States and traces are fundamental concepts in noncommutative geometry, providing a bridge between abstract algebraic structures and numerical data. They allow us to extract meaningful information from noncommutative spaces, serving as quantum analogues of probability measures and integration.
These tools play a crucial role in quantum mechanics, statistical physics, and operator algebra theory. States capture the probabilistic nature of quantum systems, while traces enable the development of noncommutative measure theory and integration, essential for understanding quantum phenomena and algebraic structures.
States on C*-algebras
States are a fundamental concept in the study of C*-algebras, providing a way to extract numerical information from abstract noncommutative spaces
They serve as noncommutative analogues of probability measures, allowing for the formulation of and noncommutative probability theory
Positive linear functionals
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Positive linear functionals are linear maps φ:A→C from a A to the complex numbers that satisfy φ(a∗a)≥0 for all a∈A
They capture the notion of positivity in the noncommutative setting
The set of positive linear functionals forms a cone in the dual space of A
Examples:
Evaluation functionals at points of the spectrum of a commutative C*-algebra
Vector states φ(a)=⟨ξ,aξ⟩ for a unit vector ξ in a Hilbert space representation of A
Normalized states
A state on a C*-algebra A is a φ that is normalized, i.e., φ(1)=1
States correspond to noncommutative probability measures, assigning "probabilities" to noncommutative events
The normalization condition ensures that the total probability is equal to 1
Examples:
Tracial states on matrix algebras, given by φ(a)=n1Tr(a) for a∈Mn(C)
Pure states on the algebra of continuous functions on a compact Hausdorff space, given by point evaluations
Convex set of states
The set of states on a C*-algebra A forms a convex subset of the dual space A∗
Convex combinations of states, i.e., λφ1+(1−λ)φ2 for λ∈[0,1] and states φ1,φ2, are again states
The convex structure reflects the possibility of forming mixtures of noncommutative probability measures
Extreme points of the state space are called pure states and correspond to irreducible representations of A
Pure vs mixed states
A state φ is called pure if it cannot be written as a non-trivial convex combination of other states
Pure states represent the most basic building blocks of the state space and correspond to irreducible representations of the C*-algebra
Mixed states are convex combinations of pure states and describe statistical ensembles or mixtures of pure states
Examples:
Vector states φ(a)=⟨ξ,aξ⟩ for a unit vector ξ in an irreducible representation are pure states
The tracial state on Mn(C) is a , as it can be decomposed into a convex combination of pure states corresponding to rank-one projections
GNS construction from states
The Gelfand-Naimark-Segal (GNS) construction is a powerful tool that associates a Hilbert space representation to each state on a C*-algebra
Given a state φ on A, the GNS construction yields a triple (πφ,Hφ,Ωφ) consisting of:
A *-representation πφ:A→B(Hφ) on a Hilbert space Hφ
A cyclic vector Ωφ∈Hφ such that φ(a)=⟨Ωφ,πφ(a)Ωφ⟩ for all a∈A
The GNS representation is unique up to unitary equivalence and captures all the information contained in the state
Pure states correspond to irreducible GNS representations, while mixed states yield reducible representations
Traces on von Neumann algebras
Traces are a special class of states on von Neumann algebras that are compatible with the additional structure present in these algebras
They play a crucial role in the classification of factors and the study of noncommutative measure theory
Definition and properties of traces
A trace on a M is a linear functional τ:M+→[0,∞] defined on the positive cone M+ of M that satisfies:
Positivity: τ(a)≥0 for all a∈M+
Additivity: τ(a+b)=τ(a)+τ(b) for all a,b∈M+
Homogeneity: τ(λa)=λτ(a) for all a∈M+ and λ≥0
Trace property: τ(ab)=τ(ba) for all a,b∈M
Traces are normally extended to the whole algebra M by linearity
Examples:
The standard trace Tr on the algebra of bounded operators on a Hilbert space
The normalized trace n1Tr on the matrix algebra Mn(C)
Tracial states
A tracial state on a von Neumann algebra M is a state τ that satisfies the trace property τ(ab)=τ(ba) for all a,b∈M
Tracial states are a special class of traces that are normalized, i.e., τ(1)=1
The set of tracial states forms a convex subset of the state space
Examples:
The normalized trace n1Tr on Mn(C)
The unique tracial state on the hyperfinite II1 factor R
Faithfulness and normality of traces
A trace τ is called faithful if τ(a∗a)=0 implies a=0 for all a∈M
Faithfulness ensures that the trace does not "ignore" any non-zero elements of the algebra
A trace τ is called normal if it is continuous with respect to the ultraweak topology on M
Normality is a technical condition that ensures compatibility with the von Neumann algebra structure
Normal faithful traces are of particular importance in the study of von Neumann algebras
Uniqueness of traces on factors
A factor is a von Neumann algebra with trivial center, i.e., Z(M)=C1
Factors are the building blocks of von Neumann algebras and are classified into types I, II1, II∞, and III
On factors of type I and II1, there exists a unique normal faithful tracial state (up to scalar multiples)
This uniqueness result highlights the special role of traces in the study of factors
Examples:
The normalized trace on Mn(C) is the unique tracial state (type I factor)
The hyperfinite II1 factor R admits a unique tracial state
Relationship between states and traces
States and traces are closely related concepts in the theory of operator algebras
Understanding their interplay is crucial for the study of noncommutative geometry and quantum statistical mechanics
Extension of states to normal states
Given a state φ on a C*-algebra A, there may exist multiple extensions of φ to a normal state on the enveloping von Neumann algebra A∗∗
