Yaki Sternfeld

DIMENSION, SUPERPOSITION OF FUNCTIONS AND SEPARATION OF POINTS, IN COMPACT METRIC SPACES

AbstractIt is proved that a compact metric spaceX isn-dimensional (n≧2) if and only if there exist 2n+1 functionsϕ1,ϕ2, ...,ϕ2n+1 inC(X) so that eachf ∈C(X) is representable as

Uniform separation of points and measures and representation by sums of algebras

f(x) = \sum\limits_{i = 1}^{2n + 1} {g(\varphi _i (x))} with g_i \in C(R), 1 \leqq i \leqq 2n + 1

Monotone basic embeddings of hereditarily indecomposable continua

. Equivalently, it is shown that dimX=n if and only ofC(X) is the algebraic sum of 2n+1 subalgebras, each of which is isomorphic toC(0,1). The properties of families {ϕi}i=1/2n+1 which satisfy the above are studied, and they are characterized in terms of their ability to separate the points ofX in some strong sense.

Dense subgroups ofC(K)-Stone-Weierstrass-type theorems for groups

Let X be a metric continuum and let C(X) denote the space of subcontinua of X with the Hausdorff metric. We settle a longstanding problem showing that if dimX = 2 then dimC(X) = ∞. The special structure and properties of hereditarily indecomposable continua are applied in the proof.

Extension of mappings of Bing spaces into ANRs

LetX andYi, 1 ≦i ≦k, be compact metric spaces, and letρi:X →Yi be continuous functions. The familyF={ρi}i1/k is said to be ameasure separating family if there exists someλ > 0 such that for every measureμ inC(X)*, ‖μoρi−1 ‖ ≧λ ‖μ ‖ holds for some 1 ≦i ≦k.F is auniformly (point) separating family if the above holds for the purely atomic measures inC(X)*. It is known that fork ≦ 2 the two concepts are equivalent. In this note we present examples which show that fork ≧ 3 measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: letA1,A2, ...,Ak be closed subalgebras ofC(X); when isA1 +A2 + ... +Ak equal to or dense inC(X)?

Approximation from the topological viewpoint

Abstract A continuous mapping ƒ : X → Y is called k-stable if for every metric space E that contains Y there exists a neighborhood U of ƒ in C(X, E) such that dim g(X) ⩾ k for all g in U. The paper is devoted to the study of k-stable maps on compact metric spaces.

Affine Invariant Subspaces of C( C )

Abstract Let {ϑi}i = 1k be monotone maps on a hereditarily indecomposable continuum X. It is proved that the following are equivalent: 1. (i) The product map ϑ = (ϑ1, ϑ2, …, ϑk) is light. 2. (ii) ϑ is an embedding. 3. (iii) Each ƒ in C(X, R) is representable as ƒ = ∑ i = 1 k g i oϑ i with ϑ ϵ C(ϑi(X), R). This is applied to prove the following result which is related to the Chogoshvili conjecture: Let n ⩾ 2 and let X be an n-dimensional hereditarily indecomposable continuum. X can be embedded in a separable Hilbert space H such that: 1. (i) The restriction to X of the continuous linear functionals of H forms a dense subset of C(X, R). 2. (ii) There exists an orthonormal basis B for H such that the restriction to X of each 2-dimensional B-coordinate projection of H factors through some 1-dimensional space and as a result has no stable values in R2. In particular the n-dimensional B-coordinate projections have no stable values on X.

Finite dimensional polish spaces are extreme boundaries of convex bodies in Euclidean space

LetB be a subgroup ofC(K) which separates points and contains the constants. An elementh∈C(R) operates onB iff∈B implies thath∘f∈B. An elementh∈C(R) is condensing if its operation onB implies the density ofB inC(K). Similar notation applies to subgroupsB ofC(K, G) whereG is a metrizable group. We study the setD(G) of condensing functions inC(G, G) whenG is the additive group of a real Banach space and in particular whenG=Rn.