We introduce and investigate a quantitative version of Steinhaus' property (S) for Banach spaces, called the uniform property (S). A Banach space X is said to have uniform (S) if for every pair of distinct unit vectors x,y∈ X and every a>0, the difference of the perturbed norms sup_z≤ a|x+z-y+z| is bounded below by a positive function of a and x-y. We compute this modulus exactly for the spaces L_1(μ) with atomless measure μ, U_L_1(μ)(d;a)=(4a2+d∧ 1)d, The class of spaces with uniform (S) is stable under ultrapowers, Bochner-L_1 constructions, and contains all Gurarispaces as well as Banach lattices of almost universal disposition. In particular, every Banach space embeds isometrically into a non-strictly convex Banach space of the same density having uniform (S). We further exhibit an explicit equivalent renorming of ℓ_1(Γ), x_S=(x_1^2+x_2^2)^1/2, which endows ℓ_1(Γ) and all its ultrapowers with uniform (S). These results settle, in ZFC, several open questions about the quantitative geometry of property (S) posed by Kochanek and the second-named author.

In a companion paper (Studia Math., 2023), we proved for every λ∈(1,2] the existence of a (λ^+)-injective renorming of ℓ_∞ that is not λ-injective, thereby establishing a forgotten theorem of Pełczyński in that range. The complementary range λ∈(2,∞) was left open. In the present paper, we resolve this remaining case: for every λ>2 we construct a Banach space that is (λ^+)-injective but not λ-injective, completing Pełczyński's theorem for all λ>1. The construction uses a single device: the `zero-sum' subspace Σ_N(Y)⊂ Z_∞^N, which multiplies the relative projection constant by μ_N=2-2/N while preserving non-attainment. Iterating this operation reduces the problem to the range (1,2] already covered by the companion paper. Since the ambient spaces arising in the iteration are finite ℓ_∞-sums of ℓ_∞, the resulting examples may be realised as subspaces of ℓ_∞. We also prove that if two Banach spaces are each isometrically isomorphic to their own square and each is isometric to a 1-complemented subspace of the other, then their Banach–Mazur distance is at most 9+6√(3). Consequently, we obtain the estimate dist(L_∞[0,1],ℓ_∞)≤ 9+6√(3), thereby improving a recent result of Korpalski and Plebanek.

We study finitely generated maximal right ideals of the Banach algebra ℬ(E) of bounded operators on a complex Banach space E. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when E is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals Lift(T)={TU:U∈ℬ(E,E^n)}, where T∈ℬ(E^n,E) for some n∈ℕ, we identify the exact operator-theoretic obstruction. The ideal Lift(T) contains the finite-rank operators precisely when T is surjective, and it equals ℬ(E) precisely when T is right invertible. If T is surjective but not right invertible, then Lift(T) is maximal exactly when the row operator [T S] is right invertible for every S∈ℬ(E)∖Lift(T). We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of ℓ_1, KB-spaces, Lebesgue spaces L_p(μ) for 1⩽ p<∞, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, ℓ_1(Γ)-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces ℓ_1(Γ)⊕ H with H a separable Hilbert space.

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family of Banach algebras and a Banach sequence lattice on , the -sum carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that , we prove that this -sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate We show that the factor is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity.As applications, we recover the classical formula for arbitrary (possibly uncountable) index sets, extend it to weighted -spaces, and characterize amenability for Orlicz sequence algebra sums. We also record how these unconditional criteria give obstructions within the conditional framework of James-type -sums.Finally, we investigate weak amenability of -sums. We prove that weak amenability passes to summands, that -sums of commutative weakly amenable algebras are weakly amenable, and-contrasting sharply with the Johnson amenability picture-that for , the -sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the -type regime (), we establish two-sided estimates for weak amenability constants with constants depending only on .

We examine the analyticity of the class of separable Banach spaces possessing the π-property, defined in terms of convergence along a filter. Our results establish that this class is Σ^1_3 whenever the underlying filter is analytic (as a subset of the Cantor set Δ). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is Σ^1_2 with respect to any admissible Polish topology on the family of closed subspaces of C(Δ).

Let K be a compact Hausdorff space and let E be an infinite-dimensional real Banach space. We prove that there is no continuous bijection h C_p(K)→ E_w whose inverse is continuous at h(0). Consequently, C_p(K) and C_w(L) are not homeomorphic for any infinite compact Hausdorff spaces K and L. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning C_p([0,1]) and weak Banach spaces.

