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 $\_^\infty$ 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 ''B''₀-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 $\begin invA \to invA \\ u \mapsto u^ \end$ 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 $\_^\infty$ 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

* 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 $A=C^(S^1)$ 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 $\left\, \varphi \right\, _ = \left \, \varphi^ \right \, _, \qquad \varphi \in A,$ where $\left \, \varphi^ \right \, _ = \sup_ \left , \varphi^(x) \right ,$ denotes the supremum of the absolute value of the $n$th derivative $\varphi^$. Then, by the product rule for differentiation, we have $\begin \, \varphi \psi \, _ &= \left \, \sum_^ \varphi^ \psi^ \right \, _ \\ &\leq \sum_^ \, \varphi \, _ \, \psi \, _ \\ &\leq \sum_^ \, \varphi \, '_ \, \psi \, '_ \\ &= 2^n\, \varphi \, '_ \, \psi \, '_, \end$ where $= \frac,$ denotes the binomial coefficient and $\, \cdot \, '_ = \max_ \, \cdot \, _.$ The primed seminorms are submultiplicative after re-scaling by $C_n=2^n$.
* 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 $\, \varphi \, _n = \max_ , \varphi(k), .$ 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 $\mathrm(\Complex)$ 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 $U= \ \subseteq G$ such that: $\bigcup_^ U^n = G.$ 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 $\ell(g) = \min \.$ Then $\ell(gh ) \leq \ell(g) + \ell(h)$, and $\ell(e) = 0$, since we define $U^ = \$. Let $A$ be the $\Complex$-vector space $S(G) = \biggr\,$ where the seminorms $\, \cdot \, _$ are defined by $\, \varphi \, _ = \, \ell^d \varphi \, _ =\sum_ \ell(g)^d , \varphi(g), .$ $A$ is an $m$-convex Fréchet algebra for the convolution#Convolutions on groups, convolution multiplication $\varphi * \psi (g) = \sum_ \varphi(h) \psi(h^g),$ $A$ 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 ,1= \bigcap_ L^p ,1/math> is an example of a commutative non-$m$-convex Fréchet algebra with discontinuous inversion. The topology is given by $L^p$ norms $\, f \, _p = \left ( \int_0^1 , f(t) , ^p dt \right )^, \qquad f \in A,$ and multiplication is given by convolution of functions with respect to Lebesgue measure on ,1/math>.

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.; This conjecture is perhaps the most famous open problem in the theory of topological algebras.

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