*1.2. Overview*

We briefly introduce the backgrounds of our works in Section 2. In the short Sections 3–8, we construct the Banach ∗-probability space (LS, *τ*<sup>0</sup>) and study *weighted-semicircular elements Qp*,*<sup>j</sup>* and corresponding *semicircular elements* <sup>Θ</sup>*p*,*<sup>j</sup>* in (LS, *<sup>τ</sup>*<sup>0</sup>), for all *p* ∈ P, *j* ∈ Z.

In Section 9, we define a free-probabilistic sub-structure LS = -LS, *τ*0 of the Banach ∗-probability space -LS, *τ*0 , having possible non-zero free distributions, and study free-probabilistic properties of LS. Then, *truncated linear functionals of τ*0 on LS and truncated free-probabilistic information on LS are studied. The main results illustrate how our truncations distort the original free distributions on LS (and hence, on LS).

In Section 10, we study *free sums X* of LS having their free distribution, the (weighted-)semicircular law(s), under truncation. Note that, in general, if free sums *X* have more than one summand as operators, then *X* cannot be (weighted-)semicircular in LS. However, certain truncations make them be.

In Section 11, we investigate a type of truncation (compared with those of Sections 9 and 10). In particular, certain truncations inducing so-called *prime-neighborhoods* are considered. The *unions* of such prime-neighborhoods provide corresponding distorted free probability on LS (different from that of Sections 9 and 10).
