We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^{*}$ of a Banach space X, then ${\ell }^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\ell }^1$ contains a copy of ${\ell }^1$ that is complemented in ${\ell }^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1[0,1]$ contains a copy of ${\ell }^1$ that is complemented in $L^1[0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of $c_0$ which simultaneously...

Characterizations are obtained for those pairs of weight functions u and v for which the operators $Tf\left(x\right)={ʃ}_{a\left(x\right)}^{b\left(x\right)}f\left(t\right)dt$ with a and b certain non-negative functions are bounded from $L_u^p(0,\infty )$ to $L_v^q(0,\infty )$, 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

We prove that the elliptic maximal function maps the Sobolev space W4,eta(R2) into L4(R2) for all eta &gt; 1/6. The main ingredients of the proof are an analysis of the intersectiQn properties of elliptic annuli and a combinatorial method of Kolasa and Wolff.

Let θ : ℳ → 𝓝 be a zero-product preserving linear map between algebras. We show that under some mild conditions θ is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C*-algebras and W*-algebras.

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\overline{A}$ and $\overline{B}$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ{0, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and...

The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will...

A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...

Let $f\in V_r\left(\right){\cup }_r\left(\right)$, where, for 1 ≤ r < ∞, $V_r\left(\right)$ (resp., ${}_r\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

Nigel J. Kalton was one of the most eminent guests participating in the Józef Marcinkiewicz Centenary Conference. His contribution to the scientific aspect of the meeting was very essential. Nigel was going to prepare a paper based on his plenary lecture. The editors are completely sure that the paper would be a real ornament of the Proceedings. Unfortunately, Nigel's sudden death totally destroyed editors' hopes and plans. Every mathematician knows how unique were Nigel's mathematical achievements....