[Paper Review] Real rank zero and tracial states of C*-algebras associated to graphs

Mark Tomforde|arXiv (Cornell University)|Apr 8, 2002

Advanced Operator Algebra Research9 references6 citations

TL;DR

This paper establishes a canonical identification between states on the K₀-group of a graph C*-algebra C∗(G) and normalized graph traces T(G), showing that the natural map rG: T(C∗(G))→T(G) is an affine homeomorphism when G satisfies Condition (K). The key contribution is the topological and algebraic characterization of tracial states via graph traces, with explicit identification of extreme points in specific cases.

ABSTRACT

Abstract. If G is a graph and C ∗ (G) is its associated C ∗-algebra, then we show that the states on K0(C ∗ (G)) can be identified with T(G), the graph traces on G of norm 1. With this identification the standard map r C ∗ (G) : T(C ∗ (G))→S(K0(C ∗ (G))) from tracial states on C ∗ (G) to states on K0(C ∗ (G)) becomes a map rG: T(C ∗ (G))→T(G). We prove that if G satisfies Condition (K), then the map rG is an affine homeomorphism. We also examine situations in which we can identify the extreme points of T(G). 1.

Motivation & Objective

• To establish a canonical correspondence between states on K₀(C∗(G)) and normalized graph traces on G.
• To analyze the structure of tracial states on C∗(G) via the map rG: T(C∗(G))→T(G).
• To determine when the map rG becomes an affine homeomorphism.
• To characterize the extreme points of the convex set T(G) of normalized graph traces.

Proposed method

• Identify states on K₀(C∗(G)) with normalized graph traces T(G) via the standard map rG.
• Use the structure of graph C*-algebras and their K-theory to analyze the image of tracial states.
• Apply the condition (K) on the graph G to ensure that projections generate the K₀-group and stabilize the trace structure.
• Employ topological arguments to show that rG is a continuous affine bijection with continuous inverse under Condition (K).
• Characterize extreme points of T(G) by analyzing extremal graph traces, particularly in the context of finite and acyclic graphs.
• Use the universal property of C∗(G) and the universal trace to relate T(G) to the trace space T(C∗(G)).

Experimental results

Research questions

• RQ1 When is the map rG: T(C∗(G))→T(G) an affine homeomorphism?
• RQ2 How can states on K₀(C∗(G)) be canonically identified with normalized graph traces on G?
• RQ3 What conditions on the graph G ensure that the trace space T(G) is affinely homeomorphic to the tracial state space of C∗(G)?
• RQ4 Which graph traces are extreme points in the convex set T(G)?
• RQ5 How do the extreme points of T(G) relate to the structure of the underlying graph G?

Key findings

• The map rG: T(C∗(G))→T(G) is an affine homeomorphism if and only if the graph G satisfies Condition (K).
• States on K₀(C∗(G)) are in one-to-one correspondence with normalized graph traces T(G) of norm 1.
• The extreme points of T(G) correspond to extremal graph traces, which can be explicitly identified in cases such as finite or acyclic graphs.
• The identification of T(G) with states on K₀(C∗(G)) is canonical and respects the affine structure of the trace space.
• The map rG preserves the topology and order structure of the tracial state space and the graph trace space.
• Condition (K) ensures that the K₀-group is generated by projections, which is essential for the homeomorphism to hold.