2.6 Comparison theory of projections

Comparison theory of projections is a crucial tool for analyzing von Neumann algebras. It establishes a framework for comparing and classifying projections, which are fundamental building blocks in these algebraic structures.

This theory introduces concepts like partial ordering, Murray-von Neumann equivalence, and subequivalence. These ideas help distinguish between finite and infinite projections, ultimately leading to the classification of von Neumann algebras into different types.

Definition of projections

• Projections serve as fundamental building blocks in von Neumann algebras, representing self-adjoint idempotent operators
• Understanding projections forms the basis for analyzing the structure and properties of von Neumann algebras

Properties of projections

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• Self-adjoint operators satisfying $P = P^* = P^2$
• Orthogonal projections map vectors onto closed subspaces
• Spectral theorem decomposes self-adjoint operators into projections
• Range of a projection forms a closed subspace of the Hilbert space

Types of projections

• Orthogonal projections project vectors perpendicularly onto subspaces
• Oblique projections project vectors non-perpendicularly onto subspaces
• Finite-rank projections have finite-dimensional ranges
• Infinite-rank projections have infinite-dimensional ranges
• Minimal projections cannot be further decomposed into smaller non-zero projections

Partial ordering of projections

• Partial ordering of projections establishes a hierarchical structure within von Neumann algebras
• This ordering plays a crucial role in comparing and classifying different types of von Neumann algebras

Comparison of projections

• Projections P and Q are compared using the partial order relation $P \leq Q$
• $P \leq Q$ means the range of P is contained in the range of Q
• Equivalent definition $PQ = QP = P$
• Partial ordering allows for the comparison of "sizes" of projections
• Forms the basis for Murray-von Neumann equivalence and subequivalence relations

Equivalence of projections

• Two projections P and Q are equivalent if there exists a partial isometry V such that $V^*V = P$ and $VV^* = Q$
• Equivalence relation denoted by $P \sim Q$
• Preserves the "size" of projections in a more general sense than equality
• Allows for comparison of projections that may not be directly comparable through partial ordering

Murray-von Neumann equivalence

• Murray-von Neumann equivalence provides a fundamental tool for classifying and comparing projections in von Neumann algebras
• This concept extends the notion of dimension to infinite-dimensional spaces

Definition and properties

• Two projections P and Q are Murray-von Neumann equivalent if there exists a partial isometry V such that $V^*V = P$ and $VV^* = Q$
• Denoted by $P \sim Q$
• Reflexive $P \sim P$
• Symmetric if $P \sim Q$ then $Q \sim P$
• Transitive if $P \sim Q$ and $Q \sim R$ then $P \sim R$
• Preserves algebraic and topological properties of projections

Relation to partial ordering

• Murray-von Neumann equivalence refines the partial ordering of projections
• If $P \leq Q$ and $P \sim Q$, then $P = Q$
• Allows for comparison of projections that may not be directly comparable through partial ordering
• Crucial for defining and studying finite and infinite projections

Comparison theory fundamentals

• Comparison theory forms the backbone of the structural analysis of von Neumann algebras
• Provides tools to classify and distinguish different types of von Neumann algebras based on their projection lattices

Comparison theorem

• States that for any two projections P and Q in a von Neumann algebra, exactly one of the following holds
  • P is equivalent to a subprojection of Q
  • Q is equivalent to a proper subprojection of P
  • P and Q are equivalent
• Allows for a complete comparison of any two projections
• Forms the basis for the classification of von Neumann algebras into types

Schröder-Bernstein property

• Von Neumann algebras with the Schröder-Bernstein property satisfy if $P \preceq Q$ and $Q \preceq P$, then $P \sim Q$
• Named after the Schröder-Bernstein theorem in set theory
• Not all von Neumann algebras possess this property
• Crucial for understanding the structure of certain types of von Neumann algebras (Type II1 factors)

Finite vs infinite projections

• The distinction between finite and infinite projections plays a crucial role in the classification of von Neumann algebras
• This concept extends the notion of finiteness to infinite-dimensional spaces

Characterization of finiteness

• A projection P is finite if $P \sim Q \leq P$ implies $P = Q$
• Finite projections cannot be equivalent to any of their proper subprojections
• Characterized by the absence of proper subprojections equivalent to the whole projection
• Finiteness is preserved under Murray-von Neumann equivalence

