Abstract
In this paper, we study âsemicircular-likeâ elements in free product Banach *-algebras induced by Haar-measurable functions over p-adic number fields , for primes p. And we investigate how the free distributions of operators generated by our mutually-free weighted-semicircular elements are close enough to (or far from) those of free reduced words generated by arbitrary mutually-free semicircular elements.
Highlights
The main purposes of this paper are (i) to establish weighted-semicircular elements and corresponding semicircular elements in a certain Banach *-probability space induced by measurable functions over p-adic number fields p, for primes p, (ii) to consider free-distributional data of weighted-semicircular elements, and those of operators generated by free, weighted-semicircular elements under free product, and (iii) to investigate how the free-distributional data of the operators generated by weighted-semicircular elements are close to those of operators generated by semicircular elements
The main results of the topic (iii) illustrate how the weights of our weighted-semicircular elements distort the semicircular law, and such distortions are measured by so-called the weight-ratios of weighted-semicircular elements
In [5], primes are regarded as linear functionals acting on arithmetic functions
Summary
The main purposes of this paper are (i) to establish weighted-semicircular elements and corresponding semicircular elements in a certain Banach *-probability space induced by measurable functions over p-adic number fields p , for primes p, (ii) to consider free-distributional data of weighted-semicircular elements, and those of operators generated by free, weighted-semicircular elements under free product, and (iii) to investigate how the free-distributional data of the operators generated by weighted-semicircular elements are close to those of operators generated by semicircular elements. The main results of the topic (iii) illustrate how the weights of our weighted-semicircular elements distort (or affect) the semicircular law, and such distortions are measured by so-called the weight-ratios of weighted-semicircular elements
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