3.6 Amenability in von Neumann algebras

Amenability in von Neumann algebras extends the concept from group theory to operator algebras. It characterizes structures with certain averaging properties, crucial for understanding their structural properties and classification.

This topic bridges group theory, functional analysis, and operator algebras. It explores equivalent formulations, including the Følner condition and invariant means, and examines amenable groups, injective von Neumann algebras, and related approximation properties.

Definition of amenability

• Amenability characterizes mathematical structures exhibiting certain averaging properties
• Concept originated in group theory but extends to von Neumann algebras and other mathematical objects
• Plays crucial role in understanding structural properties of von Neumann algebras

Historical context

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• Introduced by John von Neumann in 1929 to study Banach-Tarski paradox
• Initially defined for groups to distinguish between "paradoxical" and "non-paradoxical" groups
• Extended to von Neumann algebras by Alain Connes in the 1970s
• Bridges group theory, functional analysis, and operator algebras

Equivalent formulations

• Existence of a finitely additive, left-invariant probability measure on the group
• Følner condition: existence of almost invariant finite subsets
• Fixed point property for affine actions on compact convex sets
• Existence of an invariant mean on the space of bounded functions
• Reiter's condition: existence of approximately invariant vectors in $L^p$ spaces

Følner condition

• Provides geometric interpretation of amenability for discrete groups
• Generalizes to locally compact groups and von Neumann algebras
• Crucial in proving amenability of certain classes of groups and algebras

Finite subsets

• Følner sequence consists of finite subsets $F_n$ of a group G
• $\frac{|gF_n \Delta F_n|}{|F_n|} \to 0$ as $n \to \infty$ for all $g \in G$
• Measures how much a set changes when shifted by group elements
• Existence of Følner sequence equivalent to amenability for discrete groups
• Generalizes to continuous setting using Haar measure for locally compact groups

Approximation property

• Følner condition implies existence of approximately invariant finite-dimensional subspaces
• For von Neumann algebras, approximation property involves ultraweakly dense subalgebras
• Connects to hyperfiniteness and injectivity in operator algebras
• Allows construction of asymptotically invariant states or traces

Invariant means

• Linear functionals on the space of bounded functions satisfying certain invariance properties
• Generalize concept of average to infinite groups or spaces
• Crucial in defining amenability for general locally compact groups

Existence of invariant means

• Equivalent to amenability for groups and von Neumann algebras
• Proved using fixed point theorems or ultrafilter constructions
• For groups, invariant mean $m$ satisfies $m(f) = m(L_g f)$ for all $g \in G$ and bounded functions $f$
• In von Neumann algebras, invariant mean defined on the dual space $M_*$

Properties of invariant means

• Positivity: maps non-negative functions to non-negative real numbers
• Normalization: $m(1) = 1$ for constant function 1
• Left-invariance: $m(L_g f) = m(f)$ for all group elements $g$ and functions $f$
• Not unique in general, forms convex set
• Connects to ergodic theory through ergodic averages

Amenable groups

• Groups satisfying equivalent conditions of amenability
• Include all finite groups, abelian groups, and solvable groups
• Closed under various group-theoretic operations (subgroups, quotients, extensions)

Examples of amenable groups

• Finite groups: trivially amenable due to existence of normalized counting measure
• Abelian groups: amenable by Følner sequence construction
• Solvable groups: amenable by induction on derived series
• Subexponential growth groups (nilpotent groups)
• Locally finite groups
• Elementary amenable groups: smallest class containing finite and abelian groups, closed under group operations

Non-amenable groups

• Free groups on two or more generators: first example of non-amenable groups
• Groups containing free subgroups (SL(2, ℤ))
• Groups with Kazhdan's property (T) (SL(n, ℤ) for n ≥ 3)
• Gromov's random groups
• Thompson's group F (conjectured to be non-amenable)
• Demonstrate limitations of averaging techniques in group theory

Amenability in von Neumann algebras

• Generalizes group amenability to operator algebraic setting
• Characterizes von Neumann algebras with certain approximation properties
• Connects to injectivity, hyperfiniteness, and other structural properties

Operator algebraic formulation

• Existence of hypertrace: state on $B(H)$ extending the trace on the von Neumann algebra
• Approximately finite-dimensional: existence of increasing net of finite-dimensional subalgebras
• Existence of conditional expectations onto arbitrary von Neumann subalgebras
• Semidiscreteness: existence of normal conditional expectations onto finite-dimensional subalgebras

Connes' characterization

• Equivalence of amenability and injectivity for von Neumann algebras (Connes' theorem)
• Characterization in terms of completely positive maps approximating identity
• Connection to nuclearity in C*-algebras
• Implications for classification of factors and ergodic theory

