[论文解读] The MF property for amalgamated free products

Tatiana Shulman|arXiv (Cornell University)|Mar 13, 2026

Advanced Operator Algebra Research被引用 0

一句话总结

该论文证明当 A 是 MF 时，结合自由积 A *_C A 也是 MF，并给出 A *_C B 为 MF 的必要充分条件；还确定了若干类群的 MF 全群 C*-代数，并在张量积和某些扩张下给出稳定性结果。

ABSTRACT

A C*-algebra (or a group) is called MF (matricial field) if it admits finite dimensional approximate unitary representations which are approximately injective, where approximately is meant with respect to the operator norm. It is proved that for any MF C*-algebra $A$ and its C*-subalgebra $C$, $A\ast_C A$ is MF. For general amalgamated free products, $A\ast_C B$, a necessary and sufficient condition for being MF is given. It is shown that the following groups -- amalgamated free products of amenable groups, semidirect products of amenable groups by free groups, and $\mathbb Z^2 times SL_2(\mathbb Z)$ -- all have MF full group C*-algebra. It is shown that the class of MF C*-algebras is closed under maximal tensor products with $C^*(\mathbb F_n)$.

研究动机与目标

• Motivate the study of MF (matricial field) approximations in C*-algebras and groups and relate MF to amenability and related notions.
• Establish that A is MF and C ⊆ A implies A *_C A is MF, expanding the class of MF algebras.
• Provide necessary and sufficient criteria for A *_C B to be MF in the general amalgamated setting.
• Show that several group constructions (amalgamated products, semidirect products by free groups, and Z^2 ⋊ SL(2,Z)) have MF full group C*-algebras.
• Demonstrate stability of MF under maximal tensoring with C*(F_n) and under central HNN-extensions.
• Offer lifting characterizations and technical lemmas to support MF-embeddings and asymptotic liftings.

提出的方法

• Use the lifting characterization of MF by Shulman (discrete asymptotic liftings to the matrix-algebra product modulo direct sum).
• Prove MF for A *_C A by constructing asymptotic lifts of representations and applying Lemma 7 (amalgamated liftings) and Theorem 4 (equivalence of MF and liftability).
• Characterize MF for A *_C B via embeddings into ∏ M_n / ⊕ M_n that agree on C (Theorem 16).
• Extend MF results to multiple factors via induction (Theorem 18).
• Apply results to groups by realizing C*(G1 *_H G2) as MF when G1, G2 are amenable (Theorem 19).
• Utilize crossed product decompositions and tensor product arguments (Propositions 28, Theorems 24, 25) and central HNN extensions (Theorems 25, 29) to transfer MF properties.]
• research_questions2: []

实验结果

研究问题

• RQ1 Under what conditions is the amalgamated free product A *_C B MF given A, B, C and embeddings?
• RQ2 Does MF pass from MF algebras to their amalgamated free products, particularly A *_C A when C ⊆ A?
• RQ3 Which group constructions yield MF full group C*-algebras (e.g., amenable amalgamations, semidirect products with free groups, Z^2 ⋊ SL(2,Z))?
• RQ4 Is MF preserved under maximal tensor products with C*(F_n) and under central HNN-extensions?
• RQ5 What are practical criteria to verify MF for complex amalgamated products via finite-dimensional approximations?

主要发现

• If A is separable MF and C ⊆ A, then the amalgamated free product A *_C A is MF (Theorem 10).
• For separable C*-algebras A, B, C with inclusions C → A and C → B, A *_C B is MF iff there exist embeddings into ∏ M_n / ⊕ M_n that agree on C (Theorem 16).
• Amalgamated free products of amenable groups over any subgroup yield MF full group C*-algebras (Theorem 19).
• Semidirect products G ⋊ F_n with G amenable have MF full group C*-algebras (Corollary 31).
• Z^2 ⋊ SL_2(Z) has MF full group C*-algebra (Corollary 30).
• The class of MF C*-algebras is closed under maximal tensor products with C*(F_n) (Theorem 24) and under central HNN-extensions (Theorem 25).