[Paper Review] Some Examples of Blackadar and Kirchberg's MF Algebras

Don Hadwin, Junhao Shen|ArXiv.org|Jun 28, 2008

Advanced Operator Algebra Research24 references4 citations

TL;DR

This paper investigates Blackadar and Kirchberg's MF algebras by constructing new examples through minimal and maximal tensor products of MF algebras and crossed products by finite or integer groups. It proves that certain tensor products and crossed products—such as $ C_r^*(F_n) times_ heta bZ $ and $ C_r^*(F_n) ensor{max} C^*(F_m) $—are MF algebras, and crucially, shows that their BDF extension semigroups are not groups, extending Haagerup and Thorbjørnsen's foundational result on $ C_r^*(F_n) $.

ABSTRACT

In the paper, we provide some examples of MF algebras by considering minimal or maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C$^*$-algebras, whose BDF extension semigroup is not group. These examples include, for example, $ C_r^*(F_n)\otimes_{max} C^*(F_n) $, $ C_r^*(F_n)\otimes_{min} C^*(F_n) $, $C_r^*(H_1\ast H_2)$ with $2\le |H_1|

Motivation & Objective

• To extend the class of known C*-algebras whose BDF extension semigroups are not groups, beyond the foundational example of $ C_r^*(F_n) $.
• To investigate whether tensor products and crossed products of MF algebras preserve the MF property.
• To provide new examples of C*-algebras where Voiculescu’s topological free entropy dimension is well-defined due to the MF property.
• To establish conditions under which crossed products of MF algebras by $ bZ $ or finite groups remain MF algebras.
• To generalize Pimsner and Voiculescu’s results in the context of MF algebras.

Proposed method

• Use the definition of MF algebras as embeddings into ultraproducts of matrix algebras to verify the MF property for tensor products and crossed products.
• Apply the universal property of full crossed products to construct $ * $-homomorphisms from $ bA times_ heta bZ $ into matrix ultraproducts.
• Employ norm approximation techniques to show that the norms of noncommutative polynomials in generators are preserved up to small error in finite-dimensional representations.
• Leverage the existence of a sequence $ n_j o bN $ such that $ orm{\alpha(n_j)a - a} \to 0 $ for all $ a \in \bbA $, ensuring asymptotic invariance under the action.
• Use the fact that $ C_r^*(F_n) $ is an MF algebra (by Haagerup and Thorbjørnsen) as a base case to build more complex examples.
• Combine results on tensor products and crossed products to derive new examples where the BDF extension semigroup fails to be a group.

Experimental results

Research questions

• RQ1 Under what conditions is the maximal tensor product of two MF algebras an MF algebra?
• RQ2 When is the minimal tensor product of two MF algebras an MF algebra, particularly if one factor is exact or quasidiagonal?
• RQ3 When is the crossed product of an MF algebra by $ bZ $ or a finite group an MF algebra?
• RQ4 Which C*-algebras have BDF extension semigroups that are not groups, beyond $ C_r^*(F_n) $?
• RQ5 Can the MF property be preserved under crossed product constructions involving unitary actions with almost periodic behavior?

Key findings

• The maximal tensor product $ C^*(F_n) \otimes_{\text{max}} \mathcal{B} $ is an MF algebra whenever $ \mathcal{B} $ is a unital separable MF algebra and $ n \geq 2 $.
• The minimal tensor product $ \mathcal{A} \otimes_{\text{min}} \mathcal{B} $ is an MF algebra if $ \mathcal{A} $ is exact or quasidiagonal and $ \mathcal{B} $ is a unital separable MF algebra.
• The crossed product $ \mathcal{A} \rtimes_\alpha \bbZ $ is an MF algebra when $ \mathcal{A} $ is a finitely generated unital MF algebra and $ \alpha $ is a $ \bbZ $-action with a sequence $ n_j \to \infty $ such that $ \norm{\alpha(n_j)a - a} \to 0 $ for all $ a \in \mathcal{A} $.
• The BDF extension semigroup $ \operatorname{Ext}(C_r^*(F_n) \otimes_{\text{max}} C^*(F_m)) $ is not a group for $ n, m \geq 2 $.
• The BDF extension semigroup $ \operatorname{Ext}(C_r^*(F_n) \otimes_{\text{min}} \mathcal{B}) $ is not a group for $ n \geq 2 $ and $ \mathcal{B} $ a unital separable MF algebra.
• The BDF extension semigroup $ \operatorname{Ext}(C_r^*(H)) $ is not a group when $ H = H_1 * H_2 $ with $ |H_1| \geq 2 $, $ |H_2| \geq 3 $, and $ H_1, H_2 $ finite groups.