The study of semicircular elements is one of the major topics (e.g., [20,21,29,30]), not only in both commutative function theory and noncommutative free probability theory but also in various applied fields, including quantum statistical physics (e.g., [5,6,7 and 8,10,11,12]). The semicircular law, which is the free distributions of semicircular elements, is well characterized analytically, and combinatorially, in classical function theory and in free probability theory (e.g., [1,17,18,21,28,29 and 30]). In particular, it is playing a key role in free-probabilistic operator algebra theory (and hence, in quantum physics) by the (free) central limit theorem(s) (e.g., see [2,17,19,28,29 and 30]), i.e., it becomes a noncommutative analytic analogue of the classical Gaussian (or the normal) distribution (in commutative function theory).

From combinatorial approaches (e.g., [17,22,23]), the semicircular law is universally characterized by the Catalan numbers ${\left\{c_k\right\}}_{k=1}^{\infty }$, where

From the analysis on p-adic number fields ℚ_p (e.g., [26,27]), one can construct semicircular elements (e.g., [5,12]). By generalizing the constructions of [5,12], semicircular elements are constructed whenever there are |ℤ|-many orthogonal projections in a C^∗-algebra (e.g., [6,7 and 8,10] and [11]), different from earlier works (e.g., [20,25,29,30]). In this new approach, the semicircular elements are understood as Banach-space operators acting on a given C^∗-algebra, by regarding the C^∗-algebra as a Banach space equipped with its C^∗-norm (e.g., [13,14]).

Independently, the joint free distributions of mutually free, multi semicircular elements were re-characterized in [9] (See...