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 $$m \ge n$$ for each $$n$$ such that $$\left\| a b \right\|_n \leq C_n \left\| a \right\|_m \left\|b \right\|_m$$ 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: $$\| a b \|_n \leq \| a \|_n \| b \|_n$$ 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, $$inv A$$ may not be an open set. If $$inv A$$ 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 $$inv A^+$$, or the set of quasi invertibles may take the place of $$inv A$$.)
 * 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 $$m \in \N$$ 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 $$n \in \N$$. 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
\| \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{align}$$ where $${n \choose i} = \frac{n!},$$ denotes the binomial coefficient and $$\| \cdot \|'_{n} = \max_{k \leq n} \| \cdot \|_{k}.$$ The primed seminorms are submultiplicative after re-scaling by $$C_n=2^n$$.
 * Zero multiplication. If $$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{align}
 * Sequences on $$\N$$. Let $$\Complex^\N$$ be the space of complex-valued sequences on the natural numbers $$\N$$. Define an increasing family of seminorms on $$\Complex^\N$$ by $$\| \varphi \|_n = \max_{k\leq n} |\varphi(k)|.$$ With pointwise multiplication, $$\Complex^\N$$ 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 $$1(k) = 1, k \in \N$$ is in $$A$$.
 * Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, $$C(\Complex)$$, the algebra of all continuous functions on the complex plane $$\Complex$$, or to the algebra $$\mathrm{Hol}(\Complex)$$ of holomorphic functions on $$\Complex$$.
 * 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 $$\Complex$$-vector space $$S(G) = \biggr\{ \varphi : G \to \Complex \,\,\biggl|\,\, \| \varphi \|_{d} < \infty,\quad d = 0,1, 2, \dots \biggr\},$$ 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.

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. For a long time, this conjecture was perhaps the most famous open problem in the theory of topological algebras. Michael's Conjecture was solved completely and affirmatively in 2022.