ABSTRACT
In this chapter, we study how the Clifford algebra https://www.w3.org/1998/Math/MathML"> C = l i m → C k + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032637211/d7ba22f2-120e-401b-8c2f-40c3344e6436/content/unequ79-01.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> acts on a unital C *-probability space (A, φ), where A is a unital C *-algebra with its unity (or multiplication identity) 1 A satisfying φ (1 A ) = 1 (i.e., (A, φ) is a noncommutative analog of a classical probability space). Since the Clifford algebra 𝒞 is a ℝ-Banach algebra, instead of acting 𝒞 on a fixed C *-algebra (over ℂ) artificially, we act the Clifford-group C *-algebra (in short, the CG algebra) M 𝒢 on A under the tensor product of C *-algebras and construct a suitable linear functional https://www.w3.org/1998/Math/MathML"> ψ = denote τ ⊗ φ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032637211/d7ba22f2-120e-401b-8c2f-40c3344e6436/content/unequ79-01a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on the tensor-product C *-algebra M 𝒢 ⊗ A, where τ is the canonical trace on the group C *-algebra M 𝒢 of the Clifford group 𝒢, inducing the group C *-probability space (M 𝒢 , τ), our Clifford-group C *-probability space (in short, the CG C *-probability space). By analyzing the free-probabilistic information on this new C *-probability space, https://www.w3.org/1998/Math/MathML"> ( M 𝒢 ⊗ A , ψ = τ ⊗ φ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032637211/d7ba22f2-120e-401b-8c2f-40c3344e6436/content/unequ80-02.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
we consider how the Clifford group 𝒢 affect the original free-distributional data on (A, φ). In particular, we are interested in how the Clifford algebra 𝒢 deform the semicircular law on (A, φ).
