Best approximation in von Neumann algebras

Extract

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.