Pure infiniteness and primary factorisation

Abstract: We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}<em>E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E</em>{\operatorname{R}}$ does not have the uniform PFP.

• The paper establishes that no real or complex indecomposable Banach space admits the primary factorisation property.
• It demonstrates that Calkin-type quotients exhibit pure infiniteness when non-scalar, while proper infiniteness is not guaranteed.
• It introduces uniform factorisation conditions via ultrapower analysis, raising open questions on non-uniform dichotomies in these algebras.

Pure Infiniteness and Primary Factorisation in Banach Spaces

Overview and Main Results

This paper investigates the interplay between the primary factorisation property (PFP) of Banach spaces and ring-theoretic notions of infiniteness—primarily Dedekind, proper, and pure infiniteness—for operator algebras and certain Calkin-type quotients. The central claims establish strong obstructions on the coexistence of indecomposability and the PFP and clarify the limits under which PFP induces rich infiniteness phenomena in operator algebras. Notably, no real or complex indecomposable Banach space admits the PFP, and for Banach spaces with PFP, associated Calkin-type quotients manifest pure infiniteness precisely when the quotient is non-scalar.

The Primary Factorisation Property and Its Consequences

Definitions and Setting

Let $E$ be a (real or complex) Banach space, and $B(E)$ its algebra of bounded linear operators. The PFP specifies that for every $T \in B(E)$, the identity operator factors through $T$ or through $Id_E - T$. That is, there exist $U, V \in B(E)$ such that either $U T V = Id_E$ or $U (Id_E - T) V = Id_E$.

A fundamental subset associated to this dichotomy is $$M = \{ T \in B(E) : Id_E \text{ does not factor through } T \}$$, which is always a closed two-sided ideal. Under PFP, $M$ becomes the unique maximal ideal of $B(E)$0; thus, the quotient $B(E)$1 is a simple unital Banach algebra.

Obstruction to Indecomposability

A pivotal result is that PFP and indecomposability are mutually exclusive. The proof leverages Fredholm index continuity and maximal ideal structure, ultimately showing that any hypothetical indecomposable Banach space with the PFP would force all Fredholm operators to have index zero, contradicting the existence of injective Fredholm operators with positive codimension in infinite-dimensional settings. This resolves open questions about PFP for a host of classical non-decomposable spaces, including hereditarily indecomposable Banach spaces, Tarbard's Bourgain–Delbaen space $B(E)$2, certain $B(E)$3 spaces with few operators, and the Gowers–Maurey prime space.

Dedekind and Proper Infiniteness

For infinite-dimensional Banach spaces with PFP, the algebra $B(E)$4 is always Dedekind-infinite: there exists an idempotent Murray–von Neumann equivalent to the identity but not equal to it. However, proper infiniteness fails to be guaranteed in general. For example, $B(E)$5 is not properly infinite despite $B(E)$6 possessing the PFP. This distinction traces to subtleties in the ideal structure and the inability of certain quotient algebras to inherit sufficient orthogonality and equivalence among idempotents.

Pure Infiniteness and Calkin-Type Quotients

Pure Infiniteness for $B(E)$7

In the complex case, $B(E)$8 is characterized as purely infinite if and only if $B(E)$9 is not isomorphic to the scalars. For every nonzero element in $T \in B(E)$0, the identity factors through it; this matches the Daws–Horváth definition of pure infiniteness in Banach algebras, which implies simplicity and precludes nontrivial finite-dimensional quotients. In particular, if $T \in B(E)$1 contains two complemented copies of itself inside a complemented subspace, $T \in B(E)$2 is non-scalar and purely infinite.

Ultrapower Phenomena and Quantitative Factorisation

The behavior of ultrapowers of Calkin-type quotients captures the quantitative strictness of PFP and the uniform PFP (UPFP). The ultrapower $T \in B(E)$3 is purely infinite if and only if the infimum of products of norms for factorizations of the identity through nonzero elements is uniformly bounded on the unit sphere. The UPFP ensures such uniformity up to a dichotomy: for each $T \in B(E)$4 on the unit sphere, $T \in B(E)$5 or $T \in B(E)$6 admits a bounded factorization, but not necessarily both.

This delineates the open problem: Does there exist a Banach space $T \in B(E)$7 with non-scalar $T \in B(E)$8 and the PFP (or UPFP) such that some ultrapower $T \in B(E)$9 fails to be purely infinite? The scalar boundary cases are shown to be insufficient, emphasizing a gap in understanding "one-sided" versus "two-sided" uniform factorisation control in quotient norms.

Detailed Analysis of Classes and Examples

Classical Sequence Spaces

For $T$0 or $T$1, $T$2, with arbitrary cardinality, the maximal ideal is the ideal $T$3 of $T$4-compact operators, and $T$5 is always non-scalar and admits constant-one pure infiniteness for the quotient. For these spaces, every countably incomplete ultrapower is purely infinite, and both PFP and UPFP are satisfied robustly.

James Space and Read's Space

James' quasi-reflexive space $T$6 manifests the UPFP, but with $T$7 scalar, hence ultrapowers are not purely infinite. In contrast, Read's space does not have the UPFP due to the presence of a large square-zero ideal in the Calkin quotient algebra—shown via a quantitative obstruction. The precise status of PFP (without uniformity) for Read's space is tied to an intricate complemented-copy property for weakly compact perturbations of the identity, remaining an open question. Should Read's space be shown to have the PFP without the UPFP, it would provide an explicit instance of non-uniform dichotomous factorisation.

Boundary Cases and Open Problems

The work provides boundary examples with scalar quotients—such as $T$8 and the James space—that demarcate the limits of current methods. For non-scalar UPFP and the ultrapower problem, robust positive test cases include direct sums and spaces like $T$9, which are purely infinite in all countably incomplete ultrapowers.

The sharp open problems concern the existence and analysis of Banach spaces satisfying PFP or UPFP for which the associated Calkin-type quotient is non-scalar, but the ultrapower fails pure infiniteness, corresponding to a failure of "one-sided" uniformity in the quotient norm factorisation constant.

Theoretical and Practical Implications

The results have profound implications for both the structure theory of operator algebras on Banach spaces and Banach-algebra $Id_E - T$0-theory. The identification of which spaces preclude or guarantee pure infiniteness in their operator algebra quotients guides new lines of investigation into the rigidity and diversity of Banach space geometry as reflected in their operator algebraic properties. The sharp dichotomy between scalar and non-scalar quotients and the issues arising in ultrapowers suggest rich territory for future research in both functional analysis and set-theoretic Banach space theory.

On the operator algebra side, the construction and identification of purely infinite simple Banach algebras in non-$Id_E - T$1-contexts is of independent interest, and the techniques developed here may have extensions to other "large" Banach algebra settings.

Conclusion

This work achieves a comprehensive analysis of primary factorisation phenomena, ring-theoretic infiniteness, and their quantitative and qualitative interplay across Banach spaces and associated operator algebras. The exclusion theorem for indecomposable spaces, identification of limits of proper and pure infiniteness, and deep connections to ultrapower behavior place the work as a cornerstone in the understanding of factorisation dichotomies and infiniteness in operator algebras beyond the $Id_E - T$2-algebraic setting. The precise resolution of the open problems regarding non-scalar ultrapower pure infiniteness and the non-uniform dichotomy for spaces such as Read's remain compelling targets for further investigation.