3.1 Basic properties of noncommutative C*-algebras

Noncommutative C*-algebras are a key concept in quantum mechanics. They allow for elements that don't commute under multiplication, crucial for representing quantum systems. The involution operation maps elements to their adjoints, preserving algebraic structure.

The C*-identity relates norm and involution, ensuring consistency between algebraic and topological structures. The norm function measures element "size," satisfying properties like non-negativity and the triangle inequality. These basics set the stage for exploring more complex C*-algebra structures.

Basic Properties

Noncommutativity and Involution

• Noncommutativity defines C*-algebras allows for elements that do not commute under multiplication
• Multiplication of elements a and b may yield different results depending on order (ab ≠ ba)
• Noncommutativity crucial for representing quantum mechanical systems accurately
• Involution operation maps each element a to its adjoint a* preserves algebraic structure
• Involution satisfies properties: (a*)* = a, (ab)* = ba, and (αa + βb)* = α̅a* + β̅b* for all elements a, b and scalars α, β
• Involution generalizes complex conjugation in complex number field to abstract algebraic setting

C-identity and Norm Properties

• C*-identity fundamental property relates norm and involution: $\|a^*a\| = \|a\|^2$ for all elements a
• C*-identity ensures consistency between algebraic and topological structures of C*-algebra
• Norm function assigns non-negative real number to each element measures "size" or "magnitude"
• Norm satisfies properties:
  • Non-negativity: $\|a\| \geq 0$ for all a, with $\|a\| = 0$ if and only if a = 0
  • Homogeneity: $\|\alpha a\| = |\alpha| \|a\|$ for all scalars α and elements a
  • Triangle inequality: $\|a + b\| \leq \|a\| + \|b\|$ for all elements a and b
  • Submultiplicativity: $\|ab\| \leq \|a\| \|b\|$ for all elements a and b
• Norm induces metric topology on C*-algebra enables analysis of convergence and continuity

Special Elements

Self-adjoint and Positive Elements

• Self-adjoint elements satisfy a = a* represent observables in quantum mechanics
• Self-adjoint elements form real subspace of C*-algebra
• Spectral theorem applies to self-adjoint elements allows for functional calculus
• Positive elements satisfy a = b*b for some element b represent physically measurable quantities
• Positive elements form convex cone in C*-algebra
• Partial order defined on self-adjoint elements using positive elements: a ≤ b if and only if b - a positive

Unitization and Approximate Identity

• Unitization process adds unit element to non-unital C*-algebra
• Unitization A̅ of C*-algebra A contains A as closed two-sided ideal
• Unitization preserves C*-algebra structure extends operations from A to A̅
• Approximate identity sequence of positive elements (eλ) with norm ≤ 1 approximates unit element
• Approximate identity satisfies: lim $\|aeλ - a\| = 0$ and lim $\|eλa - a\| = 0$ for all elements a
• Existence of approximate identity guaranteed for all C*-algebras (not necessarily commutative)

Spectral Theory

Spectrum and Spectral Radius

• Spectrum of element a denoted σ(a) set of complex numbers λ such that a - λ1 not invertible
• Spectrum always non-empty compact subset of complex plane for C*-algebras
• Spectral radius r(a) maximum absolute value of elements in spectrum: r(a) = sup{|λ| : λ ∈ σ(a)}
• Spectral radius formula: r(a) = lim $\|a^n\|^{1/n}$ as n approaches infinity
• Spectral mapping theorem relates spectrum of f(a) to f(σ(a)) for continuous functions f
• Gelfand-Naimark theorem establishes isometric -isomorphism between commutative C-algebras and C(X) for compact Hausdorff space X

Functional Calculus and Spectral Projections

• Functional calculus allows application of continuous functions to normal elements
• For normal element a and continuous function f, f(a) defined using spectral theorem
• Spectral projections associated with Borel subsets of spectrum
• Spectral projections form complete Boolean algebra of projections
• Spectral measure assigns projection to each Borel subset of spectrum
• Spectral integral representation: a = ∫σ(a) λ dE(λ) where E spectral measure associated with a