2.4 Applications to function spaces

Continuous function spaces are crucial in C*-algebra theory. They form commutative C*-algebras when equipped with pointwise operations and supremum norms. Understanding these spaces helps build a foundation for more complex algebraic structures.

The Stone-Weierstrass theorem and spectral theory for normal operators are key applications. These tools allow us to approximate functions and decompose operators, bridging the gap between abstract algebra and concrete function spaces.

Continuous Function Spaces

Continuous Functions and Their Properties

• C(X) denotes the space of continuous functions on a topological space X
• Continuous functions map between topological spaces while preserving topological properties
• Properties of C(X) include completeness under uniform convergence and separability for certain X
• C(X) forms a commutative C*-algebra when equipped with pointwise operations and supremum norm
• Supremum norm for f in C(X) defined as $\|f\|_\infty = \sup_{x \in X} |f(x)|$

Vanishing Functions and Stone-Weierstrass Theorem

• C_0(X) represents the space of continuous functions vanishing at infinity on a locally compact Hausdorff space X
• Functions in C_0(X) approach zero as x approaches infinity or the boundary of X
• C_0(X) forms a closed subalgebra of C(X) and a C*-algebra in its own right
• Stone-Weierstrass theorem provides conditions for density of subalgebras in C(X)
• Theorem states a subalgebra A of C(X) dense in C(X) if A separates points and contains constant functions
• Applications of Stone-Weierstrass theorem include approximation of continuous functions by polynomials

Measurable Function Spaces and Operators

Essentially Bounded Measurable Functions

• L∞(X) denotes the space of essentially bounded measurable functions on a measure space X
• Essentially bounded functions have finite essential supremum
• Essential supremum defined as $\|f\|_\infty = \inf\{M \geq 0 : |f(x)| \leq M \text{ almost everywhere}\}$
• L∞(X) forms a commutative C*-algebra with pointwise operations and essential supremum norm
• Relationship between L∞(X) and C(X) includes L∞(X) as a completion of C(X) in certain cases

Spectral Theory and Functional Calculus

• Spectral theorem for normal operators generalizes diagonalization of normal matrices
• Normal operators in Hilbert spaces characterized by commuting with their adjoint
• Spectral theorem decomposes normal operators into integrals over their spectrum
• Functional calculus allows application of functions to operators
• Continuous functional calculus extends continuous functions to normal operators
• Borel functional calculus generalizes to measurable functions for self-adjoint operators
• Applications of functional calculus include solving operator equations and defining operator functions