The set of normal extensions forms a weak*-compact convex subset of the state space of A∗∗
The GNS representation of a state φ can be used to construct a normal extension via the vector state associated to the cyclic vector
Restrictions of traces to C*-subalgebras
A trace τ on a von Neumann algebra M can be restricted to a C*-subalgebra A⊂M, yielding a tracial state on A
The restriction map is a continuous affine map from the space of traces on M to the space of tracial states on A
This procedure allows for the study of traces on C*-algebras and their relation to the larger von Neumann algebra
Correspondence for finite-dimensional algebras
For finite-dimensional C*-algebras A, there is a one-to-one correspondence between traces and tracial states
Every trace on A is automatically normal and faithful, and can be normalized to obtain a tracial state
Conversely, every tracial state arises as the normalization of a unique trace
This correspondence simplifies the study of traces and states in the finite-dimensional setting
Example: On the matrix algebra Mn(C), the normalized trace n1Tr is the unique tracial state, and all traces are scalar multiples of the standard trace Tr
Applications and examples
States and traces find numerous applications in various branches of mathematics and physics
They provide a framework for noncommutative probability, quantum mechanics, and the study of operator algebras
Quantum mechanics and density matrices
In quantum mechanics, states on the C*-algebra of bounded operators on a Hilbert space describe the statistical properties of quantum systems
Density matrices, i.e., positive trace-class operators with unit trace, correspond to normal states
Pure states are represented by rank-one projections, while mixed states correspond to convex combinations of pure states
The expectation value of an observable a in a state φ is given by φ(a)
Statistical mechanics and KMS states
In quantum statistical mechanics, equilibrium states are described by KMS (Kubo-Martin-Schwinger) states
KMS states are characterized by a generalization of the Gibbs condition, involving the modular automorphism group associated to the state
The KMS condition is a noncommutative analogue of the DLR (Dobrushin-Lanford-Ruelle) equations in classical statistical mechanics
KMS states play a crucial role in the study of phase transitions and the classification of factors
Noncommutative measure theory
Traces on von Neumann algebras provide a framework for noncommutative measure theory
The algebra of measurable operators associated to a trace generalizes the classical L∞ space
Noncommutative Lp spaces can be constructed using the trace, yielding a noncommutative analogue of the classical Lp spaces
This allows for the development of noncommutative probability and integration theory
Dixmier traces and noncommutative geometry
Dixmier traces are a special class of singular traces on von Neumann algebras, introduced by Jacques Dixmier
They are used to construct noncommutative analogues of integration on manifolds and play a key role in ' noncommutative geometry program
Dixmier traces are defined on the ideal of compact operators and vanish on trace-class operators
They capture the asymptotic behavior of the eigenvalues of compact operators and are used to define noncommutative residues and zeta functions
Advanced topics
The theory of states and traces is a rich and active area of research, with connections to various branches of mathematics and physics
Advanced topics include the study of weights, modular theory, and the classification of von Neumann algebras
Weights and operator-valued weights
Weights are a generalization of states and traces, allowing for infinite values
A weight on a C*-algebra A is a function φ:A+→[0,∞] that is additive and homogeneous
Operator-valued weights are a further generalization, taking values in the positive cone of a von Neumann algebra
Weights and operator-valued weights are used in the study of non-finite von Neumann algebras and the construction of crossed products
Tomita-Takesaki theory and modular automorphism groups
Tomita-Takesaki theory is a powerful tool in the study of von Neumann algebras and their states
Given a faithful normal state φ on a von Neumann algebra M, the modular operator Δφ and the modular conjugation Jφ are constructed from the GNS representation of φ
The modular automorphism group σtφ(a)=ΔφitaΔφ−it describes the time evolution of the system in the state φ
Modular theory plays a crucial role in the classification of type III factors and the study of KMS states
Connes' classification of type III factors
Alain Connes achieved a complete classification of type III factors using modular theory and the Tomita-Takesaki construction
Type III factors are divided into subtypes IIIλ for λ∈[0,1], based on the behavior of their modular automorphism groups
The classification relies on the study of the flow of weights and the Connes invariant S(M)
This groundbreaking result opened up new avenues in the study of von Neumann algebras and their applications to physics
Noncommutative Lp spaces and noncommutative integration
Noncommutative Lp spaces, introduced by Irving Segal and Hideki Kosaki, generalize the classical Lp spaces to the setting of von Neumann algebras
Given a normal faithful trace τ on a von Neumann algebra M, the noncommutative Lp space Lp(M,τ) is defined as the completion of M with respect to the norm ∥a∥p=τ(∣a∣p)1/p
Noncommutative Lp spaces provide a framework for noncommutative integration theory and the study of noncommutative martingales
They have applications in quantum probability, operator space theory, and the study of noncommutative analogues of classical function spaces
Key Terms to Review (16)
Alain Connes: Alain Connes is a French mathematician known for his foundational work in noncommutative geometry, a field that extends classical geometry to accommodate the behavior of spaces where commutativity fails. His contributions have led to new understandings of various mathematical structures and their applications, bridging concepts from algebra, topology, and physics.