We develop a unified framework for locating natural properties of algebraic and analytic structures within the Borel hierarchy. Objects are presented as quotients of a universal generator and definability is read directly from the quotient data. For separable Banach-type structures (Banach algebras, C^*-algebras, Banach lattices, TROs) the kernel space is Polish under the Wijsman topology, and the quotient-norm functional K↦x+K is continuous, yielding a uniform definability scheme whose Borel ranks are bounded by quantifier alternation depth. For countable algebraic structures (groups, rings, lattices) we work on compact Polish spaces of congruences where atomic predicates are clopen. We obtain explicit Borel upper bounds: in the unital C^*-algebra coding based on C^*_max(F_∞), stable finiteness is closed, nuclearity is Borel, simplicity is G_δ, AF-ness lies in Π^0_3, nuclear dimension ≤ n lies in Π^0_3, and for fixed exact D, D-absorption is analytic. For countable groups, soficity is G_δ; for abelian groups, slenderness is Π^0_3. We give an internal Borel coding of the K_0-assignment in the quotient/Wijsman framework; for each fixed coordinate the corresponding section is F_σ, and suspension together with Bott periodicity yields Borel codings of all higher K-groups. We also show that several bounds are optimal (Σ^0_2- and Π^0_2-complete). To calibrate the method's reach, we exhibit a Π^1_1-complete property (separable dual in the commutative C^*-setting), provably outside the Borel hierarchy.

Let A be a unital C^* -algebra. An A-multiplier cover is a C^* -algebra E together with a faithful non-degenerate * -homomorphism from A to M(E). We preorder such covers by A-preserving unital completely positive maps between their multiplier algebras. We prove that Hamana’s injective envelope I(A) is a greatest cover in this preorder and that the maximal rigid covers are precisely those whose multiplier algebra is canonically * -isomorphic to I(A). Consequently, a maximal rigid cover is greatest, rather than merely maximal among rigid covers. For A=C(X) , we further classify these covers: after the canonical identification with C(G(X)), where G(X) is the Gleason cover, their underlying ideals are exactly the algebras C_0(U) for dense open C^* -embedded subsets U of G(X). Dense cozero subsets provide an important special case.

We introduce a norm-controlled notion of semiprojectivity for Banach lattices, requiring liftability of contractive lattice homomorphisms through inductive limits of closed ideals with arbitrarily small loss of norm control. Our main result establishes that, for a compact metric space X, the Banach lattice C(X) is semiprojective if and only if X is an absolute neighbourhood retract. Notably, this characterisation is strictly more permissive than its C^*-algebraic counterpart: by a theorem of Sand Thiel, C(X) is semiprojective in the category of C^*-algebras and *-homomorphisms if and only if X is an ANR of dimension at most one. The dimensional obstruction disappears in the Banach-lattice setting because lattice homomorphisms between C(K)-spaces are automatically weighted composition operators, and therefore no commutation relations need to be lifted. We also show that uncountable ℓ_1-sums of 1^+-projective Banach lattices with topological order units are semiprojective but need not be 1^+-projective, establishing that the two notions are genuinely distinct. On the negative side, we prove that ℓ_p and L_p([0,1]) for p ∈ (1,∞) as well as Orlicz spaces are not semiprojective.

We prove that no separable Banach algebra is universal for homomorphic embeddings of all separable Banach algebras, whether embeddings are merely bounded or required to be contractive. The same holds in the commutative category.The proof is short and straightforward. For each closed subspace X of c0 we build a separable commutative Banach algebra AX whose multiplication records the canonical pairing between X⁎ and X. Any injective homomorphism of AX into a Banach algebra B forces the identity operator on X⁎ to factor through B. Thus a universal separable Banach algebra would be complementably universal for the family of duals of subspaces of c0, contrary to a theorem of Johnson and Szankowski.

We study Raja's covering index Θ_X(n) for classical L_p-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we give a short elementary argument showing that Θ_H(2)=1/√(2), thus answering a question of Raja about the precise two-piece covering index of ℓ_2. For scalar-valued Lebesgue spaces L_p(μ), 1≤ p<∞, we construct an explicit block decomposition of the unit ball yielding the upper bound Θ_L_p(μ)(n)≤ n^-1/p for all n∈ℕ; in particular Θ_ℓ_p(n)≤ n^-1/p. For 1<p<∞, this combines with Raja's general lower bound for p-asymptotically uniformly smoothable spaces to give the sharp asymptotic estimate Θ_L_p(μ)(n)≍ n^-1/p. We also obtain uniform lower bounds Θ_L_p(μ;E)(n)≳ n^-1/p for Bochner spaces L_p(μ;E), with constants independent of the Banach space E; this shows that, at the level of power-type lower estimates, the covering index cannot distinguish between very different asymptotic geometries of E and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds Θ_L_p(M,τ)(n)≳ n^-1/r for non-commutative L_p(M,τ) spaces associated with semifinite von Neumann algebras, where r=min{p,2}. We do not attempt to optimise the exponent or constants in the non-commutative setting.

