 2 ,A 3 =φ 1 ·(φ 2 ·φ 3 ...
96 Noncommutative Mathematics for Quantum Systems version⊗ ̃ of the tensor product defined in Example 1.8.10 and the free produc ...
Independence and L ́evy Processes in Quantum Probability 97 (φ 1 ·φ 2 )(a 1 a 2 ) = c 1 φ 1 (a 1 )φ 2 (a 2 ), (1.8.5) (φ 1 ·φ 2 ...
98 Noncommutative Mathematics for Quantum Systems The third step, which was actually completed first in both cases, see [Spe97] ...
Independence and L ́evy Processes in Quantum Probability 99 can be understood as a functor fromComAlgto the category of unital s ...
100 Noncommutative Mathematics for Quantum Systems Adual group[Voi87, Voi90] (calledH-algebraorcogroupin the category of unital ...
Independence and Levy Processes in Quantum Probability ́ 101 As the composition of the three unital∗-algebra homomorphisms ∆:B → ...
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