1.1 C*-algebras

C*-algebras are fundamental structures in Von Neumann Algebra theory, providing a mathematical framework for quantum mechanics. They generalize bounded linear operators on Hilbert spaces, combining algebraic and topological properties through the C*-identity.

These algebras possess rich internal structures, including self-adjoint and positive elements, and are connected to concrete operator algebras through representations. States on C*-algebras offer probabilistic interpretations, while morphisms capture structure-preserving maps, essential for classification and relating different algebras.

Definition of C*-algebras

• C*-algebras form a foundational structure in the study of Von Neumann Algebras, providing a mathematical framework for quantum mechanics
• These algebras generalize the concept of algebras of bounded linear operators on Hilbert spaces, crucial for understanding operator theory in functional analysis

Banach algebra properties

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• Complete normed algebra over the complex numbers
• Satisfies the submultiplicative property $\|xy\| \leq \|x\| \|y\|$ for all elements x and y
• Closed under addition, scalar multiplication, and multiplication
• Contains a multiplicative identity element (usually denoted as 1)

Involution operation

• Antilinear map * from the algebra to itself
• Satisfies $(x^*)^* = x$ for all elements x
• Reverses order of multiplication: $(xy)^* = y^*x^*$
• Preserves linear combinations: $(αx + βy)^* = \bar{α}x^* + \bar{β}y^*$ where α and β are complex numbers

C*-identity

• Fundamental property defining C*-algebras
• Satisfies $\|x^*x\| = \|x\|^2$ for all elements x
• Connects the norm and involution, ensuring compatibility between algebraic and topological structures
• Implies that the spectrum of positive elements is non-negative

Examples of C*-algebras

• C*-algebras encompass a wide range of mathematical structures, from concrete operator algebras to abstract function spaces
• Understanding these examples provides insight into the diverse applications of C*-algebras in functional analysis and quantum theory

Bounded linear operators

• B(H): algebra of all bounded linear operators on a Hilbert space H
• Norm defined as the operator norm: $\|T\| = \sup_{x \neq 0} \frac{\|Tx\|}{\|x\|}$
• Involution given by the adjoint operator: T* is the unique operator satisfying ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x, y in H
• Includes important subalgebras (compact operators, trace-class operators)

Continuous functions

• C(X): algebra of continuous complex-valued functions on a compact Hausdorff space X
• Sup norm: $\|f\| = \sup_{x \in X} |f(x)|$
• Involution defined as complex conjugation: f*(x) = f(x)̄
• Serves as a prototype for commutative C*-algebras

Matrix algebras

• Mn(C): algebra of n × n complex matrices
• Norm given by the operator norm induced by the Euclidean norm on Cn
• Involution defined as the conjugate transpose: A* = Āᵀ
• Finite-dimensional C*-algebras are direct sums of matrix algebras

Structure of C*-algebras

• C*-algebras possess rich internal structure, allowing for powerful analytical techniques
• Understanding this structure bridges abstract algebra and functional analysis in the context of Von Neumann Algebras

Self-adjoint elements

• Elements satisfying x = x*
• Form a real vector space within the C*-algebra
• Spectral theorem applies, allowing diagonalization in the operator case
• Generate the entire algebra: every element can be written as x = a + ib with a and b self-adjoint

Positive elements

• Self-adjoint elements with non-negative spectrum
• Can be characterized as elements of the form x*x
• Form a convex cone in the C*-algebra
• Play a crucial role in defining states and representations

Spectrum of elements

• Set of complex numbers λ such that (x - λ1) is not invertible
• Always non-empty and compact for elements in a C*-algebra
• Spectral radius formula: $r(x) = \lim_{n \to \infty} \|x^n\|^{1/n}$
• For normal elements (xx* = x*x), the norm equals the spectral radius

Representations of C*-algebras

• Representations connect abstract C*-algebras to concrete operator algebras on Hilbert spaces
• These constructions form the backbone of the relationship between C*-algebras and Von Neumann Algebras

Gelfand-Naimark theorem

• Every C*-algebra is isometrically *-isomorphic to a closed *-subalgebra of B(H) for some Hilbert space H
• Provides a concrete realization of abstract C*-algebras as operator algebras
• Proof uses the direct sum of all cyclic representations
• Demonstrates the universality of B(H) as a "mother of all C*-algebras"

GNS construction

• Constructs a *-representation from a positive linear functional
• Process:
  1. Define a pre-Hilbert space structure on the algebra
  2. Complete to obtain a Hilbert space
  3. Represent algebra elements as operators via left multiplication
• Yields cyclic representations, fundamental building blocks of all representations

Irreducible representations

• Representations with no non-trivial invariant subspaces
• Characterize simple C*-algebras: those with only trivial closed two-sided ideals
• Correspond to pure states via the GNS construction
• Schur's lemma: intertwining operators between irreducible representations are scalar multiples of the identity

States and functionals

• States provide a probabilistic interpretation of C*-algebras, crucial for applications in quantum theory
• The study of states connects C*-algebras to measure theory and convex analysis

Positive linear functionals

• Linear maps φ: A → C satisfying φ(x*x) ≥ 0 for all x in A
• Form a convex cone in the dual space of the C*-algebra
• Bounded with norm equal to φ(1) for unital C*-algebras
• Riesz representation theorem relates positive linear functionals to measures in the commutative case

Pure states vs mixed states

• Pure states: extreme points of the state space
• Cannot be written as non-trivial convex combinations of other states
• Mixed states: convex combinations of pure states
• Quantum superposition principle reflected in the convex structure of the state space

