2.8 Topological algebras

Topological algebras blend algebraic structures with topology, enabling the study of continuity and convergence in algebraic contexts. They typically combine a ring or algebra over a field with a topology, allowing for the exploration of concepts like limits and continuity of algebraic operations.

This topic connects algebra and topology, providing a framework for analyzing algebraic structures with topological properties. It's crucial for understanding advanced concepts in functional analysis, operator theory, and noncommutative geometry, bridging pure mathematics with applications in physics and engineering.

Definition of topological algebras

• A topological algebra is a mathematical structure that combines an algebraic structure with a topology, allowing for the study of continuity and convergence in algebraic contexts
• The algebraic structure is typically a ring or an algebra over a field, equipped with operations such as addition, multiplication, and scalar multiplication
• The topology on the algebra is a collection of open sets that satisfies certain axioms, enabling the study of concepts like convergence, continuity, and limits

Algebraic structure

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• The algebraic structure of a topological algebra is usually a ring or an algebra over a field, which consists of a set with binary operations (addition and multiplication) and unary operations (scalar multiplication)
• The binary operations must satisfy properties such as associativity, commutativity (for addition), and distributivity of multiplication over addition
• The scalar multiplication operation allows elements of the algebra to be multiplied by elements of the underlying field, satisfying properties like distributivity and compatibility with field operations

Topology on algebras

• The topology on an algebra is a collection of subsets, called open sets, that satisfy certain axioms: the empty set and the entire algebra are open, arbitrary unions of open sets are open, and finite intersections of open sets are open
• The topology allows for the study of concepts such as convergence of sequences, continuity of functions, and limits of nets or filters
• The choice of topology can vary depending on the specific algebra and the desired properties, leading to different classes of topological algebras (Banach algebras, Fréchet algebras, etc.)

Compatibility of operations

• For an algebra to be a topological algebra, the algebraic operations must be compatible with the topology, meaning they should be continuous functions with respect to the topology
• Continuity of addition means that if two sequences converge to elements $a$ and $b$ respectively, then their sum sequence converges to $a + b$
• Continuity of multiplication means that if two sequences converge to elements $a$ and $b$ respectively, then their product sequence converges to $a \cdot b$
• Continuity of scalar multiplication ensures that if a sequence of scalars converges to $\lambda$ and a sequence of algebra elements converges to $a$, then their scalar product sequence converges to $\lambda \cdot a$

Examples of topological algebras

Banach algebras

• Banach algebras are topological algebras where the topology is induced by a complete norm, which satisfies the additional property $\|ab\| \leq \|a\| \|b\|$ for all elements $a$ and $b$
• Examples of Banach algebras include the space of continuous functions on a compact Hausdorff space with the supremum norm ($C(X)$), and the space of bounded linear operators on a Hilbert space with the operator norm ($B(H)$)
• Banach algebras have a rich theory, including the study of spectra, invertibility, and functional calculus

Frechet algebras

• Fréchet algebras are topological algebras where the topology is induced by a countable family of seminorms, making the algebra a complete metrizable locally convex space
• Examples of Fréchet algebras include the space of smooth functions on a compact manifold with the family of $C^k$ seminorms ($C^{\infty}(M)$), and the space of rapidly decreasing sequences with the family of weighted $\ell^1$ seminorms ($s(\mathbb{N})$)
• Fréchet algebras allow for the study of more general notions of convergence and continuity compared to Banach algebras

Locally convex algebras

• Locally convex algebras are topological algebras where the topology is induced by a family of seminorms, making the algebra a locally convex topological vector space
• This class of algebras includes Banach algebras and Fréchet algebras as special cases, but also allows for more general topologies that may not be metrizable or complete
• Examples of locally convex algebras include the space of compactly supported smooth functions on a manifold with the family of $C^k$ seminorms ($C_c^{\infty}(M)$), and the space of holomorphic functions on a domain with the topology of uniform convergence on compact subsets

Homomorphisms of topological algebras

Continuous homomorphisms

• A homomorphism between two topological algebras $A$ and $B$ is a map $\varphi: A \to B$ that preserves the algebraic operations, i.e., $\varphi(a + b) = \varphi(a) + \varphi(b)$, $\varphi(ab) = \varphi(a)\varphi(b)$, and $\varphi(\lambda a) = \lambda \varphi(a)$ for all $a, b \in A$ and $\lambda$ in the underlying field
• For topological algebras, we are interested in continuous homomorphisms, which are homomorphisms that are continuous functions with respect to the topologies on $A$ and $B$
• Continuous homomorphisms allow for the study of relationships between different topological algebras and the preservation of topological and algebraic properties

