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We characterize the condition (Ω) for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions (P Ω) and (P Ω ¯ ¯) for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space { f ∈ E (X) | P (D) f = 0 } satisfies (Ω) for any differential operator P(D) and any open convex set X ⊆ R d . [ABSTRACT FROM AUTHOR]

We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces the map L (Y , E) → L (X , E) , T ↦ T ∘ ι is surjective, meaning that each continuous linear map X → E can be extended to a continuous linear map Y → E via ι , under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24]. [ABSTRACT FROM AUTHOR]

We study the six diagrams generated by the first three Schechter interpolators Δ 2 (f) = f ′ ′ (1 / 2) / 2 ! , Δ 1 (f) = f ′ (1 / 2) , Δ 0 (f) = f (1 / 2) acting on the Calderón space associated to the pair (ℓ ∞ , ℓ 1) . We will study the remarkable and somehow unexpected properties of all the spaces appearing in those diagrams: two new spaces (and their duals), two Orlicz spaces (and their duals) in addition to the third order Rochberg space, the standard Kalton-Peck space Z 2 and, of course, the Hilbert space ℓ 2 . We will also deal with a nice test case: that of weighted ℓ 2 spaces, in which case all involved spaces are Hilbert spaces. [ABSTRACT FROM AUTHOR]

It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules. [ABSTRACT FROM AUTHOR]

We study those operators on a Hilbert space that can be lifted or extended to any twisted Hilbert space. We prove that these form an ideal of operators which contains all the Schatten classes. We characterize those multiplication operators on ℓ p that are liftable/extensible through centralizers. [ABSTRACT FROM AUTHOR]

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of C∗–algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests. [ABSTRACT FROM AUTHOR]

Let F be an endofunctor of a category C. We prove isomorphism theorems for F -coalgebras under condition that the underlying category C is exact; that is, regular with exact sequences. Also, F is not assumed to preserve pullbacks.

We give a complete solution to the Borel–Ritt problem in non-uniform spaces A (M) - (S) of ultraholomorphic functions of Beurling type, where S is an unbounded sector of the Riemann surface of the logarithm and M is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on A (M) - (S) in terms of the aperture of the sector S and the weight sequence M. Our work improves previous results by Thilliez (Results Math 44:169–188, 2003) and Schmets and Valdivia (Stud Math 143:221–250, 2000). [ABSTRACT FROM AUTHOR]

Let A, X be two Banach algebras and let X be an algebraic Banach A-module equipped with a bounded bilinear map Θ : X × X → A which is compatible with the A-module operations of X. Then the ℓ 1 -direct sum A × X endowed with the multiplication (a , x) (b , y) = (a b + Θ (x , y) , a y + x b + x y) (a , b ∈ A , x , y ∈ X) is a Banach algebra, denoted by A ⊠ Θ X and will be called a bi-amalgamated Banach algebra. Many known Banach algebras such as (generalized) module extension Banach algebras, Lau product Banach algebras, generalized matrix Banach algebras have this general framework. The main aim of this paper is to investigate biprojectivity and biflatness of A ⊠ Θ X . Our results extend several results in the literature and provide simple direct proofs for some known results. In particular, we characterize the biprojectivity and biflatness of certain classes of the module extension Banach algebras and generalized matrix Banach algebras. Some unsolved questions are also included. [ABSTRACT FROM AUTHOR]

We prove a change of base theorem for operator space modules over C*-algebras, analogous to the change of rings for algebraic modules. We demonstrate how this can be used to show that the category of (right) matrix normed modules and completely bounded module maps has enough injectives. [ABSTRACT FROM AUTHOR]

We study spaces of vector-valued quasianalytic functions and solve the first Cousin problem in these spaces. [ABSTRACT FROM AUTHOR]

Let $${T}$$ be a homomorphism from a Banach algebra $${B}$$ to a Banach algebra $${A}$$ . The Cartesian product space $${A\times B}$$ with $${T}$$ -Lau multiplication and $${\ell^1}$$ -norm becomes a new Banach algebra $${A\times _T B}$$ . We investigate the notions such as approximate amenability, pseudo amenability, $${\phi}$$ -pseudo amenability, $${\phi}$$ -biflatness and $${\phi}$$ -biprojectivity for Banach algebra $${A\times_T B}$$ . We also present an example to show that approximate amenability of $${A}$$ and $${B}$$ is not stable for $${A\times _TB}$$ . Finally we characterize the double centralizer algebra of $${A\times _T B}$$ and present an application of this characterization. [ABSTRACT FROM AUTHOR]

We study the construction of twisted sums of Schreier-like and Tsirelson spaces and their strictly singular character. [ABSTRACT FROM AUTHOR]