Borel Measure: A Borel measure is a type of measure defined on the Borel sigma-algebra of a topological space, capturing the intuitive concept of length, area, or volume. It extends the idea of measuring subsets of real numbers to more complex spaces and provides a foundation for probability theory and analysis, especially in the context of defining states and traces in noncommutative geometry.
C*-algebra: A c*-algebra is a complex algebra of bounded linear operators on a Hilbert space that is closed under the operation of taking adjoints and is also closed in the norm topology. This structure allows the integration of algebraic, topological, and analytical properties, making it essential in both functional analysis and noncommutative geometry.
Cyclic trace: A cyclic trace is a specific type of trace function that is used in noncommutative geometry, particularly in the study of operators on Hilbert spaces. It extends the notion of a trace, which typically sums diagonal elements of a matrix or linear operator, to accommodate more complex structures like cyclic cohomology. This function retains many important properties, including linearity and continuity, making it a vital tool for examining the behavior of operators under cyclic permutations.
Dual state: A dual state refers to a framework where two different systems or modes of governance coexist within a single political structure. This concept is particularly relevant in understanding how certain states can maintain a formal legal system while simultaneously operating through informal, often extralegal, means. The idea emphasizes the complexity and contradiction in governance, especially in contexts where formal institutions struggle to exert authority effectively.
Dual trace: The dual trace is a mathematical concept used in noncommutative geometry, representing a generalization of the notion of trace for operators on Hilbert spaces. It serves as a tool to study states and traces in the context of quantum mechanics, where it provides a means to express certain properties of linear operators acting on these spaces.
Gelfand-Naimark Theorem: The Gelfand-Naimark Theorem is a fundamental result in functional analysis that establishes a deep connection between commutative C*-algebras and compact Hausdorff spaces. It states that every commutative C*-algebra can be represented as continuous functions on some compact Hausdorff space, revealing how algebraic structures relate to geometric and topological concepts.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to various fields including functional analysis, quantum mechanics, game theory, and operator algebras. His work laid the groundwork for many concepts in mathematics and theoretical physics, particularly in relation to the algebraic structures that underpin quantum theory and noncommutative geometry.
Mixed state: A mixed state refers to a statistical ensemble of different quantum states, where each state has a certain probability of being realized. This concept is crucial in quantum mechanics as it describes systems that are not in a single pure state but rather a combination of several states, allowing for the representation of uncertainty and incomplete knowledge about the system.
Normal Trace: A normal trace is a specific type of functional on a noncommutative algebra that is associated with a normal state, which is a positive linear functional that satisfies the property of being continuous with respect to the topology induced by the algebra. It provides a way to integrate over elements in noncommutative geometry and relates closely to the structure of states, helping to define expected values in quantum mechanics and various aspects of operator theory.
Positive Linear Functional: A positive linear functional is a linear map from a vector space to the real numbers that assigns non-negative values to positive elements of that space. This concept plays a crucial role in various areas, connecting algebraic structures, representations, and states in functional analysis. It highlights how certain linear functions can measure or evaluate elements within an algebraic framework while ensuring that positivity is preserved.
Pure State: A pure state is a specific type of state in a noncommutative geometry framework, characterized by its representation as a single vector in a Hilbert space. It represents the most precise information possible about a quantum system, distinguished from mixed states, which are statistical ensembles of pure states. Understanding pure states is crucial for grasping the underlying algebraic structures and the behavior of physical systems.
Quantum Field Theory: Quantum Field Theory (QFT) is a theoretical framework that combines classical field theory, quantum mechanics, and special relativity to describe how particles interact with one another and with fields. It provides the mathematical structure for understanding particle physics and is essential in formulating models that explore fundamental forces and particles.
Quantum statistical mechanics: Quantum statistical mechanics is the branch of physics that combines the principles of quantum mechanics with statistical methods to describe and predict the behavior of systems with many particles. This framework is essential for understanding phenomena in condensed matter physics, quantum gases, and other areas where quantum effects become significant at macroscopic scales. It introduces concepts like quantum states and traces, which are crucial for analyzing the statistical properties of quantum systems.
Riesz Representation Theorem: The Riesz Representation Theorem states that every continuous linear functional on a Hilbert space can be represented as an inner product with a fixed element from that space. This powerful result connects functional analysis and geometry, illustrating how linear functionals can be understood in terms of geometric structures like inner products.
Von Neumann algebra: A von Neumann algebra is a type of operator algebra that is defined as a *-subalgebra of bounded operators on a Hilbert space which is closed in the weak operator topology and contains the identity operator. This structure plays a crucial role in the study of quantum mechanics and noncommutative geometry, particularly when discussing representations, integration, and differential calculus in infinite-dimensional spaces.