Amenable group

Indirect group is a concept from the mathematical branch of harmonic analysis. It is to locally compact groups on which some averaging function, a so-called agent exists.

Definition

It is a locally compact group. On it are known to be a hair cal measure. These are understood to the room of Maßraums, ie the vector space of the limited functions, with almost all matching functions are identified.

For a function defined on an element and is given by.

A continuous linear functional is called to an agent if the following holds

• , Where the 1 is on the left side for the constant one function,
• For all with ( that is, for all)
• For all and.

The first two properties just say that state. The third property is also called Linksinvarianz.

The group is called indirectly, if there is a means.

Examples

• Compact groups are indirectly, the normalized Haar measure on one is an agent.
• Commutative locally compact groups are indirectly. An agent can not specify directly in the non - compact case, the proof requires a non- constructive fixed-point theorem.
• Quotients and closed subgroups indirect groups are indirectly.
• If H is a normal subgroup of G and if H and G / H are indirect, then G is indirectly. In particular, locally compact solvable groups are indirectly.
• The group of two elements freely generated is the prototypical example of a non - indirect group.
• A group with property T if and only indirectly, if it is compact.
• A hyperbolic group if and only indirectly, if elementary hyperbolic, ie is finite or virtually.

Permanenzeigenschaften

• Completed subgroups indirect groups are indirectly again.
• Is a closed normal subgroup of indirect group, the factor group is indirectly.
• It is a closed normal subgroup of a locally compact group and its indirect and, then also indirectly.

Importance

The representation theory of locally compact groups by means of C *-algebras is accessible for indirect groups. Identifies the group C * - algebra, the reduced group C * - algebra and the left regular representation, then by a set of Andrzej Hulanicki following statements on a locally compact group equivalent:

• Is indirectly.
• The left regular representation is an isomorphism.

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

Group C *-algebras indirect groups are nuclear, for discrete groups the converse is also true.

Comments

Invariant measures have been introduced by John von Neumann. An accessible introduction to the theory of indirect groups is the book by Fredrick Greenleaf, which also contains complete proofs above Permanenzeigenschaften. The so-called von Neumann conjecture, according to which any non- indirect group contains a subgroup isomorphic to, has been disproved in 1980 by Olshansky.