A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a ... more A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property 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.
We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is ope... more We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ 1 (N 0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ 1 (Z) nor in ℓ 1 (Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.
Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banac... more Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n ∈ N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection P N ∈ B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by P N for some N. We can now state our main conclusions as follows: • the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements; • for each pair I J of spatial ideals, there is a family {Γ L : L ∈ ∆}, where the index set ∆ has the cardinality of the continuum, such that Γ L is an uncountable chain of spatial ideals, Γ L is a closed ideal that is not spatial, and
We construct a Banach space Z such that the lattice of closed two-sided ideals of the Banach alge... more We construct a Banach space Z such that the lattice of closed two-sided ideals of the Banach algebra B(Z) of bounded operators on Z is as follows:
Leonetti proved that whenever I is an ideal on N such that there exists an uncountable family of ... more Leonetti proved that whenever I is an ideal on N such that there exists an uncountable family of sets that are not in I with the property that the intersection of any two distinct members of that family is in I, then the space c 0,I of sequences in ℓ ∞ that converge to 0 along I is not complemented. We provide a shorter proof of a more general fact that the quotient space ℓ ∞ /c 0,I does not even embed into ℓ ∞. Very recently, Leonetti ([3]) distilled a property of ideals I on the set of natural numbers (shared by ideals that are meagre when regarded as subsets of the Cantor set; see [3, Lemma 2.3], hence for example by the ideal of sets that have density 0) that gives a fairly satisfactory sufficient condition for non-complementability in ℓ ∞ , the space of all bounded scalar sequences, of the space c 0,I consisting of sequences that converge to 0 along I. Leonetti's proof is an interesting refinement of Whitley's proof of the Phillips-Sobczyk theorem ([4]; see also [1, Theorem 2.5.4]), which asserts that the space c 0 is not complemented in ℓ ∞. (All necessary terminology will be explained in subsequent paragraphs.) More specifically, Leonetti's result contributes to the problem of characterisation of those ideals I of N for which the space c 0,I is complemented in ℓ ∞ proposed by Pérez Hernández during the Winter School in Abstract Analysis 2017 held in Svratka, Czech Republic. Conspicuously, the classical space c 0 coincides with c 0,I where I is the ideal of finite sets, so it is not complemented by virtue of the above-mentioned Phillips-Sobczyk theorem. On the other hand, when I is a maximal ideal (that is when the dual filter is an ultrafilter), the subspace c 0,I has codimension 1 in ℓ ∞ , so it is complemented. (Similarly, the intersection I of finitely many maximal ideals will correspond to c 0,I having finite co-dimension in ℓ ∞ , so in particular c 0,I is complemented in this case.) As every non-principal ideal contains the ideal of finite sets and, at the same time, is contained in a maximal ideal, the problem is quite tantalising indeed. Let us record Leonetti's result formally.
Using the fact that the normalised matrix trace is the unique linear functional f on the algebra ... more Using the fact that the normalised matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA) for all n × n matrices A and B, we derive a well-known formula expressing the normalised trace of a complex matrix A as the expectation of the numerical values of A; that is the function Ax, x , where x ranges the unit sphere of C n .
We prove that a WLD subspace of the space ℓ c ∞ (Γ) consisting of all bounded, countably supporte... more We prove that a WLD subspace of the space ℓ c ∞ (Γ) consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into ℓ ∞ if and only if it does not contain isometric copies of c 0 (ω 1). Moreover, a subspace of ℓ c ∞ (ω 1) is constructed that has an unconditional basis, does not embed into ℓ ∞ , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of c 0 (ω 1)).
Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a univ... more Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a universal constant K 44.5 such that for every set algebra F and every 1-additive function f : F → R there exists a finitely-additive signed measure µ defined on F such that |f (A) − µ(A)| K for any A ∈ F. The only known lower bound for the optimal value of K was found by Pawlik [Colloq. Math., 54 (1987), 163-164], who proved that this constant is not smaller than 1.5; we improve this bound to 3 already on a non-negative 1-additive function.
We demonstrate that two Ornstein-Uhlenbeck processes, that is, solutions to certain stochastic di... more We demonstrate that two Ornstein-Uhlenbeck processes, that is, solutions to certain stochastic differential equations that are driven by a Lévy process L have equivalent laws as long as the eigenvalues of the covariance operator associated to the Wiener part of L are strictly positive. Moreover, we show that in the case where the underlying Lévy process is a purely jump process, which means that neither it has a Wiener part nor the drift, the absolute continuity of the law of one solution with respect to another forces equality of the solutions almost surely.
Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1... more Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ε)-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pe lczyński had proved for every λ > 1 the existence of a (λ + ε)-injective space (ε > 0) that is not λinjective. Unfortunately, no trace of the proof of Pe lczyński's result has been preserved. In the present paper, we establish the said theorem for λ ∈ (1, 2] by constructing an appropriate renorming of ℓ ∞. This contrasts (at least for real scalars) with the case λ = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le... more We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) concerning restricting uniformly open, continuous surjections to smaller subspaces where they remain surjective. To wit, suppose that X and Y are metric spaces and let f : X → Y be a continuous surjection. If X is complete and f is uniformly open, then X contains a closed subspace Z with the same density as Y such that f restricted to Z is still uniformly open and surjective. Moreover, if X is a Banach space, then Z may be taken to be a closed linear subspace. A counterpart of this theorem for uniform spaces is also established.
We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative ... more 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.
Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integ... more Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L 1 or C(∆), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) L p-spaces, BMO, SL ∞ , and others.
Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally fre... more Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally free object F \mathsf F , then there is a projectively universal morphism u : F → F u\colon \mathsf F\to \mathsf F , i.e., a morphism u u such that for any B ∈ C B\in \mathcal {C} and τ ∈ Mor ( B ) \tau \in \operatorname {Mor}(B) there exists an epimorphism π ∈ Mor ( F , B ) \pi \in \operatorname {Mor}(\mathsf F, B) such that π τ = u π \pi \tau = u \pi . This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] 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.
We prove that the coordinate functionals associated with filter bases in Banach spaces are contin... more 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 ([Bull. Lond. Math. Soc. 53 (2021)]) 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.
We prove that the coordinate functionals associated with filter bases in Banach spaces are contin... more 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 ([Bull. Lond. Math. Soc. 53 (2021)]) 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.
Using the fact that the normalised matrix trace is the unique linear functional f on the algebra ... more Using the fact that the normalised matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA) for all n × n matrices A and B, we derive a well-known formula expressing the normalised trace of a complex matrix A as the expectation of the numerical values of A; that is the function Ax, x , where x ranges the unit sphere of C n .
A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a ... more A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property 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.
We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is ope... more We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ 1 (N 0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ 1 (Z) nor in ℓ 1 (Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.
Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banac... more Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n ∈ N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection P N ∈ B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by P N for some N. We can now state our main conclusions as follows: • the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements; • for each pair I J of spatial ideals, there is a family {Γ L : L ∈ ∆}, where the index set ∆ has the cardinality of the continuum, such that Γ L is an uncountable chain of spatial ideals, Γ L is a closed ideal that is not spatial, and
We construct a Banach space Z such that the lattice of closed two-sided ideals of the Banach alge... more We construct a Banach space Z such that the lattice of closed two-sided ideals of the Banach algebra B(Z) of bounded operators on Z is as follows:
Leonetti proved that whenever I is an ideal on N such that there exists an uncountable family of ... more Leonetti proved that whenever I is an ideal on N such that there exists an uncountable family of sets that are not in I with the property that the intersection of any two distinct members of that family is in I, then the space c 0,I of sequences in ℓ ∞ that converge to 0 along I is not complemented. We provide a shorter proof of a more general fact that the quotient space ℓ ∞ /c 0,I does not even embed into ℓ ∞. Very recently, Leonetti ([3]) distilled a property of ideals I on the set of natural numbers (shared by ideals that are meagre when regarded as subsets of the Cantor set; see [3, Lemma 2.3], hence for example by the ideal of sets that have density 0) that gives a fairly satisfactory sufficient condition for non-complementability in ℓ ∞ , the space of all bounded scalar sequences, of the space c 0,I consisting of sequences that converge to 0 along I. Leonetti's proof is an interesting refinement of Whitley's proof of the Phillips-Sobczyk theorem ([4]; see also [1, Theorem 2.5.4]), which asserts that the space c 0 is not complemented in ℓ ∞. (All necessary terminology will be explained in subsequent paragraphs.) More specifically, Leonetti's result contributes to the problem of characterisation of those ideals I of N for which the space c 0,I is complemented in ℓ ∞ proposed by Pérez Hernández during the Winter School in Abstract Analysis 2017 held in Svratka, Czech Republic. Conspicuously, the classical space c 0 coincides with c 0,I where I is the ideal of finite sets, so it is not complemented by virtue of the above-mentioned Phillips-Sobczyk theorem. On the other hand, when I is a maximal ideal (that is when the dual filter is an ultrafilter), the subspace c 0,I has codimension 1 in ℓ ∞ , so it is complemented. (Similarly, the intersection I of finitely many maximal ideals will correspond to c 0,I having finite co-dimension in ℓ ∞ , so in particular c 0,I is complemented in this case.) As every non-principal ideal contains the ideal of finite sets and, at the same time, is contained in a maximal ideal, the problem is quite tantalising indeed. Let us record Leonetti's result formally.
