The mathematician
Irving Kaplansky is notable for proposing numerous
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
s in several branches of mathematics, including a list of ten conjectures on
Hopf algebras. They are usually known as Kaplansky's conjectures.
Group rings
Let be a
field, and a
torsion-free group. Kaplansky's ''zero divisor conjecture'' states:
* The
group ring does not contain nontrivial
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s, that is, it is a
domain.
Two related conjectures are known as, respectively, Kaplansky's ''idempotent conjecture'':
* does not contain any non-trivial
idempotents, i.e., if , then or .
and Kaplansky's ''unit conjecture'' (which was originally made by
Graham Higman and popularized by Kaplansky):
* does not contain any non-trivial
units, i.e., if in , then for some in and in .
The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a
crystallographic group, namely the
fundamental group of the
Hantzsche–Wendt manifold; see also
Fibonacci group.
A later preprint by Gardam claims that essentially the same element also gives a counter-example in characteristic 0 (finding an inverse is computationally much more involved in this setting, hence the delay between the first result and the second one).
There are proofs of both the idempotent and zero-divisor conjectures for large classes of groups. For example, the zero-divisor conjecture is known for all torsion-free
elementary amenable groups (a class including all virtually solvable groups), since their group algebras are known to be
Ore domains. It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups. Note that when
is a field of characteristic zero, then the zero-divisor conjecture is implied by the
Atiyah conjecture, which has also been established for large classes of groups.
The idempotent conjecture has a generalisation, the
Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture, for elements in the
reduced group C*-algebra. In this setting, it is known that if the
Farrell–Jones conjecture holds for , then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all
hyperbolic groups.
The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two (as witnessed by the earlier-mentioned counter-example). This conjecture is not known to follow from any analytic statement like the other two, and so the cases where it is known to hold have all been established via a direct combinatorial approach involving the so-called unique products property. By Gardam's work mentioned above, it is now known to not be true in general.
Banach algebras
This conjecture states that every
algebra homomorphism from the
Banach algebra ''C''(''X'') (continuous complex-valued functions on ''X'', where ''X'' is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
) into any other Banach algebra, is necessarily
continuous. The conjecture is equivalent to the statement that every algebra norm on ''C''(''X'') is equivalent to the usual
uniform norm. (Kaplansky himself had earlier shown that every ''complete'' algebra norm on ''C''(''X'') is equivalent to the uniform norm.)
In the mid-1970s, H. Garth Dales and J. Esterle independently proved that, ''if one furthermore assumes'' the validity of the
continuum hypothesis, there exist compact Hausdorff spaces ''X'' and discontinuous homomorphisms from ''C''(''X'') to some Banach algebra, giving counterexamples to the conjecture.
In 1976,
R. M. Solovay (building on work of H. Woodin) exhibited a model of ZFC (
Zermelo–Fraenkel set theory +
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
) in which Kaplansky's conjecture is true. Kaplansky's conjecture is thus an example of a
statement undecidable in ZFC.
Quadratic forms
In 1953, Kaplansky proposed the conjecture that finite values of
''u''-invariants can only be
powers of 2.
In 1989, the conjecture was refuted by
Alexander Merkurjev who demonstrated fields with ''u''-invariants of any even ''m''.
In 1999,
Oleg Izhboldin built a field with ''u''-invariant ''m'' = 9 that was the first example of an odd ''u''-invariant. In 2006,
Alexander Vishik demonstrated fields with ''u''-invariant
for any integer ''k'' starting from 3.
References
Further reading
*
*
*
*
* {{Cite book , last1=Dales , first1=H. G. , url=https://www.cambridge.org/core/product/identifier/9780511662256/type/book , title=An Introduction to Independence for Analysts , last2=Woodin , first2=W. H. , date=1987 , publisher=Cambridge University Press , isbn=978-0-521-33996-4 , edition=1 , doi=10.1017/cbo9780511662256
Ring theory
Conjectures
Unsolved problems in mathematics