Fréchet algebra

In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra ${\displaystyle A}$ over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ${\displaystyle (a,b)\mapsto a*b}$ for ${\displaystyle a,b\in A}$ is required to be jointly continuous. If ${\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }}$ is an increasing family^[a] of seminorms for the topology of ${\displaystyle A}$, the joint continuity of multiplication is equivalent to there being a constant ${\displaystyle C_{n}>0}$ and integer ${\displaystyle m\geq n}$ for each ${\displaystyle n}$ such that ${\displaystyle \left\|ab\right\|_{n}\leq C_{n}\left\|a\right\|_{m}\left\|b\right\|_{m}}$ for all ${\displaystyle a,b\in A}$.^[b] Fréchet algebras are also called B₀-algebras.^[1]

A Fréchet algebra is ${\displaystyle m}$-convex if there exists such a family of semi-norms for which ${\displaystyle m=n}$. In that case, by rescaling the seminorms, we may also take ${\displaystyle C_{n}=1}$ for each ${\displaystyle n}$ and the seminorms are said to be submultiplicative: ${\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}}$ for all ${\displaystyle a,b\in A.}$^[c] ${\displaystyle m}$-convex Fréchet algebras may also be called Fréchet algebras.^[2]

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

Properties

• Continuity of multiplication. Multiplication is separately continuous if ${\displaystyle a_{k}b\to ab}$ and ${\displaystyle ba_{k}\to ba}$ for every ${\displaystyle a,b\in A}$ and sequence ${\displaystyle a_{k}\to a}$ converging in the Fréchet topology of ${\displaystyle A}$. Multiplication is jointly continuous if ${\displaystyle a_{k}\to a}$ and ${\displaystyle b_{k}\to b}$ imply ${\displaystyle a_{k}b_{k}\to ab}$. 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.^[3]
• Group of invertible elements. If ${\displaystyle invA}$ is the set of invertible elements of ${\displaystyle A}$, then the inverse map $${\displaystyle {\begin{cases}invA\to invA\\u\mapsto u^{-1}\end{cases}}}$$ is continuous if and only if ${\displaystyle invA}$ is a ${\displaystyle G_{\delta }}$ set.^[4] Unlike for Banach algebras, ${\displaystyle invA}$ may not be an open set. If ${\displaystyle invA}$ is open, then ${\displaystyle A}$ is called a ${\displaystyle Q}$-algebra. (If ${\displaystyle A}$ happens to be non-unital, then we may adjoin a unit to ${\displaystyle A}$^[d] and work with ${\displaystyle invA^{+}}$, or the set of quasi invertibles^[e] may take the place of ${\displaystyle invA}$.)
• Conditions for ${\displaystyle m}$-convexity. A Fréchet algebra is ${\displaystyle m}$-convex if and only if for every, if and only if for one, increasing family ${\displaystyle \{\|\cdot \|_{n}\}_{n=0}^{\infty }}$ of seminorms which topologize ${\displaystyle A}$, for each ${\displaystyle m\in \mathbb {N} }$ there exists ${\displaystyle p\geq m}$ and ${\displaystyle C_{m}>0}$ such that $${\displaystyle \|a_{1}a_{2}\cdots a_{n}\|_{m}\leq C_{m}^{n}\|a_{1}\|_{p}\|a_{2}\|_{p}\cdots \|a_{n}\|_{p},}$$ for all ${\displaystyle a_{1},a_{2},\dots ,a_{n}\in A}$ and ${\displaystyle n\in \mathbb {N} }$.^[5] A commutative Fréchet ${\displaystyle Q}$-algebra is ${\displaystyle m}$-convex,^[6] but there exist examples of non-commutative Fréchet ${\displaystyle Q}$-algebras which are not ${\displaystyle m}$-convex.^[7]
• Properties of ${\displaystyle m}$-convex Fréchet algebras. A Fréchet algebra is ${\displaystyle m}$-convex if and only if it is a countable projective limit of Banach algebras.^[8] An element of ${\displaystyle A}$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.^[f][9][10]

