10.2 Comparison theory and projections in von Neumann algebras

Von Neumann algebras are a special class of C*-algebras with unique properties. This section dives into comparison theory and projections, key tools for understanding their structure and classification.

We'll explore Murray-von Neumann equivalence, types of projections, and central carriers. These concepts are crucial for grasping how von Neumann algebras are built and categorized.

Equivalence and Comparison

Murray-von Neumann Equivalence and Comparison

• Murray-von Neumann equivalence defines when two projections in a von Neumann algebra are considered equivalent
• Two projections p and q are Murray-von Neumann equivalent if there exists a partial isometry v such that $v^*v = p$ and $vv^* = q$
• Comparison of projections establishes a partial order on projections in a von Neumann algebra
• Projection p is smaller than or equal to q if there exists a projection r such that p is Murray-von Neumann equivalent to r and r ≤ q
• Partial isometries play a crucial role in defining equivalence and comparison of projections
• Partial isometries are operators v satisfying $v = vv^*v$
• Initial projection of a partial isometry v is defined as $v^*v$
• Final projection of a partial isometry v is defined as $vv^*$

Applications and Properties of Equivalence

• Murray-von Neumann equivalence forms an equivalence relation on projections
• Equivalence relation properties include reflexivity, symmetry, and transitivity
• Comparison of projections extends to subspaces via their corresponding projections
• Equivalence and comparison of projections are fundamental in classifying von Neumann algebras
• Hahn-Banach theorem can be used to extend partial isometries in certain cases
• Equivalence of projections preserves various algebraic and topological properties
• Applications include decomposition of von Neumann algebras into direct sums or tensor products

Projection Types

Finite and Infinite Projections

• Finite projections are not equivalent to any proper subprojection of themselves
• Projection p is finite if p ~ q ≤ p implies q = p, where ~ denotes Murray-von Neumann equivalence
• Infinite projections are equivalent to a proper subprojection of themselves
• Projection p is infinite if there exists a projection q < p such that p ~ q
• Properties of finite projections include closure under subprojections and finite sums
• Infinite projections arise naturally in infinite-dimensional Hilbert spaces
• Examples of finite projections include rank-one projections in B(H) for separable Hilbert space H
• Examples of infinite projections include the identity operator on an infinite-dimensional Hilbert space

Properly Infinite and Abelian Projections

• Properly infinite projections are infinite projections that remain infinite when restricted to any central portion
• Projection p is properly infinite if zp is infinite for every nonzero central projection z
• Abelian projections generate commutative von Neumann subalgebras
• Projection p is abelian if the von Neumann algebra pMp is commutative
• Properly infinite projections play a crucial role in the classification of type III factors
• Abelian projections are essential in the study of maximal abelian subalgebras (MASAs)
• Examples of properly infinite projections include the identity operator in type III factors
• Examples of abelian projections include minimal projections in type I factors

Carriers

Central Carriers and Their Properties

• Central carrier of a projection p is the smallest central projection z such that zp = p
• Central carrier is denoted by c(p) and satisfies c(p) = $\bigwedge \{z \in Z(M) : zp = p\}$, where Z(M) is the center of M
• Central carriers provide information about the position of a projection within the von Neumann algebra
• Properties of central carriers include c(p) ≥ p and c(p₁ + p₂) = c(p₁) ∨ c(p₂)
• Central carriers are used to decompose von Neumann algebras into direct sums
• Relationship between central carriers and comparison theory helps in classifying von Neumann algebras
• Applications of central carriers include the study of type decomposition of von Neumann algebras
• Examples of central carriers include c(p) = I for any nonzero projection p in a factor

Applications of Carriers in von Neumann Algebras

• Carriers are used to analyze the structure of von Neumann algebras and their projections
• Range projection R(x) serves as the carrier of a positive operator x
• Left and right carriers of partial isometries provide information about their domain and range
• Carriers play a role in the polar decomposition of operators in von Neumann algebras
• Applications include the study of normal states and weights on von Neumann algebras
• Carriers are essential in the analysis of conditional expectations and index theory
• Examples of applications include the construction of crossed products and the study of subfactors