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^{[lower-alpha 1]} of seminorms for the topology of $A$ , the joint continuity of multiplication is equivalent to there being a constant $C_{n} > 0$ and integer $m \geq n$ for each $n$ such that ${‖ a b ‖}_{n} \leq C_{n} {‖ a ‖}_{m} {‖ b ‖}_{m}$ for all $a, b \in A$ .^{[lower-alpha 2]} Fréchet algebras are also called B₀-algebras.^[1] 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 $a, b \in A .$ ^{[lower-alpha 3]} $m$ -convex Fréchet algebras may also be called Fréchet algebras.^[2] 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 $a_{k} b \to a b$ and $b a_{k} \to b a$ 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 $a_{k} b_{k} \to a b$ . 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 $i n v A$ is the set of invertible elements of $A$ , then the inverse map ${\begin{cases} i n v A \to i n v A \\ u \mapsto u^{- 1} \end{cases}$ is continuous if and only if $i n v A$ is a $G_{δ}$ set.^[4] Unlike for Banach algebras, $i n v A$ may not be an open set. If $i n v A$ is open, then $A$ is called a $Q$ -algebra. (If $A$ happens to be non-unital, then we may adjoin a unit to $A$ ^{[lower-alpha 4]} and work with $i n v A^{+}$ , or the set of quasi invertibles^{[lower-alpha 5]} may take the place of $i n v A$ .)
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 $‖ a_{1} a_{2} \dots a_{n} ‖_{m} \leq C_{m}^{n} ‖ a_{1} ‖_{p} ‖ a_{2} ‖_{p} \dots ‖ a_{n} ‖_{p},$ for all $a_{1}, a_{2}, \dots, a_{n} \in A$ and $n \in ℕ$ .^[5] A commutative Fréchet $Q$ -algebra is $m$ -convex,^[6] but there exist examples of non-commutative Fréchet $Q$ -algebras which are not $m$ -convex.^[7]
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.^[8] An element of $A$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.^{[lower-alpha 6]}^[9]^[10]

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 $A = C^{\infty} (S^{1})$ 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 ${‖ φ ‖}_{n} = {‖ φ^{(n)} ‖}_{\infty}, φ \in A,$ where ${‖ φ^{(n)} ‖}_{\infty} = \sup_{x \in S^{1}} | φ^{(n)} (x) |$ denotes the supremum of the absolute value of the $n$ th derivative $φ^{(n)}$ .^{[lower-alpha 7]} Then, by the product rule for differentiation, we have $\begin{aligned} ‖ φ ψ ‖_{n} & = {‖ \sum_{i = 0}^{n} (\binom{n}{i}) φ^{(i)} ψ^{(n - i)} ‖}_{\infty} \\ \leq \sum_{i = 0}^{n} (\binom{n}{i}) ‖ φ ‖_{i} ‖ ψ ‖_{n - i} \\ \leq \sum_{i = 0}^{n} (\binom{n}{i}) ‖ φ ‖'_{n} ‖ ψ ‖'_{n} \\ = 2^{n} ‖ φ ‖'_{n} ‖ ψ ‖'_{n}, \end{aligned}$ where $(\binom{n}{i}) = \frac{n!}{i! (n - i)!},$ denotes the binomial coefficient and $‖ \cdot ‖'_{n} = \max_{k \leq n} ‖ \cdot ‖_{k} .$ The primed seminorms are submultiplicative after re-scaling by $C_{n} = 2^{n}$ .
Sequences on $ℕ$ . Let $ℂ^{ℕ}$ be the space of complex-valued sequences on the natural numbers $ℕ$ . Define an increasing family of seminorms on $ℂ^{ℕ}$ by $‖ φ ‖_{n} = \max_{k \leq n} | φ (k) | .$ With pointwise multiplication, $ℂ^{ℕ}$ is a commutative Fréchet algebra. In fact, each seminorm is submultiplicative $‖ φ ψ ‖_{n} \leq ‖ φ ‖_{n} ‖ ψ ‖_{n}$ for $φ, ψ \in A$ . 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, $C (ℂ)$ , the algebra of all continuous functions on the complex plane $ℂ$ , or to the algebra $H o l (ℂ)$ of holomorphic functions on $ℂ$ .
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: $⋃_{n = 0}^{\infty} U^{n} = G .$ Without loss of generality, we may also assume that the identity element $e$ of $G$ is contained in $U$ . Define a function $ℓ : G \to [0, \infty)$ by $ℓ (g) = \min {n ∣ g \in U^{n}} .$ Then $ℓ (g h) \leq ℓ (g) + ℓ (h)$ , and $ℓ (e) = 0$ , since we define $U^{0} = {e}$ .^{[lower-alpha 8]} Let $A$ be the $ℂ$ -vector space $S (G) = {φ : G \to ℂ | ‖ φ ‖_{d} < \infty, d = 0, 1, 2, \dots},$ where the seminorms $‖ \cdot ‖_{d}$ are defined by $‖ φ ‖_{d} = ‖ ℓ^{d} φ ‖_{1} = \sum_{g \in G} ℓ (g)^{d} | φ (g) | .$ ^{[lower-alpha 9]} $A$ is an $m$ -convex Fréchet algebra for the convolution multiplication $φ * ψ (g) = \sum_{h \in G} φ (h) ψ (h^{- 1} g),$ ^{[lower-alpha 10]} $A$ 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 $A = L^{ω} [0, 1] = ⋂_{p \geq 1} L^{p} [0, 1]$ is an example of a commutative non- $m$ -convex Fréchet algebra with discontinuous inversion. The topology is given by $L^{p}$ norms $‖ f ‖_{p} = {(\int_{0}^{1} | f (t) |^{p} d t)}^{1 / p}, f \in A,$ and multiplication is given by convolution of functions with respect to Lebesgue measure on $[0, 1]$ .^[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 $m$ -convex Frechet algebra are continuous is known as Michael's Conjecture.^[16] 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.^[17]

