Fréchet algebra

Template:Use shortened footnotes In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra $A$ over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation $(a,b)\mapsto a*b$ for $a,b\in A$ is required to be jointly continuous. If $\{\Vert \cdot {\Vert }_n{\}}_{n=0}^{\mathrm{\infty }}$ is an increasing familyTemplate:Efn of seminorms for the topology of $A$, the joint continuity of multiplication is equivalent to there being a constant $C_n>0$ and integer $m\ge n$ for each $n$ such that ${\Vert ab\Vert }_n\le C_n{\Vert a\Vert }_m{\Vert b\Vert }_m$ for all $a,b\in A$.Template:Efn Fréchet algebras are also called B₀-algebras.Template:Sfnm

A Fréchet algebra is $m$-convex if there exists such a family of semi-norms for which $m=n$. In that case, by rescaling the seminorms, we may also take $C_n=1$ for each $n$ and the seminorms are said to be submultiplicative: $\Vert ab{\Vert }_n\le \Vert a{\Vert }_n\Vert b{\Vert }_n$ for all $a,b\in A.$Template:Efn $m$-convex Fréchet algebras may also be called Fréchet algebras.

A Fréchet algebra may or may not have an identity element $1_A$. If $A$ is unital, we do not require that $\Vert 1_A{\Vert }_n=1,$ as is often done for Banach algebras.

Properties

• Continuity of multiplication. Multiplication is separately continuous if $a_{k}b\to ab$ and $b{a}_k\to ba$ for every $a,b\in A$ and sequence $a_k\to a$ converging in the Fréchet topology of $A$. Multiplication is jointly continuous if $a_k\to a$ and $b_k\to b$ imply $a_k{b}_k\to ab$. Joint continuity of multiplication is part of the definition of a Fréchet algebra. For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous.
• Group of invertible elements. If $invA$ is the set of invertible elements of $A$, then the inverse map $$\{\begin{array}{l}invA\to invA\\ u\mapsto u^{-1}\end{array}$$ is continuous if and only if $invA$ is a $G_{\delta }$ set.Template:Sfn Unlike for Banach algebras, $invA$ may not be an open set. If $invA$ is open, then $A$ is called a $Q$-algebra. (If $A$ happens to be non-unital, then we may adjoin a unit to $A$Template:Efn and work with $inv{A}^{+}$, or the set of quasi invertiblesTemplate:Efn may take the place of $invA$.)
• Conditions for $m$-convexity. A Fréchet algebra is $m$-convex if and only if for every, if and only if for one, increasing family $\{\Vert \cdot {\Vert }_n{\}}_{n=0}^{\mathrm{\infty }}$ of seminorms which topologize $A$, for each $m\in \mathbb{\mathbb{N}}$ there exists $p\ge m$ and $C_m>0$ such that $$\Vert a_1{a}_2\cdots a_n{\Vert }_m\le C_m^n\Vert a_1{\Vert }_p\Vert a_2{\Vert }_p\cdots \Vert a_n{\Vert }_p,$$ for all $a_1,a_2,\dots ,a_n\in A$ and $n\in \mathbb{\mathbb{N}}$.Template:Sfn A commutative Fréchet $Q$-algebra is $m$-convex,Template:Sfn but there exist examples of non-commutative Fréchet $Q$-algebras which are not $m$-convex.Template:Sfn
• Properties of $m$-convex Fréchet algebras. A Fréchet algebra is $m$-convex if and only if it is a countable projective limit of Banach algebras.Template:Sfn An element of $A$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.Template:EfnTemplate:Sfn

