ABSTRACT
This chapter examines free calculus induced by the semicircular elements in the sense of Cho. The construction of semicircular elements is similar to, but different from the usual, or the traditional constructions of semicircular elements in free probability theory. The chapter aims to: apply the main results of Cho in free stochastic calculus on Banach -algebras; and study how mutually free weighted-semicircular, or semicircular elements act under the settings of Cho. It also aims to measure the actions among those elements under the free-stochastic-calculus models, by considering the free distributions of stochastic integrals for the elements. As applications, the free stochastic integration in terms of free stochastic motions generated by weighted-semicircular elements of Cho (2017) was considered in Cho (2016); and the differences between our weighted-semicircular laws of Cho (2017) and the semicircular law were studied in Cho (2018).
