Fréchet algebra

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 $\{\|\cdot \|_{n}\}_{n=0}^{\infty }$ is an increasing family of seminorms for the topology of $A$, the joint continuity of multiplication is equivalent to there being a constant $C_{n}>0$ and integer $m\geq n$ for each $n$ such that $\left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}$ for all $a,b\in A$. Fréchet algebras are also called B₀-algebras.

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: $\|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}$ for all $a,b\in A.$ $m$-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 $\|1_{A}\|_{n}=1,$ as is often done for Banach algebras.

Properties

• Continuity of multiplication. Multiplication is separately continuous if $a_{k}b\to ab$ and $ba_{k}\to ba$ 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
  $${\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}$$
  is continuous if and only if $invA$ is a $G_{\delta }$ set. 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 $A$ and work with $invA^{+}$, or the set of quasi invertibles 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 $\{\|\cdot \|_{n}\}_{n=0}^{\infty }$ of seminorms which topologize $A$, for each $m\in \mathbb {N}$ there exists $p\geq m$ and $C_{m}>0$ such that
  $$\|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},$$
  for all $a_{1},a_{2},\dots ,a_{n}\in A$ and $n\in \mathbb {N}$. A commutative Fréchet $Q$-algebra is $m$-convex, but there exist examples of non-commutative Fréchet $Q$-algebras which are not $m$-convex.
• 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. An element of $A$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.

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 $A=C^{\infty }(S^{1})$ 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
  $$\left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,$$
  where
  $$\left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|$$
  denotes the supremum of the absolute value of the $n$th derivative $\varphi ^{(n)}$. Then, by the product rule for differentiation, we have
  $${\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}$$
  where
  $${n \choose i}={\frac {n!}{i!(n-i)!}},$$
  denotes the binomial coefficient and
  $$\|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.$$
  The primed seminorms are submultiplicative after re-scaling by $C_{n}=2^{n}$.
• Sequences on $\mathbb {N}$. Let $\mathbb {C} ^{\mathbb {N} }$ be the space of complex-valued sequences on the natural numbers $\mathbb {N}$. Define an increasing family of seminorms on $\mathbb {C} ^{\mathbb {N} }$ by
  $$\|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.$$
  With pointwise multiplication, $\mathbb {C} ^{\mathbb {N} }$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative $\|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}$ for $\varphi ,\psi \in A$. This $m$-convex Fréchet algebra is unital, since the constant sequence $1(k)=1,k\in \mathbb {N}$ 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 $\mathrm {Hol} (\mathbb {C} )$ 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:
  $$\bigcup _{n=0}^{\infty }U^{n}=G.$$
  Without loss of generality, we may also assume that the identity element $e$ of $G$ is contained in $U$. Define a function $\ell :G\to [0,\infty )$ by
  $$\ell (g)=\min\{n\mid g\in U^{n}\}.$$
  Then $\ell (gh)\leq \ell (g)+\ell (h)$, and $\ell (e)=0$, since we define $U^{0}=\{e\}$. Let $A$ be the $\mathbb {C}$-vector space
  $$S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},$$
  where the seminorms $\|\cdot \|_{d}$ are defined by
  $$\|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.$$
  $A$ is an $m$-convex Fréchet algebra for the convolution multiplication
  $$\varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),$$
  $A$ 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
  $$A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]$$
  is an example of a commutative non-$m$-convex Fréchet algebra with discontinuous inversion. The topology is given by $L^{p}$ norms
  $$\|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,$$
  and multiplication is given by convolution of functions with respect to Lebesgue measure on $[0,1]$.

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 space or an F-space.

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

Open problems

Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an $m$-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture.

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