Indirect G space

In mathematics , the concept of indirect group action or the indirect G-space (English amenable action ) is particularly important in the theory of operator algebras .

definition

Let be a topological group and a measurable G-space , i.e. H. a measurement space with a measurable group effect . An invariant means on a linear functional ${\ displaystyle G}$ ${\ displaystyle X}$ ${\ displaystyle X}$

{\ displaystyle m \ colon L ^ {\ infty} (X) {\ overline {\ otimes}} l ^ {\ infty} (G) \ to L ^ {\ infty} (X)}

with and for , which is invariant under the action of the group , for which is valid for all and the function defined by . ${\ displaystyle m (f \ otimes 1) = f}$ ${\ displaystyle m (f_ {1} \ otimes f_ {2}) \ geq 0}$ ${\ displaystyle f_ {1}, f_ {2} \ geq 0}$ ${\ displaystyle G}$ ${\ displaystyle m (g ^ {*} f) = m (f)}$ ${\ displaystyle g \ in G, f = f_ {1} \ otimes f_ {2} \ in L ^ {\ infty} (X) \ otimes l ^ {\ infty} (G)}$ ${\ displaystyle g ^ {*} (f_ {1} \ otimes f_ {2}) (x, h) = f_ {1} \ otimes f_ {2} (g ^ {- 1} x, g ^ {- 1 }H)}$ ${\ displaystyle g ^ {*} f}$

A space is called indirect if there is an invariant means. ${\ displaystyle G}$

Examples

If there is an indirect group , then every space is indirect. ${\ displaystyle G}$ ${\ displaystyle G}$

A free effect of a group is indirect if is indirect.

{\ displaystyle G}

{\ displaystyle G}

The effect of a hyperbolic group on its edge at infinity is indirect.

Be and locally compact groups . If there is an indirect group, then the effect of each discrete subgroup is indirect. ${\ displaystyle G}$ ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle \ Gamma \ subset G}$ ${\ displaystyle G / P}$

literature

Robert Zimmer : Amenable ergodic Group actions and application to Poisson boundaries of random walks. J. Funct. Anal. 1978, 27: 350-372.
C. Anantharaman-Delaroche, J. Renault: Amenable groupoids. With a foreword by G. Skandalis and Appendix B by E. Germain. Monographies de L'Enseignement Mathématique 36, Geneva 2000.
NP Brown, N. Ozawa: C ^* -algebras and finite dimensional approximations. Graduate Studies in Mathematics, vol. 88, 2008.