Kaplansky's conjectures

Last updated

The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures.

Contents

Group rings

Let K be a field, and G a torsion-free group. Kaplansky's zero divisor conjecture states:

Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture:

and Kaplansky's unit conjecture (which was originally made by Graham Higman and popularized by Kaplansky):

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. [1] [2] [3] 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). [4]

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. [5] 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 K[G], 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-mentionned 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 Hausdorff space) 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 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. [6] [7]

In 1989, the conjecture was refuted by Alexander Merkurjev who demonstrated fields with u-invariants of any even m. [6] In 1999, Oleg Izhboldin built a field with u-invariant m = 9 that was the first example of an odd u-invariant. [8] In 2006, Alexander Vishik demonstrated fields with u-invariant for any integer k starting from 3. [9]

Related Research Articles

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

<span class="mw-page-title-main">Hodge conjecture</span> Unsolved problem in geometry

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

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 given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. Equivalently, a domain is a ring in which 0 is the only left zero divisor. A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".

In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module consisting entirely of torsion elements. A module is torsion-free if its only torsion element is the zero element.

This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

<span class="mw-page-title-main">Irving Kaplansky</span> Canadian mathematician (1917–2006)

Irving Kaplansky was a mathematician, college professor, author, and amateur musician.

In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C*-algebra is a purely analytical object.

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Shultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

In mathematics, the Kadison–Singer problem, posed in 1959, was a problem in functional analysis about whether certain extensions of certain linear functionals on certain C*-algebras were unique. The uniqueness was proved in 2013.

Oleg Tomovich Izhboldin 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.

In mathematics, for a natural number , the nth Fibonacci group, denoted or sometimes , is defined by n generators and n relations:

References

  1. ↑ Gardam, Giles (2021-02-23). "A counterexample to the unit conjecture for group rings". Annals of Mathematics. 194 (3): 967–979. arXiv: 2102.11818 . doi:10.4007/annals.2021.194.3.9. S2CID   232013430.
  2. ↑ "Interview with Giles Gardam". Mathematics Münster, University of Münster . Retrieved 2021-03-10.
  3. ↑ Erica Klarreich (April 12, 2021). "Mathematician Disproves 80-Year-Old Algebra Conjecture". Quanta Magazine . Retrieved 2021-04-13.
  4. ↑ Gardam, Giles (December 11, 2023). "Non-trivial units of complex group rings". arXiv: 2312.05240 [math.GR].
  5. ↑ Kropholler, P. H.; Linnell, P. A.; Moody, J. A. (1988). "Applications of a New $K$-Theoretic Theorem to Soluble Group Rings". Proceedings of the American Mathematical Society. 104 (3): 675–684. doi:10.2307/2046771. ISSN   0002-9939.
  6. 1 2 Merkur'ev, A. S. (1991). "Kaplansky conjecture in the theory of quadratic forms". J Math Sci. 57 (6): 3489. doi:10.1007/BF01100118. S2CID   122865942.
  7. ↑ Kaplansky, I. (1951). "Quadratic forms". J. Math. Soc. Jpn. 5 (2): 200–207. doi: 10.2969/jmsj/00520200 .
  8. ↑ Izhboldin, Oleg T. (2001). "Fields of u-Invariant 9". Annals of Mathematics. Second Series. 154 (3): 529–587. doi:10.2307/3062141. JSTOR   3062141. Zbl   0998.11015.
  9. ↑ Vishik, Alexander (2009). "Fields of u-Invariant 2^r + 1". Algebra, Arithmetic, and Geometry. Progress in Mathematics. 270: 661. doi:10.1007/978-0-8176-4747-6_22. ISBN   978-0-8176-4746-9.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.