Abstract
In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .
Highlights
In [1,2], we constructed-and-studied weighted-semicircular elements and semicircular elements induced by p-adic number fields Q p, for all p ∈ P, where P is the set of all primes in the set N of all natural numbers
We are interested in free distributions of certain free reduced words in oursemicircular elements under conditions dictated by the primes p in a “suitable” closed interval [t1, t2 ] of the set R of real numbers
In [10], primes are regarded as linear functionals acting on arithmetic functions, understood as Krein-space operators under the representation of [11]
Summary
In [1,2], we constructed-and-studied weighted-semicircular elements and semicircular elements induced by p-adic number fields Q p , for all p ∈ P , where P is the set of all primes in the set N of all natural numbers. We consider certain “truncated” free-probabilistic information of the weighted-semicircular laws and the semicircular law of [1]. We are interested in free distributions of certain free reduced words in our (weighted-)semicircular elements under conditions dictated by the primes p in a “suitable” closed interval [t1 , t2 ] of the set R of real numbers. Our results illustrate how the original (weighted-)semicircular law(s) of [1] is (resp., are) distorted by truncations on P
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