Isomorphisms vs homeomorphisms

• An isomorphism between two topological algebras is a bijective homomorphism, meaning it is both injective (one-to-one) and surjective (onto)
• A homeomorphism between two topological spaces is a bijective continuous function with a continuous inverse
• In the context of topological algebras, an isomorphism is a bijective continuous homomorphism with a continuous inverse, combining the notions of algebraic isomorphism and topological homeomorphism
• Isomorphic topological algebras can be considered equivalent in terms of their algebraic and topological structures

Automatic continuity

• Automatic continuity refers to the phenomenon where algebraic homomorphisms between certain topological algebras are automatically continuous, without explicitly requiring continuity in the definition
• This property holds for various classes of topological algebras, such as Banach algebras, Fréchet algebras, and locally multiplicatively convex algebras
• Automatic continuity results are powerful tools in the study of topological algebras, as they allow for the use of algebraic techniques to obtain topological information
• Examples of automatic continuity theorems include the Johnson-Sinclair theorem for Banach algebras and the Allan-Sinclair theorem for Fréchet algebras

Spectra of topological algebras

Spectrum of an element

• The spectrum of an element $a$ in a topological algebra $A$ is the set of all scalars $\lambda$ for which $a - \lambda 1$ is not invertible in $A$, where $1$ denotes the multiplicative identity of $A$
• In other words, $\sigma(a) = \{\lambda \in \mathbb{C} : a - \lambda 1 \text{ is not invertible in } A\}$
• The spectrum provides information about the behavior of an element under the resolvent operator $(a - \lambda 1)^{-1}$ and is a crucial tool in the study of topological algebras
• The spectrum is always a closed subset of the underlying field (usually the complex numbers) and can be empty, finite, or infinite depending on the element and the algebra

Spectral radius formula

• The spectral radius of an element $a$ in a topological algebra $A$ is defined as $r(a) = \sup\{|\lambda| : \lambda \in \sigma(a)\}$, the supremum of the absolute values of the elements in the spectrum of $a$
• The spectral radius formula relates the spectral radius to the growth of the norms of the powers of $a$: $r(a) = \lim_{n \to \infty} \|a^n\|^{1/n}$, where $\|\cdot\|$ is the norm or seminorm on $A$
• This formula allows for the computation of the spectral radius using the asymptotic behavior of the norms of the powers of an element
• The spectral radius is a key concept in the study of the stability and growth of elements in a topological algebra

Spectral mapping theorem

• The spectral mapping theorem relates the spectrum of an element $a$ in a topological algebra $A$ to the spectrum of its image under a continuous homomorphism $\varphi: A \to B$
• Specifically, if $\varphi: A \to B$ is a continuous homomorphism between topological algebras and $a \in A$, then $\sigma(\varphi(a)) = \varphi(\sigma(a))$, where $\varphi(\sigma(a)) = \{\varphi(\lambda) : \lambda \in \sigma(a)\}$
• This theorem allows for the study of the spectrum of an element in one algebra by considering its image in another algebra under a continuous homomorphism
• The spectral mapping theorem is a powerful tool in the analysis of topological algebras and their relationships through continuous homomorphisms

Ideals in topological algebras

Closed ideals

• An ideal in a topological algebra $A$ is a subspace $I \subseteq A$ that is closed under multiplication by elements of $A$ from both sides, i.e., $AI \subseteq I$ and $IA \subseteq I$
• A closed ideal is an ideal that is also a closed subset of $A$ with respect to the topology on $A$
• Closed ideals are important in the study of topological algebras because they allow for the construction of quotient algebras and the analysis of the structure of the algebra
• Examples of closed ideals include the kernel of a continuous homomorphism and the closure of any ideal in a Banach algebra

Quotient algebras

• Given a topological algebra $A$ and a closed ideal $I$, the quotient algebra $A/I$ is the set of cosets $\{a + I : a \in A\}$ with the induced operations $(a + I) + (b + I) = (a + b) + I$, $(a + I)(b + I) = (ab) + I$, and $\lambda(a + I) = (\lambda a) + I$
• The quotient algebra $A/I$ inherits a natural topology, called the quotient topology, which makes it a topological algebra
• Quotient algebras allow for the study of the structure of a topological algebra by "collapsing" certain elements and focusing on the remaining structure
• The properties of the quotient algebra $A/I$ are related to the properties of $A$ and $I$, and the quotient map $\pi: A \to A/I$ is a continuous homomorphism

Closure of ideals

• In a topological algebra $A$, the closure of an ideal $I$ is the smallest closed ideal containing $I$, denoted by $\overline{I}$
• The closure of an ideal can be constructed by taking the topological closure of $I$ in $A$, i.e., $\overline{I} = \{a \in A : \exists (a_n) \subseteq I \text{ such that } a_n \to a\}$
• The closure operation is idempotent, meaning $\overline{\overline{I}} = \overline{I}$, and it preserves inclusion, i.e., if $I \subseteq J$, then $\overline{I} \subseteq \overline{J}$
• The closure of an ideal is useful in the study of topological algebras because it allows for the construction of quotient algebras and the analysis of the structure of the algebra using closed ideals

