[Paper Review] A selective version of Lin's Theorem.

Scott Atkinson|arXiv (Cornell University)|Aug 14, 2018

Advanced Operator Algebra Research40 references3 citations

TL;DR

This paper establishes a selective version of Lin's Theorem by proving tracial stability for specific graph products of C*-algebras, using a newly introduced pincushion class of finite graphs. The key contribution is showing that the full C*-algebra of any right-angled Artin group is quasidiagonal, hence admitting an amenable trace, and that such traces are locally finite-dimensional under tracial stability conditions.

ABSTRACT

We prove a selective version of Lin's Theorem for nearly commuting operators. This is accomplished by establishing the tracial stability of a certain of graph products of $C^*$-algebras. This result involves the development of the pincushion class of finite graphs. We are further motivated by some approximation properties of right-angled Artin groups. In particular, we show that the full $C^*$-algebra of any right-angled Artin group is quasidiagonal and thus has an amenable trace. Tracial stability is then applied in showing when these amenable traces are in fact locally finite dimensional.

Motivation & Objective

• To develop a selective version of Lin's Theorem for nearly commuting operators.
• To establish tracial stability for certain graph products of C*-algebras.
• To investigate approximation properties of right-angled Artin groups via their C*-algebras.
• To determine when amenable traces on full C*-algebras of right-angled Artin groups are locally finite-dimensional.

Proposed method

• Introduce the pincushion class of finite graphs to analyze structural properties of graph products.
• Apply tracial stability techniques to C*-algebras associated with graph products.
• Use operator algebra methods to analyze nearly commuting operators in the context of C*-algebra representations.
• Leverage properties of right-angled Artin groups to deduce quasidiagonality of their full C*-algebras.
• Connect quasidiagonality to the existence of amenable traces via C*-algebraic approximation techniques.
• Apply tracial stability to characterize when amenable traces are locally finite-dimensional.

Experimental results

Research questions

• RQ1 Under what conditions does tracial stability hold for graph products of C*-algebras?
• RQ2 How can the pincushion class of finite graphs be used to analyze C*-algebraic structures?
• RQ3 Is the full C*-algebra of any right-angled Artin group quasidiagonal?
• RQ4 When is an amenable trace on such a C*-algebra locally finite-dimensional?
• RQ5 What role do approximation properties of right-angled Artin groups play in the structure of their C*-algebras?

Key findings

• The full C*-algebra of any right-angled Artin group is quasidiagonal.
• This quasidiagonality implies the existence of an amenable trace on the C*-algebra.
• Tracial stability ensures that certain amenable traces are locally finite-dimensional.
• The pincushion class of finite graphs provides a structural tool for analyzing graph product C*-algebras.
• The selective version of Lin's Theorem is established via tracial stability in the context of these graph products.
• The results demonstrate a connection between combinatorial graph properties and C*-algebraic approximation phenomena.