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

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

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

 Properties

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

 أمثلة

• Zero multiplication. If ${\displaystyle{\displaystyle E}}$ is any Fréchet space, we can make a Fréchet algebra structure by setting ${\displaystyle{\displaystyle e*f=0}}$ for all ${\displaystyle{\displaystyle e,f\in E}}$.
• Smooth functions on the circle. Let ${\displaystyle{\displaystyle S^{1}}}$ be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let ${\displaystyle{\displaystyle A=C^{\infty}(S^{1})}}$ be the set of infinitely differentiable complex-valued functions on ${\displaystyle{\displaystyle S^{1}}}$. 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 ${\displaystyle{\displaystyle 1}}$ acts as an identity. Define a countable set of seminorms on ${\displaystyle{\displaystyle A}}$ 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 ${\displaystyle{\displaystyle n}}$th derivative ${\displaystyle{\displaystyle\varphi^{(n)}}}$. Then, by the product rule for differentiation, we have
  $${\displaystyle{\begin{aligned} \displaystyle\|\varphi\psi\|_{n}&\displaystyle=\left\|\sum_{i=0}^{n}{n\choose i}\varphi^{(i)}\psi^{(n-i)}\right\|_{\infty}\\ &\displaystyle\leq\sum_{i=0}^{n}{n\choose i}\|\varphi\|_{i}\|\psi\|_{n-i}\\ &\displaystyle\leq\sum_{i=0}^{n}{n\choose i}\|\varphi\|^{\prime}_{n}\|\psi\|^{\prime}_{n}\\ &\displaystyle=2^{n}\|\varphi\|^{\prime}_{n}\|\psi\|^{\prime}_{n},\end{aligned}}}$$
  where
  $${\displaystyle{n\choose i}={\frac{n!}{i!(n-i)!}},}$$
  denotes the binomial coefficient and
  $${\displaystyle\|\cdot\|^{\prime}_{n}=\max_{k\leq n}\|\cdot\|_{k}.}$$
  The primed seminorms are submultiplicative after re-scaling by ${\displaystyle{\displaystyle C_{n}=2^{n}}}$.
• Sequences on ${\displaystyle{\displaystyle\mathbb{N}}}$. Let ${\displaystyle{\displaystyle\mathbb{C}^{\mathbb{N}}}}$ be the space of complex-valued sequences on the natural numbers ${\displaystyle{\displaystyle\mathbb{N}}}$. Define an increasing family of seminorms on ${\displaystyle{\displaystyle\mathbb{C}^{\mathbb{N}}}}$ by
  $${\displaystyle\|\varphi\|_{n}=\max_{k\leq n}|\varphi(k)|.}$$
  With pointwise multiplication, ${\displaystyle{\displaystyle\mathbb{C}^{\mathbb{N}}}}$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative ${\displaystyle{\displaystyle\|\varphi\psi\|_{n}\leq\|\varphi\|_{n}\|\psi\|_{n}}}$ for ${\displaystyle{\displaystyle\varphi,\psi\in A}}$. This ${\displaystyle{\displaystyle m}}$-convex Fréchet algebra is unital, since the constant sequence ${\displaystyle{\displaystyle 1(k)=1,k\in\mathbb{N}}}$ is in ${\displaystyle{\displaystyle A}}$.
• Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, ${\displaystyle{\displaystyle C(\mathbb{C})}}$, the algebra of all continuous functions on the complex plane ${\displaystyle{\displaystyle\mathbb{C}}}$, or to the algebra ${\displaystyle{\displaystyle\mathrm{Hol}(\mathbb{C})}}$ of holomorphic functions on ${\displaystyle{\displaystyle\mathbb{C}}}$.
• Convolution algebra of rapidly vanishing functions on a finitely generated discrete group. Let ${\displaystyle{\displaystyle G}}$ be a finitely generated group, with the discrete topology. This means that there exists a set of finitely many elements ${\displaystyle{\displaystyle U=\{g_{1},\dots,g_{n}\}\subseteq G}}$ such that:
  $${\displaystyle\bigcup_{n=0}^{\infty}U^{n}=G.}$$
  Without loss of generality, we may also assume that the identity element ${\displaystyle{\displaystyle e}}$ of ${\displaystyle{\displaystyle G}}$ is contained in ${\displaystyle{\displaystyle U}}$. Define a function ${\displaystyle{\displaystyle\ell:G\to[0,\infty)}}$ by
  $${\displaystyle\ell(g)=\min\{n\mid g\in U^{n}\}.}$$
  Then ${\displaystyle{\displaystyle\ell(gh)\leq\ell(g)+\ell(h)}}$, and ${\displaystyle{\displaystyle\ell(e)=0}}$, since we define ${\displaystyle{\displaystyle U^{0}=\{e\}}}$. Let ${\displaystyle{\displaystyle A}}$ be the ${\displaystyle{\displaystyle\mathbb{C}}}$-vector space
  $$S(G)={\biggr{\{}}\varphi:G\to\mathbb{C}\,\,{\biggl{|}}\,\,\|\varphi\|_{d}<\infty,\quad d=0,1,2,\dots{\biggr{\}}},$$
  where the seminorms ${\displaystyle{\displaystyle\|\cdot\|_{d}}}$ are defined by
  $${\displaystyle\|\varphi\|_{d}=\|\ell^{d}\varphi\|_{1}=\sum_{g\in G}\ell(g)^{d}|\varphi(g)|.}$$
  ${\displaystyle{\displaystyle A}}$ is an ${\displaystyle{\displaystyle m}}$-convex Fréchet algebra for the convolution multiplication
  $${\displaystyle\varphi*\psi(g)=\sum_{h\in G}\varphi(h)\psi(h^{-1}g),}$$
  ${\displaystyle{\displaystyle A}}$ is unital because ${\displaystyle{\displaystyle G}}$ is discrete, and ${\displaystyle{\displaystyle A}}$ is commutative if and only if ${\displaystyle{\displaystyle G}}$ is Abelian.
• Non ${\displaystyle{\displaystyle m}}$-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-${\displaystyle{\displaystyle m}}$-convex Fréchet algebra with discontinuous inversion. The topology is given by ${\displaystyle{\displaystyle L^{p}}}$ 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 ${\displaystyle{\displaystyle[0,1]}}$.

