C*-algebras are a powerful mathematical tool, combining algebra and topology. They're the backbone of non-commutative geometry and quantum physics, offering a framework to study infinite-dimensional spaces and quantum systems.
This section lays the groundwork by defining C*-algebras and showcasing key examples. We'll see how these structures bridge classical and quantum realms, setting the stage for deeper exploration of their properties and applications.
Basic Definitions
Fundamental Concepts of C-algebras
- C*-algebra comprises a Banach algebra A over complex numbers equipped with an involution operation
- Involution denotes a map * : A → A satisfying (x*)* = x, (x + y)* = x* + y*, and (λx)* = λ̄x* for all x, y ∈ A and λ ∈ C
- Norm on a C*-algebra measures the "size" of elements, satisfying properties like ||x + y|| ≤ ||x|| + ||y|| and ||λx|| = |λ| ||x|| for x, y ∈ A and λ ∈ C
- Completeness ensures every Cauchy sequence in the algebra converges to an element within the algebra
- C*-identity states ||x*x|| = ||x||² for all elements x in the algebra, linking algebraic and topological structures
Properties and Operations
- Involution operation extends complex conjugation to algebra elements
- For scalars: (λ)* = λ̄ (complex conjugate)
- For matrices: A* = (A^T)^- (conjugate transpose)
- Norm satisfies additional properties in C*-algebras:
- ||x*|| = ||x|| for all x ∈ A
- ||xy|| ≤ ||x|| ||y|| for all x, y ∈ A
- Completeness allows for infinite series and limits to be well-defined
- Ensures existence of limits for convergent sequences
- Enables powerful analytical tools like functional calculus
Algebraic and Topological Structures
- C*-algebras combine algebraic structure (addition, multiplication, scalar multiplication) with topological structure (norm, completeness)
- Algebraic operations interact with involution:
- (xy)* = yx for all x, y ∈ A
- (x + y)* = x* + y* for all x, y ∈ A
- Topological structure interacts with algebraic operations:
- Multiplication continuous in both variables
- Involution continuous operation
- C*-identity ||x*x|| = ||x||² links algebraic (multiplication, involution) and topological (norm) structures
- Implies automatic continuity of -homomorphisms between C-algebras
Commutative Examples
Function Algebras
- Commutative C*-algebra defined as C*-algebra where xy = yx for all elements x and y
- C(X) denotes continuous complex-valued functions on compact Hausdorff space X
- Addition and multiplication defined pointwise
- Involution given by complex conjugation: f*(x) = f(x)̄
- Norm defined as supremum norm: ||f|| = sup{|f(x)| : x ∈ X}
- C₀(X) represents continuous complex-valued functions on locally compact Hausdorff space X vanishing at infinity
- Functions approach 0 as x approaches ∞
- Norm and operations defined similarly to C(X)
Specific Function Spaces
- L∞(X, μ) consists of essentially bounded measurable functions on measure space (X, μ)
- Quotient algebra of bounded measurable functions modulo functions vanishing almost everywhere
- Involution given by complex conjugation
- Norm defined as essential supremum
- C*-algebra of bounded continuous functions on metric space X
- Denoted Cb(X)
- Equipped with supremum norm
- Isomorphic to C(βX) where βX represents Stone-Čech compactification of X
Non-commutative Examples
Matrix Algebras
- Mn(C) denotes n × n complex matrices
- Addition and multiplication defined as usual matrix operations
- Involution given by conjugate transpose: A* = (A^T)^-
- Norm defined as operator norm: ||A|| = sup{||Ax|| : x ∈ C^n, ||x|| = 1}
- B(H) represents bounded linear operators on Hilbert space H
- Generalization of Mn(C) to infinite-dimensional setting
- Involution given by adjoint operator
- Norm defined as operator norm
- Finite-dimensional C*-algebras isomorphic to direct sums of matrix algebras
- A ≅ Mn₁(C) ⊕ Mn₂(C) ⊕ ... ⊕ Mnk(C)
Operator Algebras
- K(H) denotes compact operators on Hilbert space H
- Closure of finite rank operators in operator norm topology
- Proper ideal in B(H) when H infinite-dimensional
- Involution and norm inherited from B(H)
- CAR algebra (Canonical Anticommutation Relations)
- Generated by elements satisfying a_i a_j* + a_j* a_i = δ_ij
- Models fermionic systems in quantum mechanics
- Group C*-algebras
- C*(G) constructed from locally compact group G
- Completion of group algebra L¹(G) with respect to appropriate norm