Using the fact that the normalised matrix trace is the unique linear functional f on the algebra ... more Using the fact that the normalised matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA) for all n × n matrices A and B, we derive a well-known formula expressing the normalised trace of a complex matrix A as the expectation of the numerical values of A; that is the function Ax, x , where x ranges the unit sphere of C n .
We prove that a WLD subspace of the space ℓ c ∞ (Γ) consisting of all bounded, countably supporte... more We prove that a WLD subspace of the space ℓ c ∞ (Γ) consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into ℓ ∞ if and only if it does not contain isometric copies of c 0 (ω 1). Moreover, a subspace of ℓ c ∞ (ω 1) is constructed that has an unconditional basis, does not embed into ℓ ∞ , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of c 0 (ω 1)).
Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a univ... more Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a universal constant K 44.5 such that for every set algebra F and every 1-additive function f : F → R there exists a finitely-additive signed measure µ defined on F such that |f (A) − µ(A)| K for any A ∈ F. The only known lower bound for the optimal value of K was found by Pawlik [Colloq. Math., 54 (1987), 163-164], who proved that this constant is not smaller than 1.5; we improve this bound to 3 already on a non-negative 1-additive function.
We demonstrate that two Ornstein-Uhlenbeck processes, that is, solutions to certain stochastic di... more We demonstrate that two Ornstein-Uhlenbeck processes, that is, solutions to certain stochastic differential equations that are driven by a Lévy process L have equivalent laws as long as the eigenvalues of the covariance operator associated to the Wiener part of L are strictly positive. Moreover, we show that in the case where the underlying Lévy process is a purely jump process, which means that neither it has a Wiener part nor the drift, the absolute continuity of the law of one solution with respect to another forces equality of the solutions almost surely.
Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1... more Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ε)-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pe lczyński had proved for every λ > 1 the existence of a (λ + ε)-injective space (ε > 0) that is not λinjective. Unfortunately, no trace of the proof of Pe lczyński's result has been preserved. In the present paper, we establish the said theorem for λ ∈ (1, 2] by constructing an appropriate renorming of ℓ ∞. This contrasts (at least for real scalars) with the case λ = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le... more We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) concerning restricting uniformly open, continuous surjections to smaller subspaces where they remain surjective. To wit, suppose that X and Y are metric spaces and let f : X → Y be a continuous surjection. If X is complete and f is uniformly open, then X contains a closed subspace Z with the same density as Y such that f restricted to Z is still uniformly open and surjective. Moreover, if X is a Banach space, then Z may be taken to be a closed linear subspace. A counterpart of this theorem for uniform spaces is also established.
We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative ... more 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.
Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integ... more Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L 1 or C(∆), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) L p-spaces, BMO, SL ∞ , and others.
Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally fre... more Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally free object F \mathsf F , then there is a projectively universal morphism u : F → F u\colon \mathsf F\to \mathsf F , i.e., a morphism u u such that for any B ∈ C B\in \mathcal {C} and τ ∈ Mor ( B ) \tau \in \operatorname {Mor}(B) there exists an epimorphism π ∈ Mor ( F , B ) \pi \in \operatorname {Mor}(\mathsf F, B) such that π τ = u π \pi \tau = u \pi . This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] 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.
We prove that the coordinate functionals associated with filter bases in Banach spaces are contin... more 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 ([Bull. Lond. Math. Soc. 53 (2021)]) 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.
We prove that the coordinate functionals associated with filter bases in Banach spaces are contin... more 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 ([Bull. Lond. Math. Soc. 53 (2021)]) 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.
Using the fact that the normalised matrix trace is the unique linear functional f on the algebra ... more Using the fact that the normalised matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA) for all n × n matrices A and B, we derive a well-known formula expressing the normalised trace of a complex matrix A as the expectation of the numerical values of A; that is the function Ax, x , where x ranges the unit sphere of C n .
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