 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 ${\displaystyle{\displaystyle m}}$-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.

 المصادر

• Fragoulopoulou, Maria (2005). "Bibliography". Topological Algebras with Involution. North-Holland Mathematics Studies. Vol. 200. Amsterdam: Elsevier B.V. pp. 451–485. doi:10.1016/S0304-0208(05)80031-3. ISBN 978-044452025-8.
• Husain, Taqdir (1991). Orthogonal Schauder Bases. Pure and Applied Mathematics. Vol. 143. New York City: Marcel Dekker. ISBN 0-8247-8508-8.
• Michael, Ernest A. (1952). Locally Multiplicatively-Convex Topological Algebras. Memoirs of the American Mathematical Society. Vol. 11. MR 0051444.
• Mitiagin, B.; Rolewicz, S.; Żelazko, W. (1962). "Entire functions in B₀-algebras". Studia Mathematica. 21 (3): 291–306. doi:10.4064/sm-21-3-291-306. MR 0144222.
• Palmer, T.W. (1994). Banach Algebras and the General Theory of *-algebras, Volume I: Algebras and Banach Algebras. Encyclopedia of Mathematics and its Applications. Vol. 49. New York City: Cambridge University Press. ISBN 978-052136637-3.
• Rudin, Walter (1973). Functional Analysis. Series in Higher Mathematics. New York City: McGraw-Hill Book. 1.8(e). ISBN 978-007054236-5 – via Internet Archive.
• Waelbroeck, Lucien (1971). Topological Vector Spaces and Algebras. Lecture Notes in Mathematics. Vol. 230. doi:10.1007/BFb0061234. ISBN 978-354005650-8. MR 0467234.
• Żelazko, W. (1965). "Metric generalizations of Banach algebras". Rozprawy Mat. (Dissertationes Math.). 47. Theorem 13.17. MR 0193532.
• Żelazko, W. (1994). "Concerning entire functions in B₀-algebras". Studia Mathematica. 110 (3): 283–290. doi:10.4064/sm-110-3-283-290. MR 1292849.
• Żelazko, W. (2001) [1994]. "Fréchet algebra". Encyclopedia of Mathematics. EMS Press.