Indirect group

Indirect group is a term from the mathematical branch of harmonic analysis . These are locally compact groups on which a certain averaging function, a so-called mean, exists.

The term was introduced in 1929 by John von Neumann , who noted that the Banach-Tarski paradox could be explained by the impossibility of a remedy on non-Abelian free groups . As a result, it turned out that the mediability of locally compact groups is equivalent to numerous fundamental properties from harmonic analysis: the Følner criterion , the fixed point property or the condition that the regular representation contains the trivial representation weakly.

definition

Let it be a locally compact group. As is well known, there is a hairy measure . One understands the space of the measure space , i.e. H. the vector space of bounded functions, identifying matching functions almost everywhere. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle L^{\infty }(G)}$ ${\displaystyle L^{\infty }}$ ${\displaystyle (G,\mu )}$

For on -defined function and an element is by defined. ${\displaystyle G}$ ${\displaystyle f:G\rightarrow X}$ ${\displaystyle s\in G}$ ${\displaystyle f_{s}:G\rightarrow X}$ ${\displaystyle f_{s}(t):=f(s^{-1}t)}$

A continuous linear functional is a means to , if true ${\displaystyle m:L^{\infty }(G)\rightarrow \mathbb {C} }$ ${\displaystyle G}$

• ${\displaystyle m(1)=1}$, where the 1 on the left stands for the constant one function,
• ${\displaystyle m(f)\geq 0}$for all with (i.e. for all ), ${\displaystyle f\in L^{\infty }(G)}$ ${\displaystyle f\geq 0}$ ${\displaystyle f(t)\geq 0}$ ${\displaystyle t\in G}$
• ${\displaystyle m(f_{s})\,=\,m(f)}$for everyone and . ${\displaystyle f\in L^{\infty }(G)}$ ${\displaystyle s\in G}$

The first two properties just state that a state is. The third property is also called left invariance. ${\displaystyle m}$

The group is called indirect if there is a means to open. ${\displaystyle G}$ ${\displaystyle G}$

Examples

• Compact groups are indirect, the Haar measure standardized to 1 is a mean.
• Commutative locally compact groups are indirect. A means cannot be given directly in the non-compact case; the proof requires a non-constructive fixed point theorem .
• Locally compact resolvable groups are indirect.
• The group freely generated by two elements is the prototypical example of a non-indirect group. ${\displaystyle \mathbb {F} _{2}}$
• A group with property T is indirect if and only if it is compact.
• A hyperbolic group is indirect if and only if it is elementary hyperbolic, i.e. H. is finite or virtual . ${\displaystyle \mathbb {Z} }$

Permanent properties

• Closed subgroups of indirect groups are again indirect.
• If there is a closed normal divisor of an indirect group , the factor group is also indirect. ${\displaystyle H\subset G}$ ${\displaystyle G}$ ${\displaystyle G/H}$
• Let it be a closed normal divisor of a locally compact group and and be indirect, then is also indirect. ${\displaystyle H\subset G}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G/H}$ ${\displaystyle G}$

meaning

The representation theory of locally compact groups using C * -algebras is more accessible for indirect groups. If the group C * algebra , the reduced group C * algebra and the left-regular representation denote, the following statements about a locally compact group are equivalent according to a theorem by Andrzej Hulanicki : ${\displaystyle C^{*}(G)}$ ${\displaystyle C_{r}^{*}(G)}$ ${\displaystyle \lambda :C^{*}(G)\rightarrow C_{r}^{*}(G)}$ ${\displaystyle G}$

• ${\displaystyle G}$ is indirect.
• The left-regular representation is an isomorphism . ${\displaystyle \lambda :C^{*}(G)\rightarrow C_{r}^{*}(G)}$

A generalization of this theorem says that the entangled product of a C * -algebra and a locally compact group coincides with the reduced version of the entangled product.

Group C * algebras of indirect groups are nuclear ; the converse applies to discrete groups.

Remarks

Invariant measures were introduced by John von Neumann . An easily accessible introduction to the theory of indirect groups is the book by Fredrick Greenleaf , there is also complete evidence of the above permanence properties. The so-called Von Neumann conjecture , according to which every non-indirect group contains a subgroup that is too isomorphic, was refuted by Alexander Olschanski in 1980 . ${\displaystyle \mathbb {F} _{2}}$

literature

• A. Paterson: Amenability. Mathematical Surveys and Monographs, 29. American Mathematical Society, Providence, RI, 1988. ISBN 0-8218-1529-6

Web links

• Juschenko: Lecture Notes on "Amenability"
• Ozawa: Amenable actions and applications

Individual evidence

1. ↑ Gert K. Pedersen: C * -Algebras and their Automorphism Groups (= LMS Monographs. Vol. 14). Academic Press, London et al. 1979, ISBN 0-12-549450-5 , 7.3.3.
2. ↑ Kenneth R. Davidson: C * -Algebras by Example (= Fields Institute Monographs. Vol. 6). American Mathematical Society, Providence RI 1996, ISBN 0-8218-0599-1 , Corollary VII.2.2.
3. ↑ Kenneth R. Davidson: C * -Algebras by Example (= Fields Institute Monographs. Vol. 6). American Mathematical Society, Providence RI 1996, ISBN 0-8218-0599-1 , Example VII.2.4.
4. ↑ Gert K. Pedersen: C * -Algebras and their Automorphism Groups (= LMS Monographs. Vol. 14). Academic Press, London et al. 1979, ISBN 0-12-549450-5 , Theorem 7.3.9.
5. ^ Andrzej Hulanicki : Means and Følner conditions on locally compact groups. In: Studia Mathematica . Vol. 27, No. 2, 1966, pp. 87-104, online .
6. ↑ Gert K. Pedersen: C * -Algebras and their Automorphism Groups (= LMS Monographs. Vol. 14). Academic Press, London et al. 1979, ISBN 0-12-549450-5 , Theorem 7.7.7.
7. ↑ Christopher Lance: On Nuclear C * -Algebras. In: Journal of Functional Analysis. Vol. 12, No. 2, 1973, pp. 157-176, doi : 10.1016 / 0022-1236 (73) 90021-9 , Theorem 4.2.
8. ↑ John von Neumann : To the general theory of the mass. In: Fundamenta Mathematicae . Vol. 13, 1929, pp. 73-116, online ; Addendum to the work “On the general theory of measure”. Vol. 13, 1929, p. 333, online .
9. ↑ Fredrick P. Greenleaf: Invariant Means on Topological Groups and their Applications (= Van Nostrand Mathematical Studies. Vol. 16, ZDB -ID 793375-7 ). Van Nostrand Reinhold, New York et al. 1969, ISBN 0-442-02857-1 .
10. ↑ Александр Ю. Ольшанский: К Вопросу о Существовании инвариантного Среднего на Группе. In: Успехи Математических Наук. Vol. 35, No. 4 = 214, 1980, ISSN  0042-1316 , pp. 199-200, online , ( On questions about the existence of invariant agents on a group. ).