Examples

• Zero multiplication. If $E$ is any Fréchet space, we can make a Fréchet algebra structure by setting $e*f=0$ for all $e,f\in E$.
• Smooth functions on the circle. Let $S^1$ be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let $A=C^{\mathrm{\infty }}(S^1)$ be the set of infinitely differentiable complex-valued functions on $S^1$. This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation.) It is commutative, and the constant function $1$ acts as an identity. Define a countable set of seminorms on $A$ by $${\Vert \phi \Vert }_n={\Vert {\phi }^{(n)}\Vert }_{\mathrm{\infty }},\phi \in A,$$ where $${\Vert {\phi }^{(n)}\Vert }_{\mathrm{\infty }}=\underset{x\in S^1}{\sup }|{\phi }^{(n)}(x)|$$ denotes the supremum of the absolute value of the $n$th derivative ${\phi }^{(n)}$.Template:Efn Then, by the product rule for differentiation, we have $$\begin{array}{rl}\Vert \phi \psi {\Vert }_n& ={\Vert \sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i}){\phi }^{(i)}{\psi }^{(n-i)}\Vert }_{\mathrm{\infty }}\\ & \le \sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})\Vert \phi {\Vert }_i\Vert \psi {\Vert }_{n-i}\\ & \le \sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})\Vert \phi \Vert {\text{'}}_n\Vert \psi \Vert {\text{'}}_n\\ & =2^n\Vert \phi \Vert {\text{'}}_n\Vert \psi \Vert {\text{'}}_n,\end{array}$$ where $$(\genfrac{}{}{0ex}{}{n}{i})=\frac{n!}{i!(n-i)!},$$ denotes the binomial coefficient and $$\Vert \cdot \Vert {\text{'}}_n=\underset{k\le n}{\max }\Vert \cdot {\Vert }_k.$$ The primed seminorms are submultiplicative after re-scaling by $C_n=2^n$.
• Sequences on $\mathbb{\mathbb{N}}$. Let ${\mathbb{ℂ}}^{\mathbb{\mathbb{N}}}$ be the space of complex-valued sequences on the natural numbers $\mathbb{\mathbb{N}}$. Define an increasing family of seminorms on ${\mathbb{ℂ}}^{\mathbb{\mathbb{N}}}$ by $$\Vert \phi {\Vert }_n=\underset{k\le n}{\max }|\phi (k)|.$$ With pointwise multiplication, ${\mathbb{ℂ}}^{\mathbb{\mathbb{N}}}$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative $\Vert \phi \psi {\Vert }_n\le \Vert \phi {\Vert }_n\Vert \psi {\Vert }_n$ for $\phi ,\psi \in A$. This $m$-convex Fréchet algebra is unital, since the constant sequence $1(k)=1,k\in \mathbb{\mathbb{N}}$ is in $A$.
• Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, $C(\mathbb{ℂ})$, the algebra of all continuous functions on the complex plane $\mathbb{ℂ}$, or to the algebra $\mathrm{H}\mathrm{o}\mathrm{l}(\mathbb{ℂ})$ of holomorphic functions on $\mathbb{ℂ}$.
• Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let $G$ be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements $U=\{g_1,\dots ,g_n\}\subseteq G$ such that: $$\bigcup _{n=0}^{\mathrm{\infty }}{U}^n=G.$$ Without loss of generality, we may also assume that the identity element $e$ of $G$ is contained in $U$. Define a function $\ell :G\to [0,\mathrm{\infty })$ by $$\ell (g)=\min \{n\mid g\in U^n\}.$$ Then $\ell (gh)\le \ell (g)+\ell (h)$, and $\ell (e)=0$, since we define $U^0=\{e\}$.Template:Efn Let $A$ be the $\mathbb{ℂ}$-vector space $$S(G)=\{\phi :G\to \mathbb{ℂ}|\Vert \phi {\Vert }_d<\mathrm{\infty },d=0,1,2,\dots \},$$ where the seminorms $\Vert \cdot {\Vert }_d$ are defined by $$\Vert \phi {\Vert }_d=\Vert {\ell }^d\phi {\Vert }_1=\sum _{g\in G}\ell (g{)}^d|\phi (g)|.$$Template:Efn $A$ is an $m$-convex Fréchet algebra for the convolution multiplication $$\phi *\psi (g)=\sum _{h\in G}\phi (h)\psi (h^{-1}g),$$Template:Efn $A$ is unital because $G$ is discrete, and $A$ is commutative if and only if $G$ is Abelian.
• Non $m$-convex Fréchet algebras. The Aren's algebra $$A=L^{\omega }[0,1]=\bigcap _{p\ge 1}{L}^p[0,1]$$ is an example of a commutative non-$m$-convex Fréchet algebra with discontinuous inversion. The topology is given by $L^p$ norms $$\Vert f{\Vert }_p={(\underset{0}{\overset{1}{\int }}|f(t){|}^{p}dt)}^{1/p},f\in A,$$ and multiplication is given by convolution of functions with respect to Lebesgue measure on $[0,1]$.Template:Sfn

Generalizations

We can drop the requirement for the algebra to be locally convex, but still a complete metric space. In this case, the underlying space may be called a Fréchet spaceTemplate:Sfn or an F-space.Template:Sfn

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).Template:Sfnm A complete LMC algebra is called an Arens-Michael algebra.Template:Sfn

Michael's Conjecture

The question of whether all linear multiplicative functionals on an $m$-convex Frechet algebra are continuous is known as Michael's Conjecture. This conjecture is perhaps the most famous open problem in the theory of topological algebras.