Examples

• Zero multiplication. If ${\displaystyle E}$ is any Fréchet space, we can make a Fréchet algebra structure by setting ${\displaystyle e*f=0}$ for all ${\displaystyle e,f\in E}$.
• Smooth functions on the circle. Let ${\displaystyle S^{1}}$ be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let ${\displaystyle A=C^{\infty }(S^{1})}$ be the set of infinitely differentiable complex-valued functions on ${\displaystyle S^{1}}$. 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 ${\displaystyle 1}$ acts as an identity. Define a countable set of seminorms on ${\displaystyle A}$ by $${\displaystyle \left\|\varphi \right\|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,}$$ where $${\displaystyle \left\|\varphi ^{(n)}\right\|_{\infty }=\sup _{x\in {S^{1}}}\left|\varphi ^{(n)}(x)\right|}$$ denotes the supremum of the absolute value of the ${\displaystyle n}$th derivative ${\displaystyle \varphi ^{(n)}}$.^[g] Then, by the product rule for differentiation, we have $${\displaystyle {\begin{aligned}\|\varphi \psi \|_{n}&=\left\|\sum _{i=0}^{n}{n \choose i}\varphi ^{(i)}\psi ^{(n-i)}\right\|_{\infty }\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|_{i}\|\psi \|_{n-i}\\&\leq \sum _{i=0}^{n}{n \choose i}\|\varphi \|'_{n}\|\psi \|'_{n}\\&=2^{n}\|\varphi \|'_{n}\|\psi \|'_{n},\end{aligned}}}$$ where $${\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}},}$$ denotes the binomial coefficient and $${\displaystyle \|\cdot \|'_{n}=\max _{k\leq n}\|\cdot \|_{k}.}$$ The primed seminorms are submultiplicative after re-scaling by ${\displaystyle C_{n}=2^{n}}$.
• Sequences on ${\displaystyle \mathbb {N} }$. Let ${\displaystyle \mathbb {C} ^{\mathbb {N} }}$ be the space of complex-valued sequences on the natural numbers ${\displaystyle \mathbb {N} }$. Define an increasing family of seminorms on ${\displaystyle \mathbb {C} ^{\mathbb {N} }}$ by $${\displaystyle \|\varphi \|_{n}=\max _{k\leq n}|\varphi (k)|.}$$ With pointwise multiplication, ${\displaystyle \mathbb {C} ^{\mathbb {N} }}$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative ${\displaystyle \|\varphi \psi \|_{n}\leq \|\varphi \|_{n}\|\psi \|_{n}}$ for ${\displaystyle \varphi ,\psi \in A}$. This ${\displaystyle m}$-convex Fréchet algebra is unital, since the constant sequence ${\displaystyle 1(k)=1,k\in \mathbb {N} }$ is in ${\displaystyle A}$.
• Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, ${\displaystyle C(\mathbb {C} )}$, the algebra of all continuous functions on the complex plane ${\displaystyle \mathbb {C} }$, or to the algebra ${\displaystyle \mathrm {Hol} (\mathbb {C} )}$ of holomorphic functions on ${\displaystyle \mathbb {C} }$.
• Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let ${\displaystyle G}$ be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements ${\displaystyle U=\{g_{1},\dots ,g_{n}\}\subseteq G}$ such that: $${\displaystyle \bigcup _{n=0}^{\infty }U^{n}=G.}$$ Without loss of generality, we may also assume that the identity element ${\displaystyle e}$ of ${\displaystyle G}$ is contained in ${\displaystyle U}$. Define a function ${\displaystyle \ell :G\to [0,\infty )}$ by $${\displaystyle \ell (g)=\min\{n\mid g\in U^{n}\}.}$$ Then ${\displaystyle \ell (gh)\leq \ell (g)+\ell (h)}$, and ${\displaystyle \ell (e)=0}$, since we define ${\displaystyle U^{0}=\{e\}}$.^[h] Let ${\displaystyle A}$ be the ${\displaystyle \mathbb {C} }$-vector space $${\displaystyle S(G)={\biggr \{}\varphi :G\to \mathbb {C} \,\,{\biggl |}\,\,\|\varphi \|_{d}<\infty ,\quad d=0,1,2,\dots {\biggr \}},}$$ where the seminorms ${\displaystyle \|\cdot \|_{d}}$ are defined by $${\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.}$$^[i] ${\displaystyle A}$ is an ${\displaystyle m}$-convex Fréchet algebra for the convolution multiplication $${\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),}$$^[j] ${\displaystyle A}$ is unital because ${\displaystyle G}$ is discrete, and ${\displaystyle A}$ is commutative if and only if ${\displaystyle G}$ is Abelian.
• Non ${\displaystyle m}$-convex Fréchet algebras. The Aren's algebra $${\displaystyle A=L^{\omega }[0,1]=\bigcap _{p\geq 1}L^{p}[0,1]}$$ is an example of a commutative non-${\displaystyle m}$-convex Fréchet algebra with discontinuous inversion. The topology is given by ${\displaystyle L^{p}}$ norms $${\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{1/p},\qquad f\in A,}$$ and multiplication is given by convolution of functions with respect to Lebesgue measure on ${\displaystyle [0,1]}$.^[11]

