Amenability

Definition

Amenability is a property of a von Neumann algebra that indicates the existence of a faithful normal state which can be approximated by states that are invariant under a given action. This concept is important in understanding how algebras behave with respect to their structure and representation theory. It also has implications in various areas, including operator algebras, noncommutative geometry, and the study of subfactors.

5 Must Know Facts For Your Next Test

1. Amenable von Neumann algebras have the property that every action on a compact space admits an invariant state.
2. The hyperfinite II$_1$ factor is a prime example of an amenable algebra, as it can be constructed from finite-dimensional algebras.
3. Amenability is closely linked to the notion of approximation properties, particularly in the context of subfactors and their relationships.
4. In the context of noncommutative Lp spaces, amenability can influence the behavior and representation of these spaces in terms of operators.
5. A fundamental result is that if a group is amenable, then the group von Neumann algebra associated with it is also amenable.

Review Questions

• How does amenability relate to the Connes cocycle derivative and its applications in operator algebras?
  • Amenability is significant when discussing the Connes cocycle derivative because it helps determine whether certain states can be approximated by invariant ones. The cocycle derivative essentially measures how far a given cocycle is from being a coboundary. If an algebra is amenable, it provides a context where this measurement can yield valuable insights into the representation theory and behavior of actions on the algebra.
• In what ways does amenability impact the structure and classification of hyperfinite factors?
  • Amenability directly impacts hyperfinite factors as these factors are defined to be type II$_1$ and have well-defined structural properties allowing them to be approximated by finite-dimensional algebras. This approximation aspect highlights their amenable nature, making hyperfinite factors essential examples when studying amenability in von Neumann algebras. The classification of these factors often relies on their amenability properties, influencing their representation and construction.
• Evaluate the implications of amenability on noncommutative Lp spaces and their applications in modern mathematics.
  • The implications of amenability on noncommutative Lp spaces are significant as they dictate how these spaces behave under various operations, including integration and limits. Amenable algebras allow for more controlled environments in which these spaces operate, leading to better representation results and potential applications in quantum mechanics and statistical mechanics. Furthermore, understanding amenability aids in bridging classical probability theory with noncommutative geometry, offering insights into how both fields intersect in modern mathematical research.

Related terms

Connes Cocycle Derivative: A tool that measures the deviation of a cocycle from being a coboundary, allowing for the analysis of amenable actions in operator algebras.

Hyperfinite Factors: Type II$_1$ factors that can be approximated by finite-dimensional algebras, which are amenable due to their well-behaved structure.

Noncommutative Lp Spaces: Generalizations of classical Lp spaces in the setting of noncommutative probability, which can exhibit amenability properties depending on the algebra.