7.2 Basic construction

The basic construction is a powerful tool in von Neumann algebra theory. It creates a larger algebra from a given inclusion of von Neumann algebras, providing insights into their structure and properties. This concept is crucial for analyzing subfactors and their indices.

Jones and spatial basic constructions offer algebraic and geometric approaches to this concept. They preserve important structural properties and allow for the study of subfactors through iterative constructions. Key operators like the Jones projection and conditional expectation play vital roles in the basic construction's framework.

Definition of basic construction

• Fundamental concept in von Neumann algebra theory introduces a larger algebra from a given inclusion of von Neumann algebras
• Provides a powerful tool for analyzing the structure and properties of von Neumann algebras and their subalgebras

Jones basic construction

Top images from around the web for Jones basic construction

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  Conditional expectation - Wikipedia View original

  Is this image relevant?

• 

  Arquitetura de von Neumann – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  Conditional expectation - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Jones basic construction

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  Conditional expectation - Wikipedia View original

  Is this image relevant?

• 

  Arquitetura de von Neumann – Wikipédia, a enciclopédia livre View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  Conditional expectation - Wikipedia View original

  Is this image relevant?

1 of 3

• Algebraic approach developed by Vaughan Jones in the 1980s
• Constructs a new von Neumann algebra $M_1$ from an inclusion $N \subset M$ of von Neumann algebras
• Utilizes a conditional expectation $E: M \to N$ to define the Jones projection $e_N$
• Generates $M_1$ as the von Neumann algebra generated by $M$ and $e_N$

Spatial basic construction

• Geometric approach to basic construction using Hilbert space representations
• Involves representing von Neumann algebras as operators on a Hilbert space $H$
• Constructs $M_1$ as the von Neumann algebra generated by $M$ and the orthogonal projection onto $\overline{NH}$
• Equivalent to Jones basic construction under suitable conditions

Properties of basic construction

• Preserves important structural properties of the original von Neumann algebras
• Allows for the study of subfactors and their indices through iterative constructions

Fundamental properties

• Tower property: $N \subset M \subset M_1 \subset M_2 \subset \cdots$ forms an increasing sequence of von Neumann algebras
• Index preservation: $[M:N] = [M_1:M]$ under suitable conditions
• Modularity: $M_1 = JNJ$ where $J$ is the modular conjugation operator
• Commutant relation: $(M_1)' \cap M = N$

Algebraic properties

• Isomorphism: $M_1 \cong \langle M, e_N \rangle$ where $\langle M, e_N \rangle$ is the von Neumann algebra generated by $M$ and $e_N$
• Trace preservation: $\tau_{M_1}(xe_Ny) = \tau_M(xy)$ for $x,y \in M$
• Conditional expectation: $E_{M_1,M}(e_N) = [M:N]^{-1}$ where $E_{M_1,M}$ is the conditional expectation from $M_1$ to $M$
• Pimsner-Popa basis: Existence of a Pimsner-Popa basis for $M$ over $N$ in $M_1$

Operators in basic construction

• Key operators play crucial roles in the structure and properties of the basic construction
• Allow for the analysis of the relationship between the original algebras and the constructed algebra

Jones projection

• Central operator in the basic construction denoted as $e_N$
• Orthogonal projection onto the closure of $NH$ in the Hilbert space $H$
• Satisfies the relations $e_Nxe_N = E_N(x)e_N$ for all $x \in M$
• Generates $M_1$ together with $M$: $M_1 = \{M, e_N\}''$

Conditional expectation

• Linear map $E: M \to N$ preserving positivity and the identity element
• Satisfies $E(axb) = aE(x)b$ for all $a,b \in N$ and $x \in M$
• Plays a crucial role in defining the Jones projection and the basic construction
• Allows for the computation of the index $[M:N]$ as $E(1)^{-1}$

Applications of basic construction

• Powerful tool in the study of von Neumann algebras and their subfactors
• Provides insights into the structure and classification of factors

Subfactor theory

• Allows for the construction of subfactor lattices and their classification
• Enables the computation of subfactor indices and their properties
• Facilitates the study of intermediate subfactors and their relative positions
• Provides a framework for analyzing the structure of subfactor planar algebras

Index theory

• Computes and analyzes the Jones index $[M:N]$ for subfactors
• Relates the index to various algebraic and geometric properties of the subfactor
• Allows for the classification of subfactors with small indices (Jones' index theorem)
• Provides connections to quantum groups and conformal field theory through index values