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^[12] or an F-space.^[13]

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).^[14] A complete LMC algebra is called an Arens-Michael algebra.^[15]

Michael's Conjecture

The question of whether all linear multiplicative functionals on an ${\displaystyle m}$-convex Frechet algebra are continuous is known as Michael's Conjecture.^[16] This conjecture is perhaps the most famous open problem in the theory of topological algebras.

Notes

1. An increasing family means that for each ${\displaystyle a\in A,}$

   ${\displaystyle \|a\|_{0}\leq \|a\|_{1}\leq \cdots \leq \|a\|_{n}\leq \cdots }$.

2. Joint continuity of multiplication means that for every absolutely convex neighborhood ${\displaystyle V}$ of zero, there is an absolutely convex neighborhood ${\displaystyle U}$ of zero for which ${\displaystyle U^{2}\subseteq V,}$ from which the seminorm inequality follows. Conversely,

   ${\displaystyle {\begin{aligned}&{}\|a_{k}b_{k}-ab\|_{n}\\&=\|a_{k}b_{k}-ab_{k}+ab_{k}-ab\|_{n}\\&\leq \|a_{k}b_{k}-ab_{k}\|_{n}+\|ab_{k}-ab\|_{n}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b_{k}\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}\\&\leq C_{n}{\biggl (}\|a_{k}-a\|_{m}\|b\|_{m}+\|a_{k}-a\|_{m}\|b_{k}-b\|_{m}+\|a\|_{m}\|b_{k}-b\|_{m}{\biggr )}.\end{aligned}}}$

3. In other words, an ${\displaystyle m}$-convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: ${\displaystyle p(fg)\leq p(f)p(g),}$ and the algebra is complete.
4. If ${\displaystyle A}$ is an algebra over a field ${\displaystyle k}$, the unitization ${\displaystyle A^{+}}$ of ${\displaystyle A}$ is the direct sum ${\displaystyle A\oplus k1}$, with multiplication defined as ${\displaystyle (a+\mu 1)(b+\lambda 1)=ab+\mu b+\lambda a+\mu \lambda 1.}$
5. If ${\displaystyle a\in A}$, then ${\displaystyle b\in A}$ is a quasi-inverse for ${\displaystyle a}$ if ${\displaystyle a+b-ab=0}$.
6. If ${\displaystyle A}$ is non-unital, replace invertible with quasi-invertible.
7. To see the completeness, let ${\displaystyle \varphi _{k}}$ be a Cauchy sequence. Then each derivative ${\displaystyle \varphi _{k}^{(l)}}$ is a Cauchy sequence in the sup norm on ${\displaystyle S^{1}}$, and hence converges uniformly to a continuous function ${\displaystyle \psi _{l}}$ on ${\displaystyle S^{1}}$. It suffices to check that ${\displaystyle \psi _{l}}$ is the ${\displaystyle l}$th derivative of ${\displaystyle \psi _{0}}$. But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have

