On proximinality and sets of operators. III: Approximation by finite rank operators on spaces of continuous functions (Q1081777)

The author continues the study concerning proximinality of \(K_ n(X,Y)\) in K(X,Y) and L(X,Y) (the last one is the space of all bounded operators between two normed spaces X and Y; the other two spaces contain all operators of rank \(\leq n\), or - respectively - all compact operators, in L(X,Y)). The main result of the paper is the following: let Q and S be locally compact Hausdorff spaces, and assume that S ''contains \(Q_ 0''\). Then, for each \(n\geq 1\), \(K_ n(C_ 0(Q),C_ 0(S))\) is proximinal in \(K(C_ 0(Q),C_ 0(S))\) iff Q is finite. Problems in this area are partially solved by the above results and related ones.