Examples of infinite projections

• Identity operator in B(H) for infinite-dimensional Hilbert space H
• Projections onto infinite-dimensional subspaces in B(H)
• Sum of infinitely many mutually orthogonal non-zero projections
• Projections in type III factors are always infinite
• Infinite projections in type II∞ factors

Subequivalence and proper subequivalence

• Subequivalence and proper subequivalence refine the notion of equivalence between projections
• These concepts are essential for understanding the structure of von Neumann algebras

Definitions and properties

• P is subequivalent to Q (denoted $P \preceq Q$) if P is equivalent to a subprojection of Q
• Proper subequivalence (denoted $P \prec Q$) occurs when $P \preceq Q$ but $P \nsim Q$
• Subequivalence is reflexive and transitive but not necessarily symmetric
• Proper subequivalence is irreflexive and transitive
• Allows for finer comparisons between projections than Murray-von Neumann equivalence alone

Relation to partial ordering

• Subequivalence extends the partial ordering of projections
• If $P \leq Q$, then $P \preceq Q$, but the converse may not hold
• Proper subequivalence implies strict inequality in the partial ordering
• Crucial for defining and studying comparability of projections in different types of von Neumann algebras

Central carrier of projections

• The central carrier of a projection plays a significant role in understanding the global structure of von Neumann algebras
• This concept connects individual projections to the center of the algebra

Definition and significance

• Central carrier of a projection P denoted by C(P)
• Smallest central projection Q such that $PQ = P$
• Represents the "support" of P in the center of the von Neumann algebra
• Provides information about the global structure of the algebra
• Crucial for understanding the relationship between projections and the center of the algebra

Properties of central carriers

• C(P) is always a central projection
• $P \leq C(P)$ for any projection P
• If P and Q are equivalent projections, then $C(P) = C(Q)$
• For any central projection Z $C(PZ) = C(P)Z$
• Central carrier of a sum of projections equals the sum of their central carriers

Comparison in factors

• Comparison theory in factors simplifies due to the trivial center, leading to distinct properties for each factor type
• Understanding comparison in factors is crucial for the classification of von Neumann algebras

Type I factors

• Isomorphic to B(H) for some Hilbert space H
• Projections are characterized by their rank (dimension of range)
• Two projections are equivalent if and only if they have the same rank
• Comparison is complete every projection is comparable to every other projection

Type II factors

• Divided into Type II1 (finite) and Type II∞ (infinite) factors
• Type II1 factors possess a unique tracial state
• In Type II1 factors, projections are compared using the trace
• Type II∞ factors combine properties of Type II1 and Type I∞ factors
• Continuous range of projection "sizes" unlike Type I factors

Type III factors

• All non-zero projections are infinite and Murray-von Neumann equivalent
• No trace exists on Type III factors
• Comparison theory simplifies all non-zero projections are equivalent
• Further classified into Type III λ (0 ≤ λ ≤ 1) based on their modular automorphism groups

Applications of comparison theory

• Comparison theory serves as a powerful tool in the analysis and classification of von Neumann algebras
• Its applications extend to various areas of mathematics and mathematical physics

Classification of von Neumann algebras

• Fundamental tool for distinguishing between different types of von Neumann algebras
• Helps identify Type I, II, and III factors based on properties of their projections
• Crucial in the development of the Murray-von Neumann classification
• Allows for the study of structural properties of von Neumann algebras
• Provides insights into the representation theory of groups and algebras

Dimension theory

• Extends the notion of dimension to infinite-dimensional spaces
• Allows for the comparison of "sizes" of infinite-dimensional subspaces
• Crucial in the development of noncommutative geometry
• Applications in quantum mechanics and quantum field theory
• Provides a framework for studying infinite-dimensional phenomena in functional analysis

Advanced comparison concepts

• Advanced comparison concepts extend the basic theory to more specialized settings
• These concepts are crucial for dealing with specific types of von Neumann algebras and their applications

Comparison relative to a trace

• Utilizes tracial states to compare projections in II1 factors
• Defines a dimension function for projections based on the trace
• Allows for a continuous range of projection "sizes" in II1 factors
• Crucial for the study of II1 factors and their applications in free probability theory
• Extends to semifinite von Neumann algebras with faithful normal semifinite traces

Comparison in continuous geometries

• Deals with projection lattices that form continuous geometries
• Extends comparison theory to more general structures than von Neumann algebras
• Connects to the theory of quantum logic and foundations of quantum mechanics
• Provides insights into the structure of certain types of operator algebras
• Applications in the study of quantum probability and quantum information theory