"Long-standing conjectures in analysis ... often turn out to be false"

Question

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?

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¹ Jim Holt. When Einstein Walked with Gödel: Excursions to the Edge of Thought. Farrar, Straus and Giroux, 2018. pp.36-50. (NYTimes Review.)

Answer 1

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.

Answer 2

If RH is "analysis", then surely Littlewood's 1914 theorem that $\pi(x)$ (the prime counting function) and $\mathrm{li}(x)$ (the logarithmic integral) alternate in size infinitely often... despite all numerical evidence at the time indicating that $\pi(x)\le \mathrm{li}(x)$.

Part of the point is that the first reversal only occurs at a rather large number. S. Skewes, a student of Littlewood, gave an effective bound in 1933 assuming RH, and a better one in 1955 unconditionally, ... both of which were ridiculously large numbers. (Just search on "Skewes' number" to see details...)

Similarly, some regularities in the behavior of $\zeta(s)$ do only "kick in" when $\log\log(\Im(s))$ is large... which we will never appraise numerically. Dunno whether this is analysis, but it is fairly genuine mathematics of some sort, and may perhaps illustrate the possibility that phenomena do occur outside the range of direct (with or without computers, quantum or not) observation (by humans). In particular, even sophisticated numerical simulation cannot reach large $\log\log T$, so the conclusions that we draw (e.g., A. Odlyzko's and other's computation of zeros of zeta up to human-enormous heights) could potentially be ... meaningless?

(In a different way, the issues raised by Nik Weaver are somewhat similar, in that they arise from palpable issues, but whose abstractions inadvertently involve phenomenological entities surprisingly beyond our easy capacity.)

Answer 3

The Riemann hypothesis is a conjecture in both analysis and number theory. Someone who tries to undermine it necessarily has to ignore the latter part or to declare it irrelevant. I am not suggesting that it is true (I do not know), only that it becomes more plausible when you take into account that its violation implies a sort of conspiracy between primes and that there are function field analogs which are actually proven.

Once we look at the conjecture from this angle it is arguably unique, and no examples from analysis make a big point.

Answer 4

In another vein, taking the question seriously, etc.:

Let's consider the intermediate value theorem: a (continuous?) function takes every intermediate value, etc. Obviously true, when we believe that functions are things whose graphs we can (easily) draw. But, false, with introduction of not-continuous functions. But, then, again, it's ok when we add that qualifier.

Similarly, the mean value theorem: if $u'=0$, then $u$ is constant. Obviously true. Well, problems with saying what the derivative of a not-so-nice function is... distributions/generalized functions... But, in the end, if a distribution has derivative(s) zero, it is (integrate-against) a constant.

Whew!

Similarly, we can go through dialectics about integrals...

Perhaps even more substantially, the early 20th-century Polish school of set theory and real analysis looked at many situations where a hypothetical simplicity of description of subsets of $\mathbb R$ really needed the continuum hypothesis... or more. I'm insufficiently expert to discuss those things, but am aware of them.

Answer 5

With $\mu(k)$ is the Möbius function, the Mertens conjecture states that the Mertens function

$$M(n)=\sum_{1\le k\le n} \mu(k)$$

is bounded below above as $$|M(n)|\le \sqrt{n}$$

The conjecture lasted 100 years, from 1885 to 1985, when it started to crumble with the work of Andrew Odlyzko and Herman te Riele.

Answer 6

Fuglede's conjecture was open for 30 years (1974-2004) only to be proven false by T. Tao for dimensions $d\geq 5$ with a counter-example arising from a set with an exponential orthonormal basis (a spectrum) in a finite abelian group which does not tile by translation. Interestingly, the largest dimension for which the conjecture is open now is $d=2$.