Injective von Neumann algebras

• Class of von Neumann algebras with strong structural properties
• Equivalent to amenability in the von Neumann algebra setting
• Crucial in classification of factors and study of operator algebraic quantum groups

Definition of injectivity

• Von Neumann algebra M injective if there exists conditional expectation $E: B(H) \to M$
• Conditional expectation E must be normal and preserve the identity
• Generalizes notion of averaging in group theory to operator algebraic setting
• Equivalent to existence of completely positive projection of norm one onto M

Relation to amenability

• Injectivity equivalent to amenability for von Neumann algebras (Connes' theorem)
• Connects group-theoretic and operator algebraic notions of amenability
• Implies hyperfiniteness for II₁ factors
• Characterizes amenable groups through their group von Neumann algebras

Amenable traces

• Generalization of amenability to arbitrary traces on C*-algebras
• Connects amenability of groups to properties of their reduced group C*-algebras
• Crucial in studying approximation properties of operator algebras

Definition and properties

• Trace τ on C*-algebra A amenable if extends to state on $A^{**}$ factoring through $A^{**}_τ$
• Equivalent to existence of approximate diagonal in $A \otimes A^{op}$
• For group C*-algebras, amenable trace equivalent to amenability of group
• Preserved under various operations (quotients, inductive limits, tensor products)

Applications in von Neumann algebras

• Characterizes amenability of II₁ factors through their canonical trace
• Connects to Connes' embedding problem for II₁ factors
• Used in classification of amenable II₁ factors (all hyperfinite)
• Implications for approximation properties of group von Neumann algebras

Hyperfiniteness vs amenability

• Closely related concepts in von Neumann algebra theory
• Coincide for II₁ factors but differ in general
• Illustrate subtle differences between approximation properties in various settings

Similarities and differences

• Both involve approximation by finite-dimensional subalgebras
• Hyperfiniteness requires increasing net of finite-dimensional subalgebras
• Amenability allows more general approximation by completely positive maps
• Equivalent for II₁ factors (Connes' theorem)
• Differ for type III factors and non-separable von Neumann algebras

Examples and counterexamples

• All hyperfinite von Neumann algebras amenable
• Amenable II₁ factor always hyperfinite (R)
• Non-hyperfinite amenable von Neumann algebras exist in type III case
• Free group factors non-amenable but embed into ultrapower of hyperfinite II₁ factor
• Continuum many non-isomorphic amenable II₁ factors constructed using free probability

Approximation properties

• Generalizations and variations of amenability for groups and operator algebras
• Capture different aspects of finite-dimensional approximations
• Crucial in understanding structure of non-amenable objects

Weak amenability

• Defined for groups and Banach algebras
• Allows bounded approximate identities instead of norm-one approximations
• Free groups weakly amenable but not amenable
• Connected to Cowling-Haagerup constant in harmonic analysis

Relative amenability

• Amenability of a subgroup or subalgebra relative to larger structure
• Defined using conditional expectations or relative invariant means
• Crucial in study of inclusions of von Neumann algebras
• Applications in subfactor theory and ergodic theory of group actions

Applications of amenability

• Wide-ranging implications across mathematics
• Connects abstract structural properties to concrete phenomena
• Crucial in classification and structural results in various fields

In ergodic theory

• Characterizes existence of invariant measures for group actions
• Connects to orbit equivalence and measured group theory
• Applications in Ornstein isomorphism theory for Bernoulli shifts
• Implications for amenable equivalence relations and measured foliations

In operator algebras

• Classification of amenable von Neumann algebras (hyperfinite II₁ factor unique)
• Injectivity and nuclearity in C*-algebra theory
• Applications in noncommutative geometry and quantum groups
• Connections to index theory and K-theory through various crossed products

Open problems

• Unresolved questions connecting amenability to other areas of mathematics
• Highlight deep connections between different mathematical structures
• Drive research in operator algebras, group theory, and ergodic theory

Connes' embedding conjecture

• Asks whether all separable II₁ factors embed into ultrapower of hyperfinite II₁ factor
• Equivalent to various statements in quantum information theory and group theory
• Connected to Kirchberg's QWEP conjecture and Tsirelson's problem
• Recent breakthrough: conjecture shown to be false (MIP* = RE result)

Other related conjectures

• Novikov conjecture: relates to amenability of Cayley graphs of groups
• Baum-Connes conjecture: connects group amenability to K-theory of group C*-algebras
• Von Neumann-Day problem: characterization of non-amenable groups without free subgroups
• Dixmier problem: uniqueness of amenable traces on C*-algebras of non-amenable groups