Preservation of primariness under $\ell_1$-, $c_0$-, and $\ell_\infty$-sums of Banach spaces

Abstract: We prove transfer principles for the uniform primary factorisation property (UPFP) from a Banach space $X$ to the vector-valued sequence spaces $\ell_1(X)$, $c_0(X)$ and $\ell_\infty(X)$. The hypotheses are either finite-cotype assumptions on $X$ or $X<sup>*$, or natural self-similarity assumptions on $X$. Consequently, under these conditions, the resulting vector-valued sequence spaces are primary. As applications, we recover the primariness of $\ell_\infty(L_p)$ for $1\leq p&lt;\infty$ without using Bourgain's localisation method, and obtain the primariness of $c_0(L_1)$. We also show that $\ell_1(Γ,L_1[0,1])$ has the UPFP for every set $Γ$, and consequently that $C[0,1]<sup>*$ has the UPFP and is primary.

• The paper establishes transfer principles showing that the uniform primary factorisation property passes from a Banach space X to ℓ₁(X), c₀(X), and ℓ∞(X) under finite-cotype or self-similarity assumptions.
• Its diagonal-reduction method approximates operators by coordinatewise diagonal operators, enabling a factorisation dichotomy and recovering primariness of ℓ∞(Lₚ) for 1 ≤ p < ∞ while proving it for c₀(L₁).
• The results show that ℓ₁(Γ,L₁[0,1]) has the UPFP for every index set Γ, and consequently C[0,1]* is primary in ZFC, while transfer to intermediate ℓₚ-sums remains open.

This paper establishes transfer principles showing that the uniform primary factorisation property (UPFP) passes from a Banach space $X$ to the vector-valued sequence spaces $\ell_1(X)$, $c_0(X)$, and $\ell_\infty(X)$, under either finite-cotype hypotheses on $X$ or $X^*$, or self-similarity assumptions on $X$ (2606.26417). As consequences, the author recovers the primariness of $\ell_\infty(L_p)$ for $1\leq p<\infty$ by elementary means, obtains the primariness of $c_0(L_1)$, proves that $\ell_1(X)$0 has the UPFP for every set $\ell_1(X)$1, and deduces that $\ell_1(X)$2 has the UPFP and is primary.

Background and main results

A Banach space is primary if, for every projection $\ell_1(X)$3 on it, at least one of $\ell_1(X)$4 or $\ell_1(X)$5 is isomorphic to the whole space. The paper works with the stronger quantitative notion: $\ell_1(X)$6 has the $\ell_1(X)$7-primary factorisation property ($\ell_1(X)$8-PFP) if for every operator $\ell_1(X)$9, the identity factors through either $c_0(X)$0 or $c_0(X)$1 with constant $c_0(X)$2; the UPFP holds when some such uniform constant exists. Primariness follows from the UPFP via Pełczyński's decomposition method.

The two principal transfer theorems are as follows. If $c_0(X)$3 has the UPFP and either (i) $c_0(X)$4 for some $c_0(X)$5 or $c_0(X)$6, or (ii) $c_0(X)$7 has finite cotype, then $c_0(X)$8 has the UPFP and hence is primary. Dually, if $c_0(X)$9 has the UPFP and either (i) $\ell_\infty(X)$0 for some $\ell_\infty(X)$1, or (ii) $\ell_\infty(X)$2 has finite cotype, then both $\ell_\infty(X)$3 and $\ell_\infty(X)$4 have the UPFP. Note the asymmetry: dual finite cotype serves the $\ell_\infty(X)$5-case while finite cotype of $\ell_\infty(X)$6 itself serves the sup-norm cases; the self-similarity hypothesis for $\ell_\infty(X)$7 requires internal structure far from $\ell_\infty(X)$8 (namely $\ell_\infty(X)$9, $X$0, or $X$1), whereas for $X$2 and $X$3 any finite-$X$4 self-similarity suffices.

Immediate corollaries include: $X$5 has the UPFP for all $X$6, recovering Wark's result for $X$7 and Müller's extension to $X$8 without Bourgain's localisation method; $X$9 has the UPFP, a case the author could not find explicitly in the literature; and $X^*$0 has the UPFP for every ordinal $X^*$1, completing the $X^*$2-case of a classification begun elsewhere by combining the Alspach–Benyamini theorem with the transfer principle.

The diagonal-reduction scheme

The proof strategy reduces an arbitrary operator $X^*$3, where $X^*$4, up to arbitrarily small perturbation, to its diagonal part after passing to suitable coordinate subsets. This proceeds in two steps: an upper-triangular reduction followed by a lower-triangular reduction, each obtained from a one-step "forward" or "backward" reduction lemma via recursive selection. The one-step lemmas are probabilistic averaging arguments over random signs: if a compression of $X^*$5 between disjointly supported blocks were uniformly bounded below, averaging $X^*$6 over sign choices would contradict a dimension-dependent estimate derived from cotype constants or from the incompatibility of external and internal sequence-space norms.

The two types of hypotheses enter differently. In the finite-cotype case, the relevant geometry of $X^*$7 (or $X^*$8) is incompatible with the scalar sequence space: for $X^*$9, finite cotype of $X$0 forces every operator $X$1 to be compact (via Pełczyński's theorem on weakly compact operators from $X$2, Gantmacher's theorem, and the Schur property), yielding norm approximation of compressions by their triangular parts. For $X$3 and $X$4, finite cotype of $X$5 gives entrywise estimates exploiting the identity $X$6. In the self-similar case, the reductions contrast incompatible geometries: for $X$7-sums, internal $X$8 ($X$9) or $\ell_\infty(L_p)$0 structure against the external $\ell_\infty(L_p)$1 norm; for $\ell_\infty(L_p)$2- and $\ell_\infty(L_p)$3-sums, internal $\ell_\infty(L_p)$4 ($\ell_\infty(L_p)$5) structure against the external supremum norm.

