4.3 Weights and traces

Weights and traces are fundamental concepts in von Neumann algebra theory. They generalize positive linear functionals, allowing for infinite values and unbounded operators. These tools are crucial for understanding noncommutative measure theory and integration in operator algebras.

Weights come in various types, including normal, semifinite, and faithful. Traces are special weights with additional symmetry properties. Both play essential roles in classifying von Neumann algebras, constructing representations, and developing noncommutative integration theories.

Definition of weights

• Weights generalize positive linear functionals in von Neumann algebras
• Crucial for understanding noncommutative measure theory and integration in operator algebras

Positive linear functionals

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• Maps from a von Neumann algebra to non-negative real numbers
• Preserve algebraic structure and positivity
• Bounded and continuous in the weak* topology
• Represent states of quantum systems (pure states)

Extended real-valued functions

• Allow infinite values, extending domain to [0, ∞]
• Necessary for handling unbounded operators in von Neumann algebras
• Preserve order structure of the algebra
• Enable description of infinite-dimensional systems

Properties of weights

• Lower semicontinuity in the weak* topology
• Additivity: $w(x + y) = w(x) + w(y)$ for positive elements x, y
• Homogeneity: $w(λx) = λw(x)$ for λ ≥ 0 and positive x
• Monotonicity: x ≤ y implies $w(x) ≤ w(y)$
• Extend to complex linear functionals on their domains

Types of weights

• Classification of weights based on specific properties
• Essential for understanding structure of von Neumann algebras

Normal weights

• Preserve suprema of increasing nets of positive operators
• Correspond to σ-weakly continuous linear functionals
• Characterized by $w(\sup_i x_i) = \sup_i w(x_i)$ for increasing nets {x_i}
• Play crucial role in the standard form of von Neumann algebras

Semifinite weights

• Have dense domains in the positive cone of the algebra
• Allow approximation of general elements by elements with finite weight
• Characterized by the existence of an increasing net of projections {p_i} with $w(p_i) < ∞$ and $\sup_i p_i = 1$
• Essential for constructing crossed products and in modular theory

Faithful weights

• Assign non-zero values to all non-zero positive elements
• Detect algebraic structure of von Neumann algebras
• Characterized by $w(x) = 0$ implies x = 0 for positive x
• Used in the construction of modular automorphism groups

Traces

• Special class of weights with additional symmetry properties
• Fundamental in the study of von Neumann algebras and noncommutative geometry

Definition of traces

• Weights satisfying $τ(x^*x) = τ(xx^*)$ for all elements x
• Generalize the notion of trace from matrix algebras
• Invariant under unitary conjugation: $τ(uxu^*) = τ(x)$ for unitaries u
• Uniquely determined by their values on projections

Properties of traces

• Finite traces form a norm-closed face in the predual of the algebra
• Tracial states define invariant expectations onto subalgebras
• Extend to linear functionals on the whole algebra
• Characterize hyperfinite factors of type II_1

Normal traces

• Traces that are also normal weights
• Correspond to ultraweakly continuous linear functionals
• Uniquely determine the type II factor structure
• Used in the construction of standard forms for von Neumann algebras

Weights vs traces

• Comparison of general weights and traces in von Neumann algebra theory
• Highlights the importance of both concepts in different contexts

Key differences

• Traces possess additional symmetry (cyclic property)
• Weights allow for infinite values, traces are often normalized
• Traces characterize type II factors, weights are essential for type III
• Weights have non-trivial modular automorphism groups, traces do not

Similarities and connections

• Both generalize positive linear functionals
• Used to define noncommutative integration theories
• Play crucial roles in the classification of von Neumann algebras
• Connect to representation theory through GNS construction

Construction of weights

• Methods for obtaining weights on von Neumann algebras
• Essential for understanding the structure of operator algebras

From positive operators

• Use spectral theorem to define weight as $w(x) = τ(hx)$ for positive h
• Allows construction of weights with desired properties
• Connects to spatial theory of von Neumann algebras
• Generalizes construction of normal states from density operators

From states and functionals

• Extend bounded functionals to weights using Radon-Nikodym derivatives
• Construct weights as suprema of increasing nets of positive functionals
• Utilize Hahn-Banach theorem for extensions to larger domains
• Relates to the theory of operator-valued weights

Radon-Nikodym theorem

• Fundamental result relating different weights and traces
• Generalizes classical measure theory to noncommutative setting

For weights

• Expresses one weight as "derivative" of another: $w_1(x) = w_2(hx)$
• h is an unbounded positive operator affiliated with the algebra
• Allows comparison and classification of weights
• Essential in the theory of noncommutative Lp spaces

