ABSTRACT
In this chapter, motivated by the main results of Part I, we study algebra, analysis, operator theory, and free probability induced by the classical Clifford algebras {𝒞k } k∈ℕ and generalize them to those induced by the limit C∗ -algebra https://www.w3.org/1998/Math/MathML"> C = lim → C k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032637211/d7ba22f2-120e-401b-8c2f-40c3344e6436/content/inequ37.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , simply called the Clifford algebra.
To do that, from Clifford algebras {𝒞k } k∈ℕ, we construct the embedded multiplicative groups {𝒢k } k∈ℕ. The construction of such groups allows us to have a group 𝒢 from the Clifford algebra 𝒞. Since this group 𝒢 is identified with the set ±ℰ = {±w : w ∈ ℰ}, where ℰ is the ℝ-basis of 𝒞, the mathematical properties on 𝒞 are inherited to those on 𝒢.
From this group 𝒢, the corresponding group C∗ -algebra ℭ = C∗ (𝒢) is constructed, and the operator theory and free probability on ℭ are studied.
