[Paper Review] On Quasidiagonal C*-algebras

Nathanial P. Brown|ArXiv.org|Aug 23, 2000

Advanced Operator Algebra Research26 references12 citations

TL;DR

This paper provides a comprehensive, self-contained survey of quasidiagonal C*-algebras, focusing on structural properties and functorial behavior under standard C*-algebra constructions. It establishes that a C*-algebra is quasidiagonal if and only if all its separable subalgebras are quasidiagonal, and proves that quasidiagonality implies stable finiteness and the existence of a tracial state in the unital case.

ABSTRACT

We give a detailed survey of the theory of quasidiagonal C*-algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about extensions, quotients, tensor products, ect. of quasidiagonal C*-algebras. We also point out how quasidiagonality is connected to some important open problems.

Motivation & Objective

• To provide a detailed, accessible survey of quasidiagonal C*-algebras for researchers in operator algebras.
• To clarify the behavior of quasidiagonality under fundamental constructions such as subalgebras, quotients, tensor products, extensions, and crossed products.
• To connect quasidiagonality to major open problems in C*-algebra theory, including the Universal Coefficient Theorem and Elliott's Classification Program.
• To establish foundational results with detailed proofs, including Voiculescu's characterization and the separability reduction principle.
• To highlight unresolved questions and guide readers toward advanced literature on related topics like BDF-theory and inductive limits.

Proposed method

• Uses Voiculescu's theorem on quasidiagonal sets of operators as a central tool for characterizing quasidiagonal C*-algebras.
• Applies Stinespring's dilation theorem and properties of completely positive maps to analyze representations and quasidiagonality.
• Employs separable reduction techniques, proving that quasidiagonality of a C*-algebra is equivalent to quasidiagonality of all its separable subalgebras.
• Utilizes essential and faithful representations to reduce general quasidiagonality questions to the separable case.
• Applies results from Dădărlat and Popa to examine the relationship between quasidiagonality, nuclearity, and the classification program.
• Introduces a nonseparable extension via an appendix, proving that a C*-algebra is quasidiagonal iff all its separable subalgebras are.

Experimental results

Research questions

• RQ1 Under which C*-algebra constructions (e.g., tensor products, quotients, extensions) is quasidiagonality preserved?
• RQ2 What is the precise relationship between quasidiagonality, nuclearity, and stable finiteness in C*-algebras?
• RQ3 To what extent can quasidiagonality be characterized via separable subalgebras?
• RQ4 How does quasidiagonality relate to the Universal Coefficient Theorem and the group structure of Ext(C*r(F2))?
• RQ5 Is every nuclear stably finite C*-algebra necessarily quasidiagonal? (The Blackadar-Kirchberg question.)

Key findings

• A C*-algebra is quasidiagonal if and only if every separable unital subalgebra is quasidiagonal, establishing a fundamental reduction to the separable case.
• Every unital quasidiagonal C*-algebra admits a tracial state, linking quasidiagonality to the existence of invariant means.
• Quasidiagonality implies stable finiteness, a key structural constraint in the classification of C*-algebras.
• Every exact quasidiagonal C*-algebra can be locally approximated by finite-dimensional C*-algebras, a result due to Dădărlat.
• The class of quasidiagonal C*-algebras is not closed under quotients, extensions, or free products in general, highlighting non-trivial functorial behavior.
• There exist simple quasidiagonal C*-algebras that are not nuclear, refuting a conjecture by Popa and showing that quasidiagonality does not imply nuclearity.