The complemented subspace problem for $C(K)$-spaces: A counterexample

TL;DR

This paper provides a counterexample in the theory of Banach spaces, showing that a 1-complemented subspace of a $C(L)$ space need not be isomorphic to any space of continuous functions, challenging previous assumptions.

Contribution

The authors construct a specific counterexample demonstrating that not all 1-complemented subspaces of $C(L)$ are isomorphic to $C(K)$ spaces, addressing a longstanding open problem.

Findings

Existence of a compact space $L$ with a non-$C(K)$ 1-complemented subspace

Counterexample refutes the conjecture that all 1-complemented subspaces are $C(K)$ spaces

Advances understanding of the structure of complemented subspaces in $C(K)$-spaces

Abstract

We prove that there is a compact space $L$ and a 1-complemented subspace of the Banach space $C(L)$ which is not isomorphic to a space of continuous functions.