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 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 algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

s. They are usually known as Kaplansky's conjectures.

Group rings

Let be a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, and a torsion-free

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

. 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 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 A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...

. 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, winning ...

and popularized by Kaplansky): * does not contain any non-trivial

unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

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 in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a

crystallographic group In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...

, namely the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

of the

Hantzsche–Wendt manifold The Hantzsche–Wendt manifold, also known as the HW manifold or didicosm, is a compact space, compact, orientable, flat manifold, flat 3-manifold, first studied by Walter Hantzsche and Hilmar Wendt in 1934. It is the only closed flat 3-manifold wi ...

; see also

Fibonacci group In mathematics, for a natural number n \ge 2, the ''n''th Fibonacci group, denoted F(2,n) or sometimes F(n), is defined by ''n'' generators a_1, a_2, \dots, a_n and ''n'' relations: * a_1 a_2 = a_3, * a_2 a_3 = a_4, * \dots * a_ a_ = a_n, * a_a_n ...

. 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

K

is a field of characteristic zero, then the zero-divisor conjecture is implied by the

Atiyah conjecture In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l^2-Betti numbers. History In 1976, Michael Atiyah introduced l^2-cohomology of manifolds with a free co-compact actio ...

, 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 group In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abs ...

s. 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 In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

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 sp ...

''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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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 t ...

. (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, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...

, 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 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hi ...

. In 1989, the conjecture was refuted by

Alexander Merkurjev Aleksandr Sergeyevich Merkurjev (, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles. Work Merk ...

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 Alexander () is a male name of Greek origin. 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 A ...

demonstrated fields with ''u''-invariant

m=2^k+1

for any integer ''k'' starting from 3.

Group rings

Banach algebras

Quadratic forms

References

Further reading