In the paper we study the existence of nonzero positive invariant elements for positive operators in Riesz spaces. The class of Riesz spaces for which the results are valid is large enough to contain all the Banach lattices with order continuous norms. All the results obtained in earlier works deal with positive operators in KB-spaces and in many of them the approach is based upon the use of Banach limits. The methods created for KB-spaces cannot be extended to our more general setting; that is...

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${[0,1]}^{\infty }$ and a Cantor set, respectively.

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{}\oplus [0,\alpha ]$, the topological sums of Cantor cubes $2^{}$, with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C\left(2^{}\oplus [0,\alpha ]\right)$ spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{NJ}\left(X\right)$, and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{NJ}\left(X\right)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{NJ}\left(X\right)$-constant, which implies that a Banach space with $C_{NJ}\left(X\right)$-constant less than 5/4 has the fixed point property.

This paper deals with a few, not widely known, aspects of Kottman's constant of a Banach space and its symmetric and finite variations. We will consider their behaviour under ultrapowers, relations with other parameters such as Whitley's or James' constant, and connection with the extension of c₀-valued Lipschitz maps.

We obtain refinement of a result of Partington on Banach spaces containing isomorphic copies of l-∞. Motivated by this result, we prove that Banach spaces containing asymptotically isometric copies of l-∞ must contain isometric copies of l-∞.

We study the spaces $H_{\mu }\left(\Omega \right)=f:\Omega \to ℂholomorphic:{\int }_0^R{\int }_0^{2\pi }\left|f\left(r{e}^{i\phi }\right)\right|d\phi d\mu \left(r\right)<\infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{\mu }\left(\Omega \right)$ is either isomorphic to l₁ or to ${\left(\sum \oplus Aₙ\right)}_{\left(1\right)}$. Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type ${ℒ}_{\infty }$ but not conversely. Moreover, ${ℒ}_{\infty }$-spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will show that...