6.2 Unitary equivalence and spatial isomorphisms

Unitary equivalence and spatial isomorphisms are key concepts in representation theory of C*-algebras. They help us understand how different representations relate to each other and preserve important algebraic and topological structures.

Isometric -isomorphisms take things a step further, preserving even more properties between C-algebras. These tools are crucial for classifying C*-algebras and analyzing their structural relationships in deeper ways.

Unitary Equivalence and Isomorphisms

Unitary Equivalence and Spatial Isomorphisms

• Unitary equivalence establishes a relationship between two representations of a C*-algebra on Hilbert spaces
• Two representations π₁ and π₂ are unitarily equivalent when a unitary operator U exists such that Uπ₁(a)U* = π₂(a) for all elements a in the C*-algebra
• Unitary equivalence preserves the algebraic and topological structure of representations
• Spatial isomorphism refers to a bijective linear map between C*-algebras that preserves the *-algebra structure
• Spatial isomorphisms maintain the norm of elements, ensuring the preservation of both algebraic and topological properties
• Unitary equivalence implies spatial isomorphism, but the converse is not always true
• Spatial isomorphisms can exist between representations on different Hilbert spaces, while unitary equivalence requires the same Hilbert space

Isometric -Isomorphisms

• Isometric *-isomorphism combines the concepts of isometry and *-homomorphism
• An isometric -isomorphism is a bijective linear map φ between C-algebras A and B that preserves the involution and multiplication operations
• φ satisfies the following properties:
  • φ(a* + λb) = φ(a)* + λφ(b) for all a, b in A and scalar λ
  • φ(ab) = φ(a)φ(b) for all a, b in A
  • ||φ(a)|| = ||a|| for all a in A
• Isometric -isomorphisms preserve the complete structure of C-algebras, including algebraic, topological, and *-algebraic properties
• These isomorphisms play a crucial role in classifying C*-algebras and understanding their structural relationships
• Applications of isometric *-isomorphisms include:
  • Identifying equivalent representations of C*-algebras
  • Studying the structure of C*-algebra extensions
  • Analyzing the behavior of C*-algebras under various operations (direct sums, tensor products)

Operators

Unitary Operators and Their Properties

• Unitary operators serve as the foundation for unitary equivalence in C*-algebras
• A unitary operator U on a Hilbert space H satisfies UU = UU = I, where I is the identity operator
• Properties of unitary operators include:
  • Preservation of inner products: ⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y in H
  • Norm-preserving: ||Ux|| = ||x|| for all x in H
  • Bijective mapping of the Hilbert space onto itself
  • Spectral properties: spectrum of U lies on the unit circle in the complex plane
• Unitary operators form a group under composition, known as the unitary group
• Examples of unitary operators:
  • Rotation operators in finite-dimensional Hilbert spaces
  • Fourier transform operator in L²(ℝ)
• Applications of unitary operators extend to quantum mechanics, signal processing, and functional analysis

Intertwining Operators and Their Role

• Intertwining operators establish connections between different representations of C*-algebras
• An operator T intertwines representations π₁ and π₂ if Tπ₁(a) = π₂(a)T for all elements a in the C*-algebra
• Properties of intertwining operators:
  • Linearity: T(αx + βy) = αT(x) + βT(y) for all x, y in the domain and scalars α, β
  • Bounded: ||T|| < ∞, ensuring continuity of the operator
  • May not be invertible or surjective in general
• Intertwining operators play crucial roles in:
  • Studying equivalence and reducibility of representations
  • Analyzing symmetries in physical systems
  • Constructing new representations from existing ones
• Examples of intertwining operators:
  • Projection operators onto invariant subspaces
  • Fourier-Plancherel operator intertwining regular and momentum representations in quantum mechanics
• The existence of a unitary intertwining operator implies unitary equivalence between representations