For traces

• Simpler form due to additional symmetry of traces
• Radon-Nikodym derivative is a positive element in the algebra
• Characterizes absolutely continuous traces
• Used in the study of type II von Neumann algebras

Polar decomposition

• Decomposition of weights and traces into positive and partial isometry parts
• Analogous to polar decomposition of operators

Of weights

• Expresses weight as composition of partial isometry and positive weight
• Allows study of support projections and kernels of weights
• Connects to theory of operator-valued weights
• Used in the analysis of KMS states in quantum statistical mechanics

Of traces

• Simpler form due to trace property
• Decomposes trace into product of partial isometry and positive trace
• Characterizes faithful normal semifinite traces
• Essential in the study of type II factors and their subfactors

Weight theory

• Advanced topics in the study of weights on von Neumann algebras
• Foundational results for modern operator algebra theory

Pedersen-Takesaki theorem

• Characterizes normal semifinite weights on von Neumann algebras
• Expresses weights as suprema of increasing nets of normal positive functionals
• Essential for understanding structure of type III factors
• Connects to Tomita-Takesaki modular theory

Connes cocycle theorem

• Relates modular automorphism groups of different weights
• Introduces Connes cocycle derivative $[Dw_1 : Dw_2]_t$
• Fundamental in the classification of type III factors
• Applications in quantum field theory and statistical mechanics

Applications of weights

• Practical uses of weight theory in various areas of mathematics and physics
• Demonstrates importance of weights beyond abstract algebra

In noncommutative integration

• Define noncommutative Lp spaces using weights
• Generalize classical integration theory to operator algebras
• Develop Fourier analysis on quantum groups
• Construct noncommutative probability spaces

In modular theory

• Define modular automorphism groups using weights
• Study KMS states in quantum statistical mechanics
• Analyze type III factors and their invariants
• Develop theory of operator-valued weights

Trace class operators

• Special class of operators related to normal traces
• Bridge between operator theory and noncommutative integration

Definition and properties

• Operators T with finite trace norm: $\|T\|_1 = \text{Tr}(|T|) < \infty$
• Form a two-sided ideal in B(H)
• Predual of B(H) identified with trace class operators
• Compact operators with absolutely summable singular values

Relation to weights

• Normal weights correspond to densely defined operators affiliated with the algebra
• Trace class operators generate normal semifinite weights
• Used in the construction of standard forms of von Neumann algebras
• Connect to theory of noncommutative Lp spaces

Measurability for weights

• Extends notion of measurability to noncommutative setting
• Essential for developing noncommutative integration theory

Definition of measurable operators

• Operators x with $w((1+|x|)^{-1}) < \infty$ for weight w
• Generalize notion of measurable functions to operator algebras
• Form a *-algebra containing the original von Neumann algebra
• Allow integration of unbounded operators with respect to weights

Measurable domains

• Subspaces of Hilbert space where operators are measurable
• Characterized by properties of spectral projections
• Essential for defining noncommutative Lp spaces
• Connect to theory of affiliated operators

Weight extensions

• Methods for extending weights to larger classes of operators
• Important for developing comprehensive integration theories

To unbounded operators

• Extend weights to densely defined positive operators
• Use spectral theory and monotone convergence
• Essential for handling non-finite weights
• Connect to theory of affiliated operators

To affiliated operators

• Extend weights to operators affiliated with the von Neumann algebra
• Use measurability and approximation by bounded operators
• Important for studying type III factors
• Relate to theory of noncommutative Lp spaces

Spatial theory of weights

• Connects weights to Hilbert space representations
• Fundamental for understanding structure of von Neumann algebras

GNS construction for weights

• Generalizes GNS construction for states to weights
• Produces Hilbert space and *-representation from a weight
• Essential for studying modular theory
• Connects to theory of KMS states in quantum statistical mechanics

Standard form of von Neumann algebras

• Canonical representation of von Neumann algebras using weights
• Incorporates modular theory and Tomita-Takesaki theory
• Unique up to spatial isomorphism
• Essential for studying subfactors and Jones index theory

Weights in classification theory

• Role of weights in classifying von Neumann algebras
• Essential for understanding structure of operator algebras

Type classification

• Use weights to distinguish between types I, II, and III factors
• Type I: admit minimal projections
• Type II: admit semifinite traces
• Type III: admit no semifinite traces, only weights

Factor classification

• Further classify type III factors using modular theory
• Type III_λ (0 ≤ λ ≤ 1) determined by Connes spectrum
• Use flow of weights to distinguish subtypes
• Connect to ergodic theory and noncommutative geometry