We revisit Bourgain's 1981 counterexample to the sequential completeness of the 'pointwise plus domination' convergence on ℓ_1 from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain-uo convergence coincides with ordinary order convergence. Motivated by Bourgain's construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence (x_n) in a vector lattice E is called Buo-Cauchy if for every strictly increasing sequence (n_k) the differences x_n_k+1-x_n_k converge to 0 in order in E. We study the resulting sequential completeness problem in several concrete Banach lattices. Our main results are twofold. On the positive side, we prove that the classical sequence lattices c_0 and ℓ_∞ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. In addition, we establish a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice Lip_b(X) of bounded Lipschitz functions on a metric space (X,d) is sequentially -complete if and only if X is uniformly discrete. On the negative side, we show that vector lattices of continuous or Lipschitz functions over metric spaces with non-isolated points fail sequential Buo-completeness. In particular, C(X), C_b(X), C_0(X) and Lip_b(X) are not sequentially Buo-complete whenever X is not uniformly discrete.

Let A be a unital C^*-algebra. We call an A-multiplier cover a pair (E,ι) consisting of a C^*-algebra E and a faithful non-degenerate *-homomorphism ι A→ M(E). Ordering such covers by A-preserving unital completely positive maps between multiplier algebras, we study those covers for which the inclusion A⊆ M(E) is rigid in Hamana's sense. We prove that Hamana's injective envelope I(A) is a maximal rigid A-multiplier cover and that, conversely, a rigid cover is maximal if and only if its multiplier algebra is canonically *-isomorphic to I(A) over A. Thus maximal rigid multiplier covers provide an order-theoretic characterisation of the injective envelope. In the commutative case A=C(X), this recovers the familiar realisation I(C(X))≅ C(G(X))≅ M(C_0(U)) for a dense cozero set U in the Gleason cover G(X), in a form inspired by Błaszczyk's concise construction of the Gleason cover.

Let X be an N-dimensional real normed space, let 1⩽ k⩽ N, and set V=Λ^kX and m= Nk. We characterise the probability measures η on the unit sphere of V for which tr(Λ^kA)=m∫ w^♯((Λ^kA)w)dη(w) holds for every A∈End(X): this is equivalent to m∫ w⊗ w^♯ dη(w)=Id_V. The cone probability measure always satisfies this condition, giving a canonical higher-trace formula for every norm. Normalised Euclidean hypersurface measure also does so under a scalar-commutant symmetry hypothesis, including spaces with a 1-symmetric basis. We further obtain atomic and polyhedral formulae and show that, within a natural power-weighted family, cone measure is the unique universally isotropic member; for hypersurface measure the first-order obstruction is precisely the degree-2 spherical harmonic component of the support function.

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterise in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

Let $\mathcal C$ be a concrete category. We prove that if $\mathcal{C}$ admits a universally free object $\mathsf F$, then there is a projectively universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such that for any $B\in \mathcal{C}$ and $\tau\in {\rm Mor}(B)$ there exists an epimorphism $\pi\in {\rm Mor}(\mathsf F, B)$ such that $\pi \tau = u \pi$. This builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math. Soc. 145 (2017)) who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc.

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ epsilon)-injective for every epsilon > 0, yet is not 2-injective, and remarked in a footnote that Pelczynski had proved for every lambda > 1 the existence of a (lambda + epsilon)-injective space (epsilon > 0) that is not lambda-injective. Unfortunately, no trace of the proof of Pelczynski's result has been preserved. In the present paper, we establish that result for lambda is an element of (1, 2] by constructing an appropriate renorming of l(infinity). This contrasts (at least for real scalars) with the case lambda = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.

We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by the two last-named authors (Kania and Swaczyna, 2021 [9]) at the expense of reducing the generality from projective to analytic. In particular, we obtain a ZFC solution to Kadets' problem of continuity of coordinate functionals associated with bases with respect to the filter of statistical convergence. Even though the automatic continuity of coordinate functionals beyond the projective class remains a mystery, we prove that a basis with respect to an arbitrary filter that has continuous coordinate functionals is also a basis with respect to a filter that is analytic.

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence (fn)n=1∞$(f_{n})_{n=1}^\infty$ so that ⟨fn,xn⟩→0$\langle f_{n},x_{n}\rangle \rightarrow 0$ as n→∞$n\rightarrow \infty$ for every weakly null sequence (xn)n=1∞$(x_{n})_{n=1}^\infty$ in X; X has property (μs)$(\mu ^{s})$ if every weak* null sequence in X∗$X^{*}$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property (μs)$(\mu ^{s})$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

It was kindly pointed out by L. G. Cordeiro as well as independently by T. Bice and W. Kubiś that the proof of Theorem 1.1 from the paper in the title is flawed. We salvage the result in the class of totally disconnected compact spaces and we propose an a