Combinatorics of free probability theory [Lecture notes] by Roland Speicher

By Roland Speicher

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An ). ii) We have for all n ∈ N, 1 ≤ k < n and all a1 , . . , ak ∈ X1 and ak+1 , . . , an ∈ X2 that kn (a1 , . . , ak , ak+1 , . . , an ) = 0. 11. We will use the following notations: A partition π ∈ P(n) is called decomposable, if there exists an interval I = {k, k + 1, . . , k + r} = {1, . . , n} (for some k ≥ 1, 0 ≤ r ≤ n − r), such that π can be written in the form π = π1 ∪ π2 , where π1 ∈ P({k, k + 1, . . , k + r}) is a partition of I and π2 ∈ P(1, . . , k − 1, k + r + 1, . . , n}) is a partition of {1, .

I (a)η ⊗ ξ1 ⊗ ξ2 ⊗ . . = (aηi ) ⊗ ξ1 ⊗ ξ2 ⊗ . . = λη ⊗ ξ1 ⊗ ξ2 ⊗ · · · + ξ o ⊗ ξ1 ⊗ ξ2 ⊗ . . = λξ1 ⊗ ξ2 ⊗ · · · + ξ o ⊗ ξ1 ⊗ ξ2 ⊗ . . ⊂ H. g. ) The representation πi of Ai is faithful, thus we have (57) Ai ∼ = πi (Ai ) ⊂ B(H). We put now (58) A := C ∗ (π1 (A1 ), . . , πm (Am )) and (59) ϕ(a) := η, aη (a ∈ A). 4. , ϕ|πi (Ai ) = ϕi . It only remains to show that π1 (A1 ), . . , πm (Am ) are free in (A, ϕ). For this consider π(a1 ) . . π(ak ) with ai ∈ Aoj(i) and j(1) = j(2) = · · · = j(k).

3. 1) The classical example which gave the name to the M¨obius inversion is due to M¨obius (1832) and comes from number theory: M¨obius showed that a relation f (n) = m|n g(n) – where n and m are integer numbers and m | n means that m is a divisor of n – can be inverted in the form g(n) = m|n f (m)µ(m/n) where µ is now the classical M¨obius function given by (76) µ(n) = (−1)k , if n is the product of k distinct primes 0, otherwise. 2) We will be interested in the case where P = N C(n). ), when we have more adequate tools at our disposition.

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