The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,
and he is cautioning "those who blithely assume the truth of the
Riemann conjecture."
Q.
What are examples of long-standing conjectures in analysis that turned out to be false?
Is Holt's adverb "often" justified?
1
Jim Holt.
When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Farrar, Straus and Giroux, 2018. pp.36-50.
(NYTimes Review.)
Best Answer
I don't know about analysis in general, but I think it's definitely fair to say "often" in functional analysis. My feeling is that we have a solid, thorough, elegant body of theory which usually leads to positive solutions rather quickly, when they exist. (The Kadison-Singer problem is a recent exception which required radically new tools for a positive solution.) Problems that stick around for a long time tend to do so not because there's a complicated positive solution but because there's a complicated counterexample. That's a gross overgeneralization but I think there's some truth to it.
The first examples I can think of are:
every separable Banach space has the approximation property and has a Schauder basis (counterexample by Enflo)
every bounded linear operator on a Banach space has a nontrivial closed invariant subspace (counterexamples by Enflo and Read)
every infinite dimensional Banach space has an infinite dimensional subspace which admits an unconditional Schauder basis (counterexample by Gowers and Maurey)
every infinite dimensional Banach space $X$ is isomorphic to $X \oplus \mathbb{R}$; if $X$ and $Y$ are Banach spaces, each linearly homeomorphic to a subspace of the other, then they are linearly homeomorphic (counterexample by Gowers)
I can't resist also mentioning some examples that I was involved with.
Dixmier's problem: every prime C*-algebra is primitive (counterexample by me)
Naimark's problem: if a C*-algebra has only one irreducible representation up to unitary equivalence, then it is isomorphic to $K(H)$ for some Hilbert space $H$ (counterexample by Akemann and me)
every pure state on $B(l^2)$ is pure on some masa (counterexample by Akemann and me)
every automorphism of the Calkin algebra is inner (counterexample by Phillips and me)
The last three require extra set-theoretic axioms, so the correct statement is that if ordinary set theory is consistent, then it is consistent that these counterexamples exist. Presumably all three are independent of the usual axioms of set theory, but this is only known of the last one, where the consistency of a positive solution was proved by Farah.