Basic construction vs standard form

• Compares two fundamental constructions in von Neumann algebra theory
• Highlights the relationships and differences between these approaches

Similarities and differences

• Both involve representing von Neumann algebras on Hilbert spaces
• Standard form provides a canonical representation, while basic construction creates a larger algebra
• Basic construction utilizes a specific conditional expectation, whereas standard form uses a faithful normal semifinite weight
• Standard form preserves the original algebra, while basic construction generates a new, larger algebra

Advantages and limitations

• Basic construction allows for iterative analysis of subfactors and their properties
• Standard form provides a unique representation up to unitary equivalence
• Basic construction may not preserve certain properties (finiteness) of the original algebras
• Standard form is useful for studying modular theory and Tomita-Takesaki theory

Tensor products in basic construction

• Explores the behavior of basic construction under tensor products
• Provides insights into the structure of composite systems in quantum theory

Tensor product of factors

• Examines how basic construction behaves when applied to tensor products of factors
• Investigates the relationship between $M_1 \otimes N_1$ and $(M \otimes N)_1$
• Studies the preservation of properties (hyperfiniteness, type classification) under tensor products
• Analyzes the behavior of conditional expectations and Jones projections in tensor product situations

Tensor product of subfactors

• Explores the basic construction for tensor products of subfactor inclusions
• Investigates the relationship between indices: $[M \otimes N : P \otimes Q] = [M:P][N:Q]$
• Studies the behavior of intermediate subfactors under tensor products
• Examines the compatibility of basic constructions with various tensor product operations

Commutants in basic construction

• Analyzes the role of commutants in the structure of basic construction
• Provides insights into the relationships between various algebras in the construction

Relative commutant

• Studies the relative commutant $N' \cap M$ and its properties in basic construction
• Investigates the relationship between $N' \cap M$ and $(M_1)' \cap M_2$
• Examines the behavior of relative commutants under iterated basic constructions
• Relates the structure of relative commutants to properties of the subfactor (depth, amenability)

Double commutant theorem

• Applies the double commutant theorem to analyze the structure of basic construction
• Investigates the relationship between $M_1 = \{M, e_N\}''$ and $(\{M, e_N\}')'$
• Studies the preservation of the double commutant property under basic construction
• Examines the role of the double commutant theorem in establishing isomorphisms in basic construction

Basic construction for finite factors

• Focuses on the behavior of basic construction in the context of finite von Neumann algebras
• Provides insights into the structure of important classes of factors

Type II1 factors

• Examines the properties of basic construction for inclusions of Type II1 factors
• Investigates the preservation of the trace and finite dimensionality in the construction
• Studies the behavior of the Jones index and its relationship to the coupling constant
• Analyzes the structure of the basic construction tower for Type II1 subfactors

Hyperfinite factors

• Explores the basic construction for inclusions of hyperfinite factors
• Investigates the preservation of hyperfiniteness under basic construction
• Studies the relationship between basic construction and approximation properties
• Examines the role of basic construction in the classification of hyperfinite subfactors

Infinite index basic construction

• Extends the concept of basic construction to inclusions with infinite index
• Explores the challenges and unique properties associated with infinite index situations

Properties and challenges

• Investigates the behavior of conditional expectations for infinite index inclusions
• Examines the structure of the basic construction algebra $M_1$ in the infinite index case
• Studies the relationship between infinite index and type classification of factors
• Analyzes the challenges in defining and working with Jones projections for infinite index inclusions

Examples and applications

• Explores concrete examples of infinite index inclusions and their basic constructions
• Investigates the role of infinite index basic construction in the study of free products
• Examines applications to the theory of amalgamated free products of von Neumann algebras
• Studies the connections between infinite index basic construction and ergodic theory

Basic construction in quantum field theory

• Explores the applications of basic construction techniques in quantum field theory
• Provides insights into the algebraic structure of quantum systems

Algebraic quantum field theory

• Investigates the role of basic construction in the local algebra approach to quantum field theory
• Examines the relationship between basic construction and the split property for von Neumann algebras
• Studies the applications of basic construction to the analysis of superselection sectors
• Explores the connections between basic construction and the Doplicher-Haag-Roberts theory

Conformal field theory

• Analyzes the applications of basic construction in the study of conformal field theories
• Investigates the relationship between basic construction and the Virasoro algebra
• Examines the role of subfactors and their basic constructions in the classification of conformal field theories
• Studies the connections between basic construction and the representation theory of loop groups