2026/07/21 by Mansi Anil Suryawanshi
#math.OA #math.FA
Let On be the odometer semigroup, and let \mathcal Aξ=C^*(S1,…,Sn,Wξ)⊆\mathcal B(\mathcal Fn2) be the C^*-algebra generated by the left creation operators and the scalar odometer map associated with a symbol ξ∈\mathcal Fn2. We show that \mathcal Aξ contains the compact operators for every scalar symbol. For an isometric scalar symbol, we prove that Wξ is Fredholm if and only if the associated inner function is a finite Blaschke product. We further show that the image of \mathcal Aξ in the Calkin algebra is canonically isomorphic to the odometer boundary quotient \mathcal Q(On). If the associated finite Blaschke product has degree d, then ind(Wξ)=-d. For d≥ 1, we obtain K0(\mathcal Aξ)≅\mathbb Z⊕\mathbb Zd(n-1) and K1(\mathcal Aξ)=0, whereas for d=0, K0(\mathcal Aξ)≅\mathbb Z2 and K1(\mathcal Aξ)≅\mathbb Z. Consequently, for fixed n≥ 2, finite Blaschke symbols of distinct degrees generate non-isomorphic C^*-algebras.