[Paper Review] Permutations of Strongly Self-Absorbing C*-algebras

Ilan Hirshberg, Wilhelm Winter|ArXiv.org|Aug 1, 2007

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes that crossed products of C*-algebras by finite group actions permuting tensor factors preserve strong self-absorption properties when the original algebra absorbs certain infinite-type UHF algebras, the Jiang-Su algebra 𝒪₂, or is approximately divisible. The key result shows that if a unital G-equivariant homomorphism exists from 𝒟⊗ⁿ into the central sequence algebra of 𝒜, then the crossed product 𝒜×G inherits 𝒟-absorption, extending permanence results beyond Rokhlin-type actions.

ABSTRACT

Let G be a finite group acting on {1,...,n}. For any C*-algebra A, this defines an action of αof G on A^{\otimes n}. We show that if A tensorially absorbs a UHF algebra of infinite type, the Jiang-Su algebra, or is approximately divisible, then A imes_α G has the corresponding property as well.

Motivation & Objective

• To extend permanence results for strong self-absorption in C*-algebras beyond actions with the Rokhlin property.
• To investigate whether crossed products by finite permutation actions preserve 𝒟-absorption for known strongly self-absorbing C*-algebras 𝒟.
• To establish conditions under which 𝒜×G inherits 𝒟-absorption when 𝒜 is 𝒟-stable or approximately divisible.
• To provide a constructive proof of unital embeddings of 𝒟 into fixed-point algebras of tensor powers under symmetric group actions.

Proposed method

• Use a G-equivariant unital homomorphism ψ: 𝒟⊗ⁿ → ℳ(𝒜)∞ ∩ 𝒜′ to lift the action structure to the central sequence algebra.
• Construct unital embeddings of 𝒟 into the fixed-point algebra (𝒟⊗ⁿ)ˢⁿ for specific 𝒟, including UHF algebras of infinite type, the Jiang-Su algebra ℤ, and Mₚ⊕M_q with p,q > n.
• Apply fiberwise analysis of C*-algebras over the simplex Δ = [0,1]ⁿ/Sₙ to show that all fibers of (ℰ⊗ⁿ)ˢⁿ are ℤ-absorbing.
• Use the fact that UHF algebras of type 2∞ or 3∞ are ℤ-absorbing (by [JS, Theorem 5]) to deduce ℤ-absorption of the fixed-point algebra.
• Leverage the structure of symmetric group actions on tensor products to show that isotropy-invariant subalgebras absorb UHF algebras.
• Apply Theorem 2.4 (from [HRW]) on finite covering dimension to conclude that (ℰ⊗ⁿ)ˢⁿ is ℤ-absorbing.

Experimental results

Research questions

• RQ1 Under what conditions does a crossed product 𝒜×G by a finite permutation action preserve 𝒟-absorption for strongly self-absorbing C*-algebras 𝒟?
• RQ2 Can 𝒟-absorption be preserved in crossed products when the group action lacks the Rokhlin property, as in symmetric group actions on tensor powers?
• RQ3 Is there a unital embedding of the Jiang-Su algebra ℤ into the fixed-point algebra (𝒟⊗ⁿ)ˢⁿ for 𝒟 = ℰ = C([0,1], M₂∞⊗M₃∞) with the symmetric group action?
• RQ4 Does the fixed-point algebra of a tensor power under the symmetric group action absorb a UHF algebra when 𝒟 is a UHF algebra of infinite type?
• RQ5 Can the permanence of approximate divisibility be extended to crossed products under permutation actions?

Key findings

• If 𝒜 is 𝒟-stable for 𝒟 a UHF algebra of infinite type, the Jiang-Su algebra ℤ, or 𝒪₂ or 𝒪∞, then 𝒜×G is 𝒟-absorbing under any finite permutation action of G on 𝒜⊗ⁿ.
• For 𝒟 = ℤ, a unital embedding of ℤ into (ℰ⊗ⁿ)ˢⁿ exists, where ℰ = {f ∈ C([0,1], M₂∞⊗M₃∞) | f(0) ∈ M₂∞⊗1, f(1) ∈ 1⊗M₃∞}, via fiberwise analysis over the simplex Δ.
• The fixed-point algebra (ℰ⊗ⁿ)ˢⁿ is ℤ-absorbing because all its fibers are ℤ-absorbing and Δ has finite covering dimension.
• The isotropy-invariant subalgebras (ℰₛⱼ⊗ᵏʲ)ˢᵏʲ absorb UHF algebras of type 2∞ or 3∞, and hence are ℤ-absorbing by [JS, Theorem 5].
• For 𝒟 = Mₚ⊕M_q with p,q > n, a unital G-equivariant homomorphism ψ: 𝒟⊗ⁿ → ℳ(𝒜)∞ ∩ 𝒜′ implies that 𝒜×G is approximately divisible.
• The result holds even when the action lacks the Rokhlin property, such as in the case of ℤ acting on ℤ⊗ⁿ via permutations, where no Rokhlin action exists due to projectionlessness.