Topological modules over topological algebras

Definition and examples

• A topological module over a topological algebra $A$ is a topological vector space $M$ together with a continuous bilinear map $A \times M \to M$, $(a, m) \mapsto am$, satisfying $(ab)m = a(bm)$, $a(m + n) = am + an$, and $(\lambda a)m = \lambda(am) = a(\lambda m)$ for all $a, b \in A$, $m, n \in M$, and scalars $\lambda$
• Examples of topological modules include topological algebras themselves (with the multiplication map), continuous function spaces over a topological space (with pointwise multiplication), and Hilbert spaces over the algebra of bounded linear operators (with operator application)
• Topological modules allow for the study of the action of a topological algebra on a topological vector space, generalizing the notion of modules over rings

Projective and injective modules

• A topological module $P$ over a topological algebra $A$ is called projective if, for any continuous surjective $A$-module homomorphism $f: M \to N$ and any continuous $A$-module homomorphism $g: P \to N$, there exists a continuous $A$-module homomorphism $h: P \to M$ such that $f \circ h = g$
• Projective modules are characterized by their ability to "lift" homomorphisms, and they play a crucial role in homological algebra and the study of resolutions
• A topological module $I$ over a topological algebra $A$ is called injective if, for any continuous injective $A$-module homomorphism $f: M \to N$ and any continuous $A$-module homomorphism $g: M \to I$, there exists a continuous $A$-module homomorphism $h: N \to I$ such that $h \circ f = g$
• Injective modules are characterized by their ability to "extend" homomorphisms, and they are dual to projective modules in many aspects

Topological tensor products

• Given two topological modules $M$ and $N$ over a topological algebra $A$, the topological tensor product $M \otimes_A N$ is a topological space that generalizes the notion of the tensor product of modules over a ring
• The topological tensor product is constructed by taking the algebraic tensor product of $M$ and $N$ over $A$ and equipping it with a suitable topology, such as the projective topology or the injective topology
• The topological tensor product satisfies a universal property that allows for the study of bilinear maps between topological modules
• Topological tensor products are used in the study of homological algebra, operator algebras, and noncommutative geometry, as they provide a way to construct new topological modules from existing ones

Completions of topological algebras

Definition of completeness

• A topological algebra $A$ is called complete if every Cauchy net (or Cauchy sequence, if $A$ is metrizable) in $A$ converges to an element of $A$
• Completeness is a desirable property for topological algebras, as it allows for the use of powerful tools from analysis, such as the Baire Category Theorem and the Uniform Boundedness Principle
• Examples of complete topological algebras include Banach algebras and Fréchet algebras
• Not all topological algebras are complete, and the completion process allows for the construction of a complete topological algebra from a given one

Completion functor

• The completion functor is a way to associate a complete topological algebra $\widehat{A}$ to a given topological algebra $A$ while preserving the algebraic and topological structure
• The completion of $A$ is constructed by taking the set of all Cauchy nets (or Cauchy sequences, if $A$ is metrizable) in $A$ and identifying nets that converge to the same limit
• The resulting space $\widehat{A}$ is a complete topological algebra, and there is a continuous homomorphism $\iota: A \to \widehat{A}$, called the canonical map, which is dense in $\widehat{A}$
• The completion functor is a left adjoint to the forgetful functor from the category of complete topological algebras to the category of topological algebras

Properties preserved by completion

• The completion process preserves many important properties of topological algebras, making it a useful tool in the study of these structures
• If $A$ is a topological algebra, then its completion $\widehat{A}$ is also a topological algebra
• If $A$ is a locally convex topological algebra, then $\widehat{A}$ is also locally convex
• If $A$ is a Banach algebra, then $\widehat{A}$ is isomorphic to $A$, as Banach algebras are already complete
• Continuous homomorphisms between topological algebras extend uniquely to continuous homomorphisms between their completions
• The spectrum of an element in a topological algebra is preserved under the completion process, i.e., $\sigma_A(a) = \sigma_{\widehat{A}}(\iota(a))$ for all $a \in A$

Applications of topological algebras

Functional analysis

• Topological algebras play a fundamental role in functional analysis, as many function spaces and operator algebras are naturally equipped with topological algebra structures
• Banach algebras, which are complete normed algebras, are central to the study of bounded linear operators on Banach spaces and the development of spectral theory
• Fréchet algebras, which are complete metrizable locally convex algebras, arise in the study of differential operators and the theory of distributions
• The study of topological algebras in functional analysis has led to