[Paper Review] Simple C*-algebras with locally finite decomposition rank

Wilhelm Winter|ArXiv.org|Feb 27, 2006

Advanced Operator Algebra Research9 references4 citations

TL;DR

This paper introduces the concept of locally finite decomposition rank for simple unital C*-algebras, showing that such algebras with real rank zero and Z-stability have tracial rank zero. This result confirms the Elliott conjecture for Z-stable ASH algebras with real rank zero and implies they are approximately homogeneous of topological dimension at most 3 under the UCT.

ABSTRACT

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C*-algebras. The concept is particularly relevant for Elliott's program to classify nuclear C*-algebras by K-theory data. We study some of its properties and show that a simple unital C*-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang-Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence, any such C*-algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homogeneous of topological dimension at most 3. Our result in particular confirms the Elliott conjecture for the class of simple unital Z-stable ASH algebras with real rank zero. Moreover, it implies that simple unital Z-stable AH algebras with real rank zero not only have slow dimension growth in the ASH sense, but even in the AH sense.

Motivation & Objective

• To extend classification results for nuclear C*-algebras beyond finite decomposition rank to the broader class of locally finite decomposition rank.
• To demonstrate that Z-stability, real rank zero, and locally finite decomposition rank imply tracial rank zero in simple unital C*-algebras.
• To confirm the Elliott conjecture for simple unital Z-stable ASH algebras with real rank zero, under the UCT.
• To show that such algebras are approximately homogeneous of topological dimension at most 3, strengthening known dimension growth results.
• To establish that locally finite decomposition rank is a mild structural condition that does not imply strong regularity properties like stable rank one or weak unperforation.

Proposed method

• Introduce and study the notion of locally finite decomposition rank as an exhaustion by C*-subalgebras of finite decomposition rank, without requiring a global bound on ranks.
• Use inductive approximation techniques to construct finite-dimensional C*-subalgebras D within A, satisfying norm and tracial control conditions.
• Apply order zero maps and properties of projections in C*-algebras with real rank zero to control commutators and spectral gaps.
• Employ a recursive construction of projections and subalgebras satisfying distance and spectral approximation conditions (e.g., ||[p_i, φ(f)]|| < δ).
• Use tracial state control via inequalities involving μ and τ(1_C_{k-1}) ≥ μ·τ(1_A - q_{k-1}) to ensure uniform lower bounds on trace of projections.
• Leverage Lin’s tracial rank zero characterization and the UCT to conclude that the algebra is approximately homogeneous of topological dimension at most 3.

Experimental results

Research questions

• RQ1 Can the classification result for C*-algebras with finite decomposition rank be extended to those with locally finite decomposition rank?
• RQ2 Does Z-stability, real rank zero, and locally finite decomposition rank imply tracial rank zero in simple unital C*-algebras?
• RQ3 To what extent does locally finite decomposition rank imply topological dimension control in Z-stable C*-algebras?
• RQ4 Can the tracial rank zero condition be established without assuming slow dimension growth, relying instead on Z-stability and local decomposition rank?
• RQ5 Do simple unital Z-stable ASH algebras with real rank zero necessarily have AH decomposition rank at most 2 and topological dimension at most 3?

Key findings

• A simple unital C*-algebra with locally finite decomposition rank, real rank zero, and Z-stability has tracial rank zero.
• Such algebras satisfy the Elliott conjecture if they additionally satisfy the UCT, confirming classification by K-theory data.
• Simple unital Z-stable ASH algebras with real rank zero have tracial rank zero and are approximately homogeneous of topological dimension at most 3.
• The class of simple unital Z-stable AH algebras with real rank zero has both slow dimension growth (in the ASH sense) and in fact in the AH sense.
• Locally finite decomposition rank is preserved under quotients, inductive limits, and hereditary subalgebras generated by projections, and implies nuclearity and quasidiagonality.
• The result removes redundancy in assumptions: tracial rank zero is now derived from Z-stability and locally finite decomposition rank, rather than requiring finite decomposition rank or slow dimension growth.