3.4 The structure of simple C*-algebras

Simple C*-algebras are the building blocks of more complex algebraic structures. They have no non-trivial closed two-sided ideals, meaning all non-zero elements generate the entire algebra. This property makes them crucial for understanding larger C*-algebraic systems.

Factors, a type of von Neumann algebra with a trivial center, are closely related to simple C*-algebras. They're classified into Types I, II, and III based on their projection lattices. This classification provides insights into the structural properties of von Neumann algebras.

Simple C-algebras and Factors

Fundamental Concepts of Simple C-algebras

• Simple C*-algebra defined as a C*-algebra with no non-trivial closed two-sided ideals
• Minimal ideals in C*-algebras represent smallest non-zero closed two-sided ideals
• Factor refers to a von Neumann algebra with trivial center, consisting only of scalar multiples of the identity operator
• Type classification of factors divides them into Type I, Type II, and Type III based on their projection lattices
  • Type I factors further subdivided into Type In (finite-dimensional) and Type I∞ (infinite-dimensional)
  • Type II factors split into Type II1 (finite) and Type II∞ (infinite)
  • Type III factors characterized by absence of non-zero finite projections

Properties and Examples of Simple C-algebras

• Simple C*-algebras possess no proper ideals, ensuring all non-zero elements generate the entire algebra
• Minimal ideals in C*-algebras often correspond to finite-dimensional representations
• Factors play crucial role in decomposition theory of von Neumann algebras
• Examples of simple C*-algebras include:
  • Matrix algebras Mn(C) (finite-dimensional, Type In factor)
  • Compact operators K(H) on a Hilbert space H (Type I∞ factor)
  • Irrational rotation C*-algebras (simple, but not factors)

Applications and Significance

• Simple C*-algebras form building blocks for more complex C*-algebraic structures
• Factors arise naturally in quantum mechanics, representing algebras of observables for quantum systems
• Type classification provides insights into structural properties of von Neumann algebras
• Understanding simple C*-algebras and factors essential for:
  • Representation theory of C*-algebras
  • Classification of von Neumann algebras
  • Quantum statistical mechanics

Equivalence and Traces

Murray-von Neumann Equivalence

• Murray-von Neumann equivalence defines relation between projections in a von Neumann algebra
• Two projections p and q considered equivalent if there exists a partial isometry v such that $v^*v = p$ and $vv^* = q$
• Equivalence relation preserves "size" of projections in infinite-dimensional settings
• Crucial for classification of factors and study of dimension theory in von Neumann algebras
• Applications include:
  • Comparison theory of projections
  • Construction of dimension functions on von Neumann algebras

Traces and Their Properties

• Trace defined as a positive linear functional τ on a C*-algebra A satisfying $τ(ab) = τ(ba)$ for all a, b ∈ A
• Traces provide method for measuring "size" of elements in C*-algebras
• Properties of traces include:
  • Faithfulness: τ(a*a) = 0 implies a = 0
  • Normality: τ(sup ai) = sup τ(ai) for increasing net of positive elements
  • Finite trace: τ(1) < ∞, where 1 denotes the identity element
• Traces play significant role in:
  • Classification of factors (Type II1 factors characterized by existence of unique normalized trace)
  • Noncommutative integration theory
  • Index theory of subfactors

Special Classes of C-algebras

Uniformly Hyperfinite (UHF) Algebras

• UHF algebra defined as inductive limit of sequence of matrix algebras
• Construction process involves:
  • Starting with sequence of matrix algebras Mn1(C), Mn2(C), ...
  • Connecting maps φk: Mnk(C) → Mnk+1(C) preserving unit and *-operations
  • Taking inductive limit of this sequence
• UHF algebras classified by supernatural numbers (formal products of prime powers)
• Properties of UHF algebras include:
  • Simplicity
  • Unique trace
  • Nuclear (approximable by finite-dimensional C*-algebras)
• Examples of UHF algebras:
  • CAR algebra (Canonical Anticommutation Relations algebra)
  • Glimm's algebra (UHF algebra of type 2∞)

Approximately Finite-dimensional (AF) Algebras

• AF algebra defined as inductive limit of sequence of finite-dimensional C*-algebras
• Construction similar to UHF algebras but allows for more general finite-dimensional algebras
• Characterized by existence of increasing sequence of finite-dimensional C*-subalgebras with dense union
• Properties of AF algebras include:
  • Nuclear
  • Representation theory closely related to their K-theory
• Classification of AF algebras achieved through:
  • Bratteli diagrams (graphical representation of construction)
  • Elliott's classification theorem (using K-theory and traces)
• Examples of AF algebras:
  • UHF algebras (special case of AF algebras)
  • Bunce-Deddens algebras
  • Irrational rotation algebras (not AF, but AT algebra, closely related)