The unicity of best approximation in a space of compact operators

Chebyshev subspaces of $\mathcal{K}(c_0,c_0)$ are studied. A $k$-dimensional non-interpolating Chebyshev subspace is constructed. The unicity of best approximation in non-Chebyshev subspaces is considered.

The L1-version of the Diliberto–Straus algorithm in C(T × S)

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

Best approximation in von Neumann algebras

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

On equidistant sets in normed linear spaces

In this note some results concerning the equidistant setE(−x, x) and the kernelMθof the metric projectionPM, whereMis a Chebyshev subspace of a normed linear spaceX, have been obtained. In particular, whenX=lp(1 <p< ∞), it has been proved that every equidistant set is closed in thebw-topology of the space. Inc0no equidistant set has this property.