Once diagonal form is reached, two elementary observations complete the argument. First, a diagonal dichotomy: if $\ell_\infty(L_p)$6 has the $\ell_\infty(L_p)$7-PFP, then for every diagonal operator $\ell_\infty(L_p)$8 on $\ell_\infty(L_p)$9, either $1\leq p<\infty$0 or $1\leq p<\infty$1 factors the identity on $1\leq p<\infty$2 with constant $1\leq p<\infty$3, by passing to an infinite subset on which the same alternative holds coordinatewise. Second, factorisation is stable under perturbation: if $1\leq p<\infty$4 factors the identity with constant $1\leq p<\infty$5 and $1\leq p<\infty$6, then $1\leq p<\infty$7 factors the identity with constant $1\leq p<\infty$8, via a Neumann-series argument. Combining these yields the UPFP for $1\leq p<\infty$9 with constant $c_0(L_1)$0 whenever $c_0(L_1)$1 has the $c_0(L_1)$2-PFP.

The author notes a technical caveat specific to $c_0(L_1)$3: the operator matrix does not determine the operator in general, so operators are required to act according to their matrices, and only the weaker entrywise form of the upper-triangular reduction is recorded there rather than norm approximation.

Uncountable $c_0(L_1)$4-sums and $c_0(L_1)$5

For uncountable index sets, the paper develops a variant replacing the two-sided triangular reduction by a one-sided finite-interference estimate combined with Hajnal's free set theorem. The summand treated is $c_0(L_1)$6, which is not covered by the finite-dual-cotype hypothesis but possesses sufficient measure-theoretic structure.

Three ingredients combine. First, Capon's localisation argument, based on Kalton's representation of operators on $c_0(L_1)$7 by families of measures $c_0(L_1)$8, shows that $c_0(L_1)$9 itself has the UPFP: for any $\ell_1(X)$00, either $\ell_1(X)$01 or $\ell_1(X)$02 on a set of positive measure, and localising on such a set makes the corresponding compression invertible with inverse norm at most $\ell_1(X)$03. Second, a finite-interference lemma shows that every operator $\ell_1(X)$04 admits a finite exceptional set $\ell_1(X)$05 and a positive-measure set $\ell_1(X)$06 with $\ell_1(X)$07; this follows from absolute continuity of the measures $\ell_1(X)$08 obtained via Kalton's theorem and a Radon–Nikodym argument, plus separability of $\ell_1(X)$09 to reduce arbitrary $\ell_1(X)$10 to a countable support. Third, Hajnal's finite free set theorem removes the finitely many exceptional coordinates simultaneously on a subset $\ell_1(X)$11 of full cardinality.

Together these give a diagonal reduction for every operator on $\ell_1(X)$12: there exist contractions $\ell_1(X)$13 with $\ell_1(X)$14, where $\ell_1(X)$15, and a diagonal $\ell_1(X)$16 with $\ell_1(X)$17. Since each $\ell_1(X)$18 is isometric to $\ell_1(X)$19, the diagonal dichotomy applies verbatim, yielding the UPFP for $\ell_1(X)$20 for every set $\ell_1(X)$21. This generalises a previous result of Acuaviva and Kania that assumed the negation of CH — the present argument is ZFC-unconditional. Finally, since $\ell_1(X)$22, the dual of $\ell_1(X)$23 has the UPFP and is primary.

Limitations and open questions

The methods are intrinsically tied to the extreme geometries of $\ell_1(X)$24 and the supremum norm, where coordinate restrictions satisfy identities such as $\ell_1(X)$25; analogous one-sided reductions for intermediate $\ell_1(X)$26-sums do not control the restricted operator norm. Consequently, the transfer principles do not cover $\ell_1(X)$27 for $\ell_1(X)$28, except through ad hoc additional structure as in the earlier work on $\ell_1(X)$29. The paper poses three open questions: whether natural broad conditions exist under which the UPFP of $\ell_1(X)$30 passes to $\ell_1(X)$31 for $\ell_1(X)$32; whether some $\ell_1(X)$33 with the PFP (or UPFP) has $\ell_1(X)$34 or $\ell_1(X)$35 failing the corresponding property; and whether some primary $\ell_1(X)$36 has a non-primary vector-valued sequence space over it. The author remarks that a counterexample to the latter questions would likely be pathological, while a positive answer appears beyond current methods. It is also conceded that a quantitative version of Enflo's primariness proof for $\ell_1(X)$37 was not verified, though it is plausible.

Conclusion

The paper provides an elementary, unified route to several primariness results previously requiring substantially heavier machinery, notably removing Bourgain's localisation method from the proofs concerning $\ell_1(X)$38 and settling the primariness of $\ell_1(X)$39 unconditionally in ZFC. Its core contribution is the observation that the UPFP transfers across $\ell_1(X)$40-, $\ell_1(X)$41-, and $\ell_1(X)$42-sums under mild geometric or self-similarity assumptions, via perturbative diagonal reduction combined with a coordinatewise factorisation dichotomy. The scope of the technique outside the extreme sequence spaces remains the principal open issue.