Masas of the Calkin Algebra

• Masas of the Calkin Algebra are maximal abelian self-adjoint subalgebras in Q(ℓ₂) defined by their projection-generated and maximal commutative properties.
• Under the Continuum Hypothesis, every projection-generated masa is *-isomorphic to C(K) for a totally disconnected compact space K without Gδ-points and weight at most the continuum.
• ZFC constructions reveal a vast family of non-liftable masas achieved via diagonal twisting, lacking conditional expectations and classical SAW* properties.

A maximal abelian self-adjoint subalgebra (masa) of the Calkin algebra $\mathcal Q(\ell_2)$ is a commutative, self-adjoint C*-subalgebra that is maximal with respect to these properties. The study of masas in the Calkin algebra is central to understanding fundamental interactions among operator theory, set theory, and C*-algebraic structure. Recent work has led to a comprehensive classification, under the Continuum Hypothesis (CH), of masas generated by projections, and has exhibited unexpectedly rich families of otherwise elusive, highly non-classical examples even in the absence of extra set-theoretic axioms (Koszmider, 6 Dec 2025).

1. The Calkin Algebra and Masas: Preliminaries

Let $\mathcal B(\ell_2)$ denote the C*-algebra of all bounded linear operators on a separable Hilbert space $\ell_2$, and $\mathcal K(\ell_2)$ its ideal of compact operators. The Calkin algebra is the quotient

$$\mathcal Q(\ell_2) = \mathcal B(\ell_2)/\mathcal K(\ell_2),$$

with the canonical quotient map $\pi:\mathcal B(\ell_2)\to\mathcal Q(\ell_2)$. A masa in $\mathcal Q(\ell_2)$ is a commutative, self-adjoint subalgebra $M\subseteq \mathcal Q(\ell_2)$ that is maximal with respect to inclusion. When a C*-subalgebra $M\simeq C(K)$ is generated by its projections, the associated compact Hausdorff Gelfand space $K$ is totally disconnected.

2. Classification of Projection-Generated Masas Under CH

Koszmider's main result establishes that, under the Continuum Hypothesis, for any compact, totally disconnected Hausdorff space $K$ of weight $w(K)\leq 2^{\aleph_0}$ with no $G_\delta$-points, there exists a masa $M\subset\mathcal Q(\ell_2)$ generated by projections and $*$-isomorphic to $C(K)$. Conversely, every projection-generated masa in $\mathcal Q(\ell_2)$ is of this form (Koszmider, 6 Dec 2025). This gives a complete $*$-isomorphic classification for such masas.

The key constraints are:

• Weight constraint: The density character (weight) of $K$ must be at most the continuum.
• No $G_\delta$-points: No point of $K$ has a countable intersection of open neighborhoods forming a neighborhood basis.

The necessity of CH arises because, in the absence of additional set-theoretic assumptions, certain C*-algebras $C(K)$ fail to embed as projection-generated masas. Existence and sharpness of such masas thus sharply depend on set-theoretic context. The condition on the absence of $G_\delta$-points is also indispensable, even in ZFC, for such $C(K)$ to appear as masas of $\mathcal Q(\ell_2)$.

3. ZFC Construction of Non-Liftable Masas

Without extra set-theoretic assumptions, one can still construct an enormous family (cardinality $2^{2^{\aleph_0}}$) of pairwise non-$*$-isomorphic projection-generated masas in $\mathcal Q(\ell_2)$. These are obtained using “diagonal twisting” techniques in a block-decomposed subalgebra $A_0\subset\mathcal B(\ell_2)$ (viewed as operators with $2\times2$-block structure indexed by countable direct sum decomposition).

For each wide family $\mathcal X$ of infinite subsets of $\mathbb N$ and a coherent choice of scalars $\alpha_X\in (71/72,1]$, one builds an “almost masa” $M_{\mathcal X,\alpha}$ in $A_0$ via intersections defined by diagonalization and compact perturbations. The wide and coherence properties ensure maximality modulo compacts and pairwise non-isomorphism (Koszmider, 6 Dec 2025). These masses have the following properties:

• Non-liftable: They cannot be realized as quotient images of commutative subalgebras of $\mathcal B(\ell_2)$.
• Not SAW*: Their Gelfand spaces are not $F$-spaces, violating a key property necessary for liftability.
• No conditional expectations: By failure of the Banach space Grothendieck property in their duals, no conditional expectation $E:\mathcal Q(\ell_2)\to\pi[M_{\mathcal X,\alpha}]$ exists.

4. Comparison to Classical Liftable Types

Prior to these developments, only three $*$-isomorphism types of projection-generated masas were established, all liftable and arising directly from commutative masas in $\mathcal B(\ell_2)$:

1. $\ell_\infty/c_0$ (atomic masa)
2. $L_\infty[0,1]$ (diffuse masa)
3. $\ell_\infty/c_0\oplus L_\infty$ (direct sum)

All three are hyper-Stonean (hence SAW*) and admit conditional expectations. Newly constructed examples, by contrast, can be non-SAW*, lack conditional expectations, and have Gelfand spaces failing to be $F$-spaces. This demonstrates that the structure of masas in the Calkin algebra is fundamentally richer than previously thought, with possibilities extending far beyond classical paradigms.

Masa Type	Gelfand Space Condition	Liftable	SAW*	Conditional Expectation
$\ell_\infty/c_0$	Stone–Čech compactification	Yes	Yes	Yes
$L_\infty[0,1]$	Standard measure space	Yes	Yes	Yes
New (ZFC-construction)	Non-$F$-space, not $G_\delta$	No	No	No

5. Methods and Set-Theoretic Dependence

The CH-enabled construction proceeds by transfinite induction of length $\omega_1$, enumerating the clopen algebra of $K$ and all self-adjoint elements of $\mathcal Q(\ell_2)$, and at each step, augmenting the Boolean algebra of projections so as to realize each clopen subset, guaranteeing maximality and control over the generated masa's algebraic and topological invariants. CH controls the process by bounding the weight and cardinals involved.

The ZFC diagonal twisting construction relies on block-diagonal operator decompositions, wide families of subsets, and independent choices across blocks. This prevents the projection-generated masa from being the quotient of a commutative subalgebra in $\mathcal B(\ell_2)$.

6. Significance and Implications

These results indicate that the projection-generated masa landscape in the Calkin algebra exhibits a spectrum of phenomena influenced by set-theoretic axioms, topological space properties, and operator-algebraic structure. Under CH, the classification is complete for totally disconnected Gelfand spaces of weight at most continuum lacking $G_\delta$-points; without additional axioms, one already has maximal multiplicity and diversity among non-liftable, non-SAW*, expectationless masas (Koszmider, 6 Dec 2025).

A plausible implication is that operator-algebraic properties such as the SAW* property, liftability, and existence of conditional expectations are deeply intertwined with the set-theoretic and combinatorial features of the underlying topological spaces.

7. Research Directions and Further Context

Research continues to probe the connections among set theory, topology, and operator algebras in the structure of the Calkin algebra. The approach yielding $2^{2^{\aleph_0}}$ non-liftable, non-expectation-admitting masas in ZFC improves upon earlier work requiring supplementary set-theoretic assumptions to produce even a single such example. The findings highlight the consistency strengths necessary for embedding general $C(K)$ as projection-generated masas, the epistemic independence phenomena in operational context, and the breadth of commutative C*-algebra representations in the non-liftable case (Koszmider, 6 Dec 2025). Further investigations may explore extensions of these phenomena to other corona algebras and classifiable C*-algebra contexts.