Banach algebras blend algebra and topology, combining an associative algebra with a complete norm. They're key in functional analysis, uniting algebraic operations with continuity. This foundation sets the stage for exploring their properties and applications.

In this section, we'll define Banach algebras, examine their core features, and look at key examples. We'll see how they pop up in various math areas, from complex analysis to operator theory.

Definition and Basic Properties

Fundamental Concepts of Banach Algebras

• Banach algebra combines algebraic and topological structures consists of an associative algebra over real or complex numbers equipped with a norm
• Norm assigns non-negative real number to each element in the algebra measures "size" or "magnitude" of elements
• Completeness property ensures every Cauchy sequence in the algebra converges to an element within the algebra
• Submultiplicativity condition requires norm of product of two elements does not exceed product of their individual norms expressed as $\|xy\| \leq \|x\| \|y\|$ for all elements x and y

Formal Definition and Key Characteristics

• Banach algebra A defined as complete normed algebra over real or complex numbers
• Algebraic operations (addition, scalar multiplication, and multiplication) must be continuous with respect to norm topology
• Norm satisfies following properties for all elements x, y in A and scalar α:
  • Non-negativity: $\|x\| \geq 0$ and $\|x\| = 0$ if and only if $x = 0$
  • Homogeneity: $\|\alpha x\| = |\alpha| \|x\|$
  • Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$
• Completeness ensures every Cauchy sequence $(x_n)$ in A converges to an element x in A
• Submultiplicativity condition $\|xy\| \leq \|x\| \|y\|$ crucial for interplay between algebraic and topological structures

Examples of Banach Algebras

Fundamental Examples

• Complex numbers C form simplest Banach algebra with usual addition and multiplication operations
• Norm for complex numbers defined as absolute value $|z|$ for any $z \in \mathbb{C}$
• Continuous functions on compact Hausdorff space X denoted C(X) form Banach algebra
  • Addition and multiplication defined pointwise
  • Norm given by supremum norm $\|f\| = \sup_{x \in X} |f(x)|$
• Bounded linear operators on Banach space X denoted B(X) constitute important Banach algebra
  • Addition and composition of operators serve as algebraic operations
  • Operator norm defined as $\|T\| = \sup_{\|x\| \leq 1} \|Tx\|$

Advanced Examples and Applications

• Sequence spaces $\ell^1$ and $\ell^\infty$ form Banach algebras with convolution product
• Group algebras L¹(G) for locally compact groups G provide rich class of Banach algebras
• Matrix algebras M_n(C) of n×n complex matrices form finite-dimensional Banach algebras
• Algebras of p-adic numbers and adeles arise in number theory and algebraic geometry

Algebra Homomorphisms

Definition and Properties

• Algebra homomorphism φ: A → B maps between two algebras preserves algebraic structure
• Homomorphism satisfies following conditions for all x, y in A and scalars α, β:
  • φ(αx + βy) = αφ(x) + βφ(y) (linearity)
  • φ(xy) = φ(x)φ(y) (multiplicativity)
• Continuity of algebra homomorphisms between Banach algebras crucial property
• Continuous algebra homomorphism φ: A → B between Banach algebras satisfies $\|\phi(x)\|_B \leq C\|x\|_A$ for some constant C > 0

Types and Applications of Homomorphisms

• Isomorphisms between Banach algebras preserve both algebraic and topological structures
• Automorphisms map Banach algebra to itself while preserving structure
• Ideals and quotient algebras defined using homomorphisms play crucial role in structure theory
• Representations of Banach algebras as operators on Hilbert spaces fundamental in spectral theory
• Gelfand transform provides important homomorphism from commutative Banach algebra to algebra of continuous functions