   ${\displaystyle {\begin{aligned}&{}\psi _{l}(x)-\psi _{l}(x_{0})\\=&{}\lim _{k\to \infty }\left(\varphi _{k}^{(l)}(x)-\varphi _{k}^{(l)}(x_{0})\right)\\=&{}\lim _{k\to \infty }\int _{x_{0}}^{x}\varphi _{k}^{(l+1)}(t)dt\\=&{}\int _{x_{0}}^{x}\psi _{l+1}(t)dt.\end{aligned}}}$

8. We can replace the generating set ${\displaystyle U}$ with ${\displaystyle U\cup U^{-1}}$, so that ${\displaystyle U=U^{-1}}$. Then ${\displaystyle \ell }$ satisfies the additional property ${\displaystyle \ell (g^{-1})=\ell (g)}$, and is a length function on ${\displaystyle G}$.
9. To see that ${\displaystyle A}$ is Fréchet space, let ${\displaystyle \varphi _{n}}$ be a Cauchy sequence. Then for each ${\displaystyle g\in G}$, ${\displaystyle \varphi _{n}(g)}$ is a Cauchy sequence in ${\displaystyle \mathbb {C} }$. Define ${\displaystyle \varphi (g)}$ to be the limit. Then

   ${\displaystyle {\begin{aligned}&\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|\\&\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi _{m}(g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|\\&\leq \|\varphi _{n}-\varphi _{m}\|_{d}+\sum _{g\in S}\ell (g)^{d}|\varphi _{m}(g)-\varphi (g)|,\end{aligned}}}$

   where the sum ranges over any finite subset ${\displaystyle S}$ of ${\displaystyle G}$. Let ${\displaystyle \epsilon >0}$, and let ${\displaystyle K_{\epsilon }>0}$ be such that ${\displaystyle \|\varphi _{n}-\varphi _{m}\|_{d}<\epsilon }$ for ${\displaystyle m,n\geq K_{\epsilon }}$. By letting ${\displaystyle m}$ run, we have

   ${\displaystyle \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|<\epsilon }$

   for ${\displaystyle n\geq K_{\epsilon }}$. Summing over all of ${\displaystyle G}$, we therefore have ${\displaystyle \left\|\varphi _{n}-\varphi \right\|_{d}<\epsilon }$ for ${\displaystyle n\geq K_{\epsilon }}$. By the estimate

   ${\displaystyle {\begin{aligned}&{}\sum _{g\in S}\ell (g)^{d}|\varphi (g)|\\&{}\leq \sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)-\varphi (g)|+\sum _{g\in S}\ell (g)^{d}|\varphi _{n}(g)|\\&{}\leq \|\varphi _{n}-\varphi \|_{d}+\|\varphi _{n}\|_{d},\end{aligned}}}$

   we obtain ${\displaystyle \|\varphi \|_{d}<\infty }$. Since this holds for each ${\displaystyle d\in \mathbb {N} }$, we have ${\displaystyle \varphi \in A}$ and ${\displaystyle \varphi _{n}\to \varphi }$ in the Fréchet topology, so ${\displaystyle A}$ is complete.

10. ${\displaystyle {\begin{aligned}&\|\varphi *\psi \|_{d}\\&\leq \sum _{g\in G}\left(\sum _{h\in G}\ell (g)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\right)\\&\leq \sum _{g,h\in G}\left(\ell (h)+\ell \left(h^{-1}g\right)\right)^{d}|\varphi (h)|\left|\psi (h^{-1}g)\right|\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{g,h\in G}\left|\ell ^{i}\varphi (h)\right|\left|\ell ^{d-i}\psi (h^{-1}g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\left(\sum _{h\in G}\left|\ell ^{i}\varphi (h)\right|\right)\left(\sum _{g\in G}\left|\ell ^{d-i}\psi (g)\right|\right)\\&=\sum _{i=0}^{d}{d \choose i}\|\varphi \|_{i}\|\psi \|_{d-i}\\&\leq 2^{d}\|\varphi \|'_{d}\|\psi \|'_{d}\end{aligned}}}$