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On positive embeddings of C(K) spaces

100%

Plebanek G.

nr 2

179-192

We investigate isomorphic embeddings T: C(K) → C(L) between Banach spaces of continuous functions. We show that if such an embedding T is a positive operator then K is the image of L under an upper semicontinuous set-function having finite values. Moreover we show that K has a π-base of sets whose closures are continuous images of compact subspaces of L. Our results imply in particular that if C(K) can be positively embedded into C(L) then some topological properties of L, such as countable tightness or Fréchetness, are inherited by K. We show that some isomorphic embeddings C(K) → C(L) can be, in a sense, reduced to positive embeddings.

Convex Corson compacta and Radon measures

100%

Plebanek G.

tom 175

nr 2

143-154

Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ(ℝ^{ω₁})$, and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$-points.

Nonseparable Radon measures and small compact spaces

100%

Plebanek G.

tom 153

nr 1

25-40

We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube $[0, 1]^κ$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of $ω_1$ null sets in $2^{ω1}$ such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is "no" for κ = ω. We also give alternative proofs of two related results due to Kunen and van Mill [18].

Countable tightness in the spaces of regular probability measures

63%

Plebanek G. , Sobota D.

nr 2

159-169

We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.

On asymptotic density and uniformly distributed sequences

63%

Frankiewicz R. , Plebanek G.

nr 1

17-26

Assuming Martin's axiom we show that if X is a dyadic space of weight at most continuum then every Radon measure on X admits a uniformly distributed sequence. This answers a problem posed by Mercourakis [10]. Our proof is based on an auxiliary result concerning finitely additive measures on ω and asymptotic density.

On Radon measures on first-countable spaces

32%

Plebanek G.

nr 2

159-164