Kaplansky Conjectures
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The mathematician
Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St And ...
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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
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 In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
does not contain nontrivial zero divisors, that is, it is a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
. Two related conjectures are known as, respectively, Kaplansky's ''idempotent conjecture'': * does not contain any non-trivial
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s, i.e., if , then or . and Kaplansky's ''unit conjecture'' (which was originally made by
Graham Higman Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory. Biography Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winnin ...
and popularized by Kaplansky): * does not contain any non-trivial
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
s, 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 for fields of positive characteristic by Giles Gardam in February 2021: he published a preprint on the
arXiv arXiv (pronounced " archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists o ...
that constructs a counterexample. The field is of characteristic 2. (see also: Fibonacci group) There are proofs of both the idempotent and zero-divisor conjectures for large classes of groups. For example, the zero-divisor conjecture is known to hold for all virtually solvable groups and more generally also for all residually torsion-free solvable groups. These solutions go through establishing first the conclusion to the Atiyah conjecture on L^2-Betti numbers, from which the zero-divisor conjecture easily follows. 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. For example, there is a torsion-free 3-dimensional
crystallographic group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...
for which it is not known whether all units are trivial. 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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
''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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
) 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 In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when ...
. (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 In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent ...
, 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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
+
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) 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 Oleg Tomovich Izhboldin (russian: Олег Томович Ижболдин; 1963 - 2000) was a Russian mathematician who was first to provide a non-trivial example of an odd u-invariant field solving a classical Kaplansky's conjecture. Oleg Iz ...
built a field with ''u''-invariant ''m'' = 9 that was the first example of an odd ''u''-invariant. In 2006,
Alexander Vishik Alexander is a male given name. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here are Aleksandar, Al ...
demonstrated fields with ''u''-invariant m=2^k+1 for any integer ''k'' starting from 3.


References

* H. G. Dales, ''Automatic continuity: a survey''. Bull. London Math. Soc. 10 (1978), no. 2, 129–183. * W. Lück, ''L2-Invariants: Theory and Applications to Geometry and K-Theory''. Berlin:Springer 2002 * D.S. Passman, ''The Algebraic Structure of Group Rings'', Pure and Applied Mathematics, Wiley-Interscience, New York, 1977. {{ISBN, 0-471-02272-1 * M. Puschnigg, ''The Kadison–Kaplansky conjecture for word-hyperbolic groups''. Invent. Math. 149 (2002), no. 1, 153–194. * H. G. Dales and W. H. Woodin, ''An introduction to independence for analysts'', Cambridge 1987 Ring theory Conjectures Unsolved problems in mathematics