6.1 Representations of C*-algebras and their properties

Representations of C*-algebras map abstract algebraic structures to concrete operators on Hilbert spaces. This powerful tool allows us to study C*-algebras through their actions on vector spaces, bridging pure algebra and functional analysis.

Key properties like non-degeneracy, cyclicity, and faithfulness shape how representations behave. Understanding these concepts is crucial for grasping how abstract C*-algebras relate to concrete operator algebras and quantum mechanics.

Representations and Hilbert Spaces

Fundamental Concepts of Representations

• Representation defines a mapping from a C*-algebra to bounded linear operators on a Hilbert space
• Hilbert space consists of a complete inner product space with complex-valued inner product
  • Allows for infinite-dimensional vector spaces
  • Includes familiar spaces like $\mathbb{R}^n$ and $\mathbb{C}^n$
• Bounded linear operator transforms between Hilbert spaces while preserving linearity and boundedness
  • Satisfies $\|T(x)\| \leq M\|x\|$ for some constant M and all vectors x
  • Examples include multiplication operators and integral operators

Properties of -Homomorphisms

• -homomorphism preserves algebraic structure between C-algebras
• Maps between C*-algebras A and B, denoted φ: A → B
• Preserves addition: φ(a + b) = φ(a) + φ(b)
• Preserves multiplication: φ(ab) = φ(a)φ(b)
• Preserves involution: φ(a*) = φ(a)*
• Maintains positivity: φ(a*a) ≥ 0 for all a in A

Properties of Representations

Types of Representations

• Non-degenerate representation ensures the image of the C*-algebra acts densely on the Hilbert space
  • For any non-zero vector v in the Hilbert space, there exists an element a in the C*-algebra such that π(a)v ≠ 0
  • Guarantees the representation captures the full structure of the C*-algebra
• Cyclic representation generates the entire Hilbert space from a single vector
  • There exists a vector v such that {π(a)v : a ∈ A} is dense in the Hilbert space
  • Simplifies the study of representations (GNS construction)
• Faithful representation preserves the norm structure of the C*-algebra
  • Injectivity: π(a) = 0 implies a = 0
  • Isometric: ‖π(a)‖ = ‖a‖ for all a in the C*-algebra
  • Allows embedding of abstract C*-algebras into concrete operator algebras

Norm Properties and Continuity

• Norm-decreasing property ensures representations do not increase the norm of elements
• Satisfies ‖π(a)‖ ≤ ‖a‖ for all elements a in the C*-algebra
• Implies continuity of the representation
• Preserves convergence of sequences and series in the C*-algebra
• Follows from the C*-identity: ‖aa‖ = ‖a‖^2 for all a in the C-algebra