Haagerup property
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| Haagerup property | |
|---|---|
| Name | Haagerup property |
| Alt | Haagerup property |
| Field | Mathematics |
| Subfield | Functional analysis; Operator algebras; Geometric group theory |
| Introduced by | Uffe Haagerup |
| Year | 1978 |
Haagerup property
The Haagerup property is a form of "a-T-menability" for locally compact groups and related objects in operator algebras that admits proper conditionally negative definite functions and proper positive definite kernels; it lies between amenability and property (T) and is central to interactions among Uffe Haagerup, Alain Connes, Nikolai Ivanovich Lobachevsky, Gromov, Mikhael Gromov, Valentin Lafforgue, Boris Tsirelson, Edward Nelson, Daniel Voiculescu. It appears in the study of von Neumann algebras, C*-algebras, and coarse geometry, influencing work by Izumi Shimada, Paul Jolissaint, Guoliang Yu, Higson Eric and others.
Definition and equivalent formulations
A second-countable locally compact group G has the Haagerup property if there exists a sequence of continuous, normalized, positive definite functions on G vanishing at infinity that converge pointwise to 1; the same notion can be expressed via existence of a proper, conditionally negative definite length function, unitary representations with almost invariant vectors which are weakly contained in the regular representation, or by existence of proper affine isometric actions on a Hilbert space. Equivalent formulations appear across literature by Uffe Haagerup, Paul Jolissaint, Gromov, Cowling Michael, Pierre Julg, Nicolas Monod, and Yoram Choi, tying the property to Herz–Schur multipliers, Schur multipliers studied by Schoenberg Isaac, and to the existence of proper cocycles for representations considered by Boris Bekka and Pierre de la Harpe.
Examples and non-examples
Classical examples include all amenable groups such as Abel, virtually abelian groups and compact groups; more striking examples are free groups like Free group of finite rank, discrete subgroups of SO(n,1), lattices in SL(2,R), and Coxeter groups; large classes arising from hyperbolic geometry and CAT(0) cube complexes studied by Frédéric Haglund and Dani Wise also have the property. Non-examples include groups with Kazhdan's property (T) such as some higher-rank lattices in SL(n,Z) for n≥3, groups constructed by Valentin Lafforgue exhibiting strong Banach property (T), and certain expanders linked to work of Margulis and Alexander Lubotzky.
Properties and permanence results
The Haagerup property is preserved under taking closed subgroups, directed unions of open subgroups, direct products, free products, and extensions under suitable hypotheses; permanence results were proved by researchers including Paul Jolissaint, Jacek Brodzki, Chris Higson, and Guoliang Yu. It is stable under induction of representations as in work of Bernard Bekka and under measured groupoid equivalence treated by Alain Connes and Jean Renault. Certain crossed product constructions for C*-algebras and von Neumann algebras reflect preservation or failure of the property, with counterexamples analyzed by Sorin Popa and Narutaka Ozawa.
Connections to other approximation properties
The Haagerup property interacts with the weak amenability of groups studied by Cowling, the completely bounded approximation property for C*-algebras investigated by Uffe Haagerup and Eric Ricard, and with the Baum–Connes conjecture framework advanced by Paul Baum and Alain Connes. Relations to property (T) of Kazhdan and to exactness and nuclearity in operator algebras (themes in work by Elliott George, Kirchberg, Connes and Brown Ozawa) clarify obstructions and compatibilities. Connections to coarse embeddability into Hilbert space, developed by Guoliang Yu and Gromov, tie the Haagerup property to uniform Roe algebra phenomena explored by John Roe.
Applications in operator algebras and geometric group theory
In operator algebras the Haagerup property yields structural consequences for reduced group C*-algebras and group von Neumann algebras: it implies absence of certain Cartan subalgebras in contexts studied by Ozawa Sorin and Popa Sorin, provides approximation properties used in classification programs of Elliott George, and contributes to calculation of K-theory in work of Higson Eric and Gennadi Kasparov. In geometric group theory the property implies coarse embeddability into Hilbert space with applications to the Novikov conjecture via Guoliang Yu and to rigidity phenomena investigated by Margulis, Zimmer Robert, and Gromov; it also informs random walk behavior on groups studied by Varopoulos, Woess Wolfgang and links to percolation and probabilistic methods used by Itai Benjamini.
History and open problems
Introduced in analytic form by Uffe Haagerup in the late 1970s, the property was popularized through connections to Connes Alain’s classification questions and to geometric group theory by Gromov and Yu Guoliang; subsequent breakthroughs involved work of Jolissaint Paul, Brown}}, Ozawa, Brodzki, and Bekka. Open problems include characterization of all groups with the property among higher-rank lattices, existence of nontrivial permanence obstructions in exotic group constructions by Lafforgue Valentin, and understanding implications for type classification of von Neumann algebras related to conjectures posed by Connes Alain and Kadison Richard. Further directions concern quantitative metrics on approximation and relations to Banach space representations investigated by Bruno Melleray and Bader],] with ongoing research by many institutions such as Institute for Advanced Study, Mathematical Sciences Research Institute, and Clay Mathematics Institute.