[Paper Review] The Countable Chain Condition for C*-Algebras

Shuhei Masumoto|arXiv (Cornell University)|May 5, 2014

Advanced Operator Algebra Research4 references3 citations

TL;DR

This paper introduces the countable chain condition (CCC) for C*-algebras via orthogonal ideals, generalizing the topological CCC. It proves that the preservation of CCC under minimal tensor products is independent of ZFC, showing Martin's Axiom implies CCC is preserved, while its negation leads to counterexamples.

ABSTRACT

In this paper, we introduce the countable chain condition for C*-algebras and study its fundamental properties. We show independence from ZFC of the statement that this condition is preserved under the tensor products of C*-algebras.

Motivation & Objective

• To generalize the topological countable chain condition (CCC) to C*-algebras using orthogonal ideals.
• To investigate whether the CCC property is preserved under minimal tensor products of C*-algebras.
• To determine the set-theoretic independence of this preservation property from ZFC.
• To explore the role of Martin's Axiom and the Suslin Hypothesis in establishing or contradicting the preservation of CCC under tensor products.

Proposed method

• Define CCC for C*-algebras as the absence of uncountable families of nonzero mutually orthogonal ideals.
• Establish equivalence between topological CCC and C*-algebraic CCC via the Gelfand-Naimark duality for C₀(X).
• Use Martin’s Axiom (MA(ω₁)) to construct filters on a poset of ideals, ensuring uncountable families with finite intersection property.
• Apply Lemma 2.6 and properties of inductive limits to show density of certain ideals under MA(ω₁).
• Prove that if A and B have CCC, then A⊗B has CCC under MA(ω₁), using finite intersection properties and ideal containment.
• Demonstrate independence from ZFC by showing that the negation of the Suslin Hypothesis implies a counterexample to the preservation of CCC under tensor products.

Experimental results

Research questions

• RQ1 Is the countable chain condition for C*-algebras, defined via orthogonal ideals, a natural generalization of the topological CCC?
• RQ2 Does the minimal tensor product of two unital CCC C*-algebras necessarily have CCC?
• RQ3 Can the preservation of CCC under tensor products be proven or disproven within ZFC?
• RQ4 What is the role of Martin’s Axiom in ensuring the preservation of CCC under tensor products?
• RQ5 Is the CCC property preserved under non-minimal tensor products, and how does the kernel of the quotient map affect this?

Key findings

• The CCC for C*-algebras is defined via the absence of uncountable families of nonzero mutually orthogonal ideals, generalizing the topological CCC.
• A C*-algebra has CCC if and only if there is no uncountable family of nonzero elements {a_λ} such that a_λ A a_μ = 0 for λ ≠ μ.
• Martin’s Axiom (MA(ω₁)) implies that the minimal tensor product of any two unital CCC C*-algebras has CCC.
• The statement that tensor products of CCC C*-algebras preserve CCC is independent of ZFC, as shown by the consistency of both the statement and its negation.
• The negation of the Suslin Hypothesis implies the existence of a counterexample to the preservation of CCC under tensor products.
• A von Neumann algebra has CCC if and only if its center is σ-finite, and such algebras preserve CCC under tensor products.