State space topology

• Weak* topology: convergence of expectation values on all observables
• Banach-Alaoglu theorem ensures compactness of the state space in this topology
• Krein-Milman theorem: state space is the closed convex hull of its pure states
• Choquet theory describes mixed states as integrals over pure states

C*-algebra morphisms

• Morphisms capture structure-preserving maps between C*-algebras
• Understanding these maps is essential for classifying and relating different C*-algebras in Von Neumann Algebra theory

*-homomorphisms

• Linear maps π: A → B preserving multiplication and involution
• Automatically continuous with $\|π(x)\| \leq \|x\|$ for all x in A
• Kernel is a closed two-sided ideal in A
• Image is a C*-subalgebra of B

Isomorphisms and automorphisms

• Isomorphisms: bijective *-homomorphisms
• Automatically isometric due to the C*-identity
• Automorphisms: isomorphisms from a C*-algebra to itself
• Form a group under composition, often studied via their generator (derivations)

Ideals and quotients

• Closed two-sided ideals I in a C*-algebra A
• Quotient A/I naturally inherits a C*-algebra structure
• Every *-homomorphism factors through its kernel via the first isomorphism theorem
• Primitive ideals: kernels of irreducible representations, crucial for studying the structure of non-commutative C*-algebras

Commutative C*-algebras

• Commutative C*-algebras form a bridge between functional analysis and topology
• Understanding these algebras provides insight into the general non-commutative case in Von Neumann Algebra theory

Gelfand transform

• Isometric *-isomorphism between a commutative C*-algebra A and C(X), where X is the spectrum of A
• Spectrum: space of non-zero *-homomorphisms from A to C (characters)
• Maps each element a in A to the function â on X defined by â(φ) = φ(a)
• Generalizes the Fourier transform and spectral theory

Spectrum of commutative C*-algebras

• Compact Hausdorff space when equipped with the weak* topology
• Points correspond to maximal ideals of the algebra
• Homeomorphic to the space of pure states
• Stone-Weierstrass theorem characterizes dense subalgebras of C(X)

Continuous functional calculus

• For normal elements x in a C*-algebra, f(x) can be defined for any continuous function f on the spectrum of x
• Extends the polynomial functional calculus
• Preserves algebraic operations and respects composition of functions
• Spectral mapping theorem: spectrum of f(x) is the image of the spectrum of x under f

Non-commutative C*-algebras

• Non-commutative C*-algebras generalize classical structures to quantum settings
• These algebras form the core of operator algebra theory in the study of Von Neumann Algebras

Von Neumann algebras

• C*-algebras closed in the strong operator topology on B(H)
• Equivalently, double commutants of self-adjoint subsets of B(H)
• Possess a predual, allowing for a notion of normal states and normal representations
• Classification into types (I, II, III) based on the structure of their projections

AF algebras

• Approximately finite-dimensional C*-algebras
• Inductive limits of sequences of finite-dimensional C*-algebras
• Classified by K-theory and Bratteli diagrams
• Include important examples (UHF algebras, Bunce-Deddens algebras)

Operator spaces

• Linear subspaces of B(H) with the inherited matrix norms
• Provide a non-commutative analogue of Banach spaces
• Haagerup tensor product generalizes the spatial tensor product of C*-algebras
• Ruan's theorem characterizes abstract operator spaces

C*-algebras in physics

• C*-algebras provide a rigorous mathematical framework for quantum theories
• Applications in physics demonstrate the power of C*-algebraic techniques in Von Neumann Algebra theory

Quantum mechanics applications

• Observables represented as self-adjoint operators in a C*-algebra
• States correspond to positive linear functionals, pure states to wave functions
• Uncertainty principle derived from non-commutativity of position and momentum operators
• Quantum dynamics described by one-parameter groups of automorphisms

Statistical mechanics connections

• KMS states model thermal equilibrium states
• Phase transitions studied via changes in the structure of equilibrium states
• Quantum statistical mechanics formulated using C*-dynamical systems
• Tomita-Takesaki theory relates modular automorphisms to time evolution

Quantum field theory usage

• Local algebras of observables in algebraic quantum field theory
• Haag-Kastler axioms formulated in terms of net of C*-algebras
• Superselection sectors described using representation theory of C*-algebras
• Operator product expansions formalized using asymptotic morphisms between C*-algebras

C*-algebras and K-theory

• K-theory provides powerful invariants for classifying C*-algebras
• These tools connect operator algebra theory to algebraic topology and index theory in Von Neumann Algebra studies

K0 and K1 groups

• K0(A): group of formal differences of isomorphism classes of projections in matrix algebras over A
• K1(A): group of connected components of invertible elements in matrix algebras over A
• Functorial: *-homomorphisms induce group homomorphisms between K-groups
• Bott periodicity: K0(SA) ≅ K1(A) where SA is the suspension of A

Bott periodicity

• Fundamental periodicity theorem in topological K-theory
• States that Ki+2(A) ≅ Ki(A) for all i
• Reduces K-theory computations to K0 and K1
• Proved using Clifford algebras and the relation between K-theory and homotopy theory

Index theory connections

• Atiyah-Singer index theorem relates analytical and topological indices
• Fredholm index of operators computed using K-theory classes
• Noncommutative geometry extends index theory to singular spaces using C*-algebras
• Connes' trace theorem connects traces on certain C*-algebras to Dixmier traces and Wodzicki residues