[Paper Review] Embedding $\mathrm{C}^*$-algebras into the Calkin algebra

Ilijas Farah, Georgios Katsimpas+1 more|arXiv (Cornell University)|Sep 29, 2018

Advanced Topology and Set Theory19 references3 citations

TL;DR

Under Martin’s Axiom, every C*-algebra of density character less than the continuum embeds into the Calkin algebra. However, it is consistent with ZFC that some such C*-algebras do not embed, showing the embedding question is independent of ZFC. This resolves a noncommutative analogue of the Boolean algebra embedding problem.

ABSTRACT

Every $\mathrm{C}^*$-algebra, regardless of its density character, can be embedded into the Calkin algebra in a forcing extension of the universe obtained without collapsing any cardinal.

Motivation & Objective

• To determine the conditions under which C*-algebras of density character less than the continuum embed into the Calkin algebra.
• To investigate the noncommutative analogue of the Boolean algebra embedding problem: which C*-algebras embed into the Calkin algebra?
• To explore the independence of the embedding property from ZFC, particularly when the Continuum Hypothesis fails.
• To analyze the role of forcing axioms like Martin’s Axiom and the Proper Forcing Axiom in determining embeddability.
• To examine the structural properties of the Calkin algebra, such as its gap spectra and ccc-ness, in relation to embedding universality.

Proposed method

• Using Martin’s Axiom to construct embeddings of C*-algebras of density character < 2^ℵ₀ into the Calkin algebra via forcing techniques and ccc properties.
• Applying ccc forcing to preserve the structure of projections and gaps in the Calkin algebra, particularly focusing on property K and productively ccc posets.
• Analyzing the poset of projections in the Calkin algebra under Martin’s Axiom to show the existence of (2^ℵ₀, 2^ℵ₀)-gaps that are not frozen by forcing.
• Leveraging known results on Tukey equivalence and the additivity of Lebesgue measure to linearize analytic gaps in the projection lattice.
• Using the fact that the Calkin algebra is not countably saturated to show that embeddings of nonseparable C*-algebras require strong set-theoretic assumptions.
• Establishing that the poset 𝔼_A associated with a C*-algebra A has property K, implying it is productively ccc and cannot freeze gaps in P(ℕ)/Fin.

Experimental results

Research questions

• RQ1 Under which set-theoretic assumptions does every C*-algebra of density character less than the continuum embed into the Calkin algebra?
• RQ2 Is the existence of such embeddings independent of ZFC, and if so, in what models does the embedding fail?
• RQ3 Can the Calkin algebra be 2^ℵ₀-universal in models where CH fails?
• RQ4 What is the role of forcing axioms like Martin’s Axiom in enabling or obstructing embeddings of C*-algebras into the Calkin algebra?
• RQ5 How do the gap structures in the projection lattice of the Calkin algebra interact with ccc forcing and the preservation of embeddability?

Key findings

• Martin’s Axiom implies that every C*-algebra of density character strictly less than the continuum embeds into the Calkin algebra.
• It is consistent with ZFC that there exists a C*-algebra of density character less than the continuum that does not embed into the Calkin algebra, showing the embedding property is independent of ZFC.
• The poset of projections in the Calkin algebra contains a (2^ℵ₀, 2^ℵ₀)-gap under Martin’s Axiom that cannot be frozen by forcing, indicating structural richness.
• The poset 𝔼_A associated with any C*-algebra A has property K, meaning it is productively ccc and cannot freeze gaps in P(ℕ)/Fin.
• The Calkin algebra is not 2^ℵ₀-universal in models where the Proper Forcing Axiom holds, as some abelian C*-algebras of density 2^ℵ₀ fail to embed.
• The gap spectrum of the Calkin algebra’s projection lattice is more complex than that of P(ℕ)/Fin, and its structure is preserved under ccc forcing when property K holds.