Let X and Y be separable Banach spaces. Suppose Y either has a shrinking basis or Y is isomorphic to C(2N) andA is a subset of weakly compact operators from X to Y which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis Z such that every T ∈ A factors through Z. Likewise, we prove that if A ⊂ L(X,C(2N)) is a set of operators whose adjoints have s...

It is shown that the weakly closed algebra generated by a weakly compact group of operators on a Banach space is reflexive and equals its second commutant. Also, an example is given to show that the generator of a monothetic weakly compact group of operators need not have a logarithm in the algebra of all bounded linear operators on the underlying space. Let X be a complex Banach space, B(X) th...

For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E/E) is defined by R(S)(x + E) = Sx + E (x ∈ E). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W (E) (here W (E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non–zero compact...

We prove from the existence of a Mahlo cardinal the consistency of the statement that 2ω = ω3 holds and every stationary subset of ω2∩cof(ω) reflects to an ordinal less than ω2 with cofinality ω1. Let us say that stationary set reflection holds at ω2 if for any stationary set S ⊆ ω2∩cof(ω), there is an ordinal α ∈ ω2∩cof(ω1) such that S∩α is stationary in α (that is, S reflects to α). In a clas...

We investigate the structure of the lattice of clones on an infinite set X . We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 2 λ many maximal (= “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a stron...

We show that if κ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order on the generalized Cantor space 2 there is an Lκ+κsentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bireducible (and, in fact, classwise Borel isomorphic) to R...

In this paper, we introduce the concept of IVF weakly m∗-continuous mappings on between IVF minimal spaces and investigate some characterizations for such mappings. Also we study the relationships IVF weakly m∗-continuous mappings and IVF M -compactness. Key ords : IVF minimal space, IVF weakly m∗-continuous, IVF M -compact, almost IVF M -compact, nearly IVF M -compact

Using the analysis developed in our earlier paper [ADK], we show that every uncountable cardinal in Gitik’s model of [Git80] in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of [Git80] is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable card...