Fréchet algebra

• 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 (Waelbroeck 1971, Chapter VII, Proposition 1, Palmer 1994, ${\displaystyle \S }$ 2.9).
• 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 (Waelbroeck 1971, Chapter VII, Proposition 2). 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 ${\displaystyle A}$[lower-alpha 4] and work with ${\displaystyle invA^{+}}$, or the set of quasi invertibles[lower-alpha 5] 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 ${\displaystyle n\in \mathbb {N} }$ (Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2). A commutative Fréchet ${\displaystyle Q}$-algebra is ${\displaystyle m}$-convex (Żelazko 1965, Theorem 13.17). But there exist examples of non-commutative Fréchet ${\displaystyle Q}$-algebras which are not ${\displaystyle m}$-convex (Żelazko 1994).

• 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 (Michael 1952, Theorem 5.1). An element of ${\displaystyle A}$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible (Michael 1952, Theorem 5.2).[lower-alpha 6] See also (Palmer 1994, Theorem 2.9.6).

• 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

${\displaystyle \|\varphi \|_{n}=\left\|\varphi ^{(n)}\right\|_{\infty },\qquad \varphi \in A,}$

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 ${\displaystyle \varphi ^{(n)}}$.[lower-alpha 7] 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 ${\displaystyle Hol(\mathbb {C} )}$ 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 ${\displaystyle U^{0}=\{e\}}$.[lower-alpha 8] 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

${\displaystyle \|\varphi \|_{d}=\|\ell ^{d}\varphi \|_{1}=\sum _{g\in G}\ell (g)^{d}|\varphi (g)|.}$[lower-alpha 9]

${\displaystyle A}$ is an ${\displaystyle m}$-convex Fréchet algebra for the convolution multiplication

${\displaystyle \varphi *\psi (g)=\sum _{h\in G}\varphi (h)\psi (h^{-1}g),}$[lower-alpha 10]

${\displaystyle A}$ 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

${\displaystyle A=L^{\omega }[0,1]=\bigcup _{p\geq 1}L^{p}[0,1]}$

is an example of a commutative non-${\displaystyle m}$-convex Fréchet algebra with discontinuous inversion. The topology is given by ${\displaystyle L^{p}}$ norms

${\displaystyle \|f\|_{p}=\left(\int _{0}^{1}|f(t)|^{p}dt\right)^{\frac {1}{p}},\qquad f\in A,}$

and multiplication is given by convolution of functions with respect to Lebesgue measure on ${\displaystyle [0,1]}$ (Fragoulopoulou 2005, Example 6.13 (2)).

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 (Waelbroeck 1971) or an F-space (Rudin 1973, 1.8(e)).

If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC) (Michael 1952, Husain 1991). A complete LMC algebra is called an Arens-Michael algebra (Fragoulopoulou 2005, Chapter 1).

Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an ${\displaystyle m}$-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture (Michael 1952, ${\displaystyle \S }$ 12, Question 1, Palmer 1994, ${\displaystyle \S }$ 3.1).