1.2 Basic properties and operations in Banach algebras

Banach algebras are complex mathematical structures that combine algebraic and topological properties. This section dives into their basic properties and operations, laying the groundwork for understanding these powerful tools in functional analysis.

We'll explore ideals, quotient algebras, direct sums, and various mappings between Banach algebras. These concepts are crucial for grasping how Banach algebras behave and interact, setting the stage for more advanced topics in the field.

Algebraic Structures

Ideals and Quotient Algebras

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• Ideal represents a special subset of a Banach algebra closed under addition and multiplication by elements of the algebra
• Two-sided ideal absorbs multiplication from both left and right sides
• Left ideal absorbs multiplication only from the left side
• Right ideal absorbs multiplication only from the right side
• Maximal ideal denotes the largest proper ideal in a Banach algebra
• Quotient algebra formed by taking the set of equivalence classes of elements modulo an ideal
• Quotient algebra inherits algebraic and topological properties from the original algebra
• Quotient norm defined as the infimum of norms of representatives in each equivalence class
• Quotient algebra becomes a Banach algebra when equipped with quotient norm

Direct Sum and Its Properties

• Direct sum combines multiple Banach algebras into a single larger algebra
• Algebraic direct sum involves finite number of algebras
• Topological direct sum extends the concept to infinite number of algebras
• Elements in direct sum represented as tuples with components from each constituent algebra
• Addition in direct sum performed component-wise
• Multiplication in direct sum also performed component-wise
• Norm in direct sum typically defined as maximum of norms of components
• Direct sum preserves properties like commutativity and associativity of constituent algebras
• Useful for constructing new algebras with desired properties

Mappings

Homomorphisms and Their Properties

• Homomorphism denotes a structure-preserving map between two Banach algebras
• Preserves algebraic operations (addition and multiplication)
• Does not necessarily preserve norm or topological properties
• Kernel of a homomorphism forms an ideal in the domain algebra
• Image of a homomorphism forms a subalgebra of the codomain algebra
• Continuous homomorphisms preserve both algebraic and topological structures
• Epimorphism refers to a surjective homomorphism
• Monomorphism denotes an injective homomorphism
• Composition of homomorphisms yields another homomorphism

Isomorphisms and Their Significance

• Isomorphism represents a bijective homomorphism between Banach algebras
• Preserves both algebraic and topological structures
• Inverse of an isomorphism also forms an isomorphism
• Isometric isomorphism preserves norms exactly
• Topological isomorphism preserves topological structure but may not preserve norms exactly
• Isomorphic Banach algebras share all algebraic and topological properties
• Useful for classifying and studying Banach algebras
• Automorphism denotes an isomorphism from an algebra to itself
• Inner automorphisms arise from conjugation by invertible elements

Constructions

Unitization Process

• Unitization adds a unit element to a non-unital Banach algebra
• Constructs a new algebra by adjoining an identity element
• Original algebra embedded as a closed ideal in the unitized algebra
• Unitization preserves most properties of the original algebra
• Norm in unitized algebra typically defined to extend the original norm
• Unitization of a commutative algebra remains commutative
• Useful for extending results from unital to non-unital algebras
• Allows application of spectral theory to non-unital algebras

Tensor Products and Their Applications

• Tensor product combines two Banach algebras to form a new, larger algebra
• Algebraic tensor product defined using linear combinations of elementary tensors
• Projective tensor product completes algebraic tensor product with respect to projective norm
• Injective tensor product uses a different norm completion
• Tensor product of commutative algebras remains commutative
• Useful for constructing new examples of Banach algebras
• Applications in quantum mechanics and representation theory
• Tensor products preserve many properties of constituent algebras (commutativity)
• Cross-norms on tensor products satisfy certain compatibility conditions with original norms