Pure infiniteness and primary factorisation

Technical Summary: Pure Infiniteness and Primary Factorisation

Problem Statement
The paper investigates the Primary Factorisation Property (PFP) in the context of Banach space theory and its interplay with ring-theoretic properties of the algebra of bounded linear operators, $B(E)$. A Banach space $E$ possesses the PFP if, for every operator $T \in B(E)$, the identity operator $Id_E$ factors through $T$ or through $Id_E - T$. This property is a strong operator-theoretic form of primariness, often used as a preliminary step in proving that a space is primary.

The central questions addressed are:

1. Can an indecomposable Banach space (one that cannot be written as the direct sum of two infinite-dimensional closed subspaces) possess the PFP?
2. What are the ring-theoretic consequences of the PFP, specifically regarding Dedekind-infiniteness, proper infiniteness, and pure infiniteness of $B(E)$ and its quotient $A_E := B(E)/M_E$, where $M_E$ is the set of operators through which the identity does not factor?
3. How do these properties behave under ultrapowers? Specifically, if $E$ has the PFP (or the uniform version, UPFP), is the ultrapower of the quotient $A_E$ purely infinite?

Methodology
The authors employ a combination of operator theory, Fredholm theory, and Banach algebra theory:

• Ideal Structure Analysis: They characterize $M_E$ as the unique maximal ideal of $B(E)$ when $E$ has the PFP. This allows the study of the quotient algebra $A_E$.
• Fredholm Theory: The authors utilize the continuity of the Fredholm index and the properties of Fredholm operators to derive contradictions regarding the structure of indecomposable spaces with PFP.
• Factorisation Dichotomy: They analyze the quantitative constants involved in factorisation. For the uniform PFP (UPFP), they examine whether the factorisation constants are uniformly bounded.
• Ultrapower Techniques: The paper applies results by Daws and Horváth regarding the pure infiniteness of ultrapowers, which depends on the uniform boundedness of the "pure infiniteness constant" $C_{pi}(a)$ on the unit sphere.
• Specific Counterexamples and Constructions: The analysis relies heavily on specific Banach spaces, including Read's space ($E_R$), the James space ($J$), classical sequence spaces ($c_0, \ell_p$), and $C[0, \omega_1]$.

Key Contributions and Results

1. Non-existence of Indecomposable PFP Spaces (Theorem A)
The paper proves that no real or complex indecomposable Banach space possesses the PFP.

• Method: The proof assumes the existence of such a space and derives a contradiction. If $E$ is indecomposable and has PFP, the quotient $B(E)/M_E$ must be a division algebra (by Gelfand-Mazur). This implies that every Fredholm operator has index zero. However, the PFP allows the construction of an operator $L$ with trivial kernel but arbitrarily large cokernel (via Lemma 3.4), leading to a non-zero Fredholm index. This contradiction establishes that indecomposability and PFP are mutually exclusive.
• Corollary: This result rules out the PFP for hereditarily indecomposable spaces, Tarbard's Bourgain–Delbaen space, spaces $C(K)$ for connected Koszmider compacta, and the Gowers–Maurey prime indecomposable space.

2. Ring-Theoretic Infiniteness (Theorem B)
The authors establish the infiniteness properties forced by the PFP:

• Dedekind-infiniteness: If $E$ is infinite-dimensional and has PFP, $B(E)$ is Dedekind-infinite. This is shown by factoring the identity through the complement of a rank-one projection, creating an idempotent equivalent to the identity but not equal to it.
• Proper Infiniteness: The PFP does not imply that $B(E)$ is properly infinite. The space $C[0, \omega_1]$ has the PFP, but $B(C[0, \omega_1])$ is not properly infinite because its quotient by $M_E$ is isomorphic to $\mathbb{C}$, and proper infiniteness is preserved by unital homomorphisms.
• Pure Infiniteness of the Quotient: For a complex space $E$ with PFP, the quotient $A_E = B(E)/M_E$ is purely infinite if and only if $A_E \not\cong \mathbb{C}$. If $A_E \not\cong \mathbb{C}$, every non-zero element in the quotient allows the identity to factor through it.

3. Ultrapowers and the Non-Scalar Problem
The paper investigates whether the ultrapower $(A_E)_U$ is purely infinite.

• Scalar Boundary Cases: The authors show that if $A_E \cong \mathbb{C}$ (as in the cases of $C[0, \omega_1]$ and the James space $J$), the ultrapower is not purely infinite. These serve as necessary boundary cases, demonstrating that the hypothesis $A_E \not\cong \mathbb{C}$ is essential for pure infiniteness of ultrapowers.
• Classical Sequence Spaces: For $E = c_0(\Gamma)$ or $\ell_p(\Gamma)$, the quotient $A_E$ is non-scalar and has a pure infiniteness constant of 1. Consequently, their ultrapowers are purely infinite.
• Read's Space ($E_R$): The paper proves that Read's space does not have the Uniform PFP (UPFP). The proof utilizes the structure of Read's codimension-one ideal $J_R$, which becomes a non-zero square-zero ideal modulo weakly compact operators. This square-zero structure is incompatible with the uniform estimates required for UPFP.
• Open Questions: The paper formulates Question 6.3 and Question 6.4, asking whether there exists a non-scalar complex Banach space with PFP (or UPFP) such that the ultrapower of its quotient is not purely infinite. The authors note that while Read's space fails UPFP, it remains an open question whether it has the non-uniform PFP. If it did, it would provide a space with PFP but without UPFP, though the resulting quotient would still be scalar.

Significance and Claims
The paper claims to resolve the relationship between indecomposability and the PFP, definitively showing they cannot coexist. This contrasts with the existence of indecomposable prime spaces (Gowers–Maurey), clarifying that primariness does not necessarily imply the stronger factorisation dichotomy of PFP.

The work isolates the quantitative gap relevant to ultrapowers, distinguishing between the qualitative factorisation dichotomy (PFP) and the uniform boundedness of factorisation constants (UPFP). The authors demonstrate that while classical sequence spaces satisfy the strongest conditions (non-scalar quotient, UPFP, purely infinite ultrapowers), other spaces like the James space and $C[0, \omega_1]$ satisfy PFP but yield scalar quotients, failing the ultrapower test.

The paper concludes by identifying Read's space as a critical test case. While it is proven not to have UPFP, the question of whether it possesses the non-uniform PFP remains open. A positive answer would provide a Banach space with PFP but without UPFP, but the authors caution that this would not resolve the non-scalar ultrapower question, as the quotient would remain scalar. The paper thus delineates the precise boundaries of current knowledge regarding the interplay between factorisation properties, ideal structures, and the infiniteness of operator algebras and their ultrapowers.