Notes

↑ An increasing family means that for each $a \in A,$
$‖ a ‖_{0} \leq ‖ a ‖_{1} \leq \dots \leq ‖ a ‖_{n} \leq \dots$ .
↑ Joint continuity of multiplication means that for every absolutely convex neighborhood $V$ of zero, there is an absolutely convex neighborhood $U$ of zero for which $U^{2} \subseteq V,$ from which the seminorm inequality follows. Conversely,
$\begin{aligned} ‖ a_{k} b_{k} - a b ‖_{n} \\ = ‖ a_{k} b_{k} - a b_{k} + a b_{k} - a b ‖_{n} \\ \leq ‖ a_{k} b_{k} - a b_{k} ‖_{n} + ‖ a b_{k} - a b ‖_{n} \\ \leq C_{n} (‖ a_{k} - a ‖_{m} ‖ b_{k} ‖_{m} + ‖ a ‖_{m} ‖ b_{k} - b ‖_{m}) \\ \leq C_{n} (‖ a_{k} - a ‖_{m} ‖ b ‖_{m} + ‖ a_{k} - a ‖_{m} ‖ b_{k} - b ‖_{m} + ‖ a ‖_{m} ‖ b_{k} - b ‖_{m}) . \end{aligned}$
↑ In other words, an $m$ -convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: $p (f g) \leq p (f) p (g),$ and the algebra is complete.
↑ If $A$ is an algebra over a field $k$ , the unitization $A^{+}$ of $A$ is the direct sum $A \oplus k 1$ , with multiplication defined as $(a + μ 1) (b + λ 1) = a b + μ b + λ a + μ λ 1 .$
↑ If $a \in A$ , then $b \in A$ is a quasi-inverse for $a$ if $a + b - a b = 0$ .
↑ If $A$ is non-unital, replace invertible with quasi-invertible.
↑ To see the completeness, let $φ_{k}$ be a Cauchy sequence. Then each derivative $φ_{k}^{(l)}$ is a Cauchy sequence in the sup norm on $S^{1}$ , and hence converges uniformly to a continuous function $ψ_{l}$ on $S^{1}$ . It suffices to check that $ψ_{l}$ is the $l$ th derivative of $ψ_{0}$ . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
$\begin{aligned} ψ_{l} (x) - ψ_{l} (x_{0}) \\ = & \lim_{k \to \infty} (φ_{k}^{(l)} (x) - φ_{k}^{(l)} (x_{0})) \\ = & \lim_{k \to \infty} \int_{x_{0}}^{x} φ_{k}^{(l + 1)} (t) d t \\ = & \int_{x_{0}}^{x} ψ_{l + 1} (t) d t . \end{aligned}$
↑ We can replace the generating set $U$ with $U \cup U^{- 1}$ , so that $U = U^{- 1}$ . Then $ℓ$ satisfies the additional property $ℓ (g^{- 1}) = ℓ (g)$ , and is a length function on $G$ .
↑ To see that $A$ is Fréchet space, let $φ_{n}$ be a Cauchy sequence. Then for each $g \in G$ , $φ_{n} (g)$ is a Cauchy sequence in $ℂ$ . Define $φ (g)$ to be the limit. Then
$\begin{aligned} \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | \\ \leq \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ_{m} (g) | + \sum_{g \in S} ℓ (g)^{d} | φ_{m} (g) - φ (g) | \\ \leq ‖ φ_{n} - φ_{m} ‖_{d} + \sum_{g \in S} ℓ (g)^{d} | φ_{m} (g) - φ (g) |, \end{aligned}$
where the sum ranges over any finite subset $S$ of $G$ . Let $ϵ > 0$ , and let $K_{ϵ} > 0$ be such that $‖ φ_{n} - φ_{m} ‖_{d} < ϵ$ for $m, n \geq K_{ϵ}$ . By letting $m$ run, we have
$\sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | < ϵ$
for $n \geq K_{ϵ}$ . Summing over all of $G$ , we therefore have ${‖ φ_{n} - φ ‖}_{d} < ϵ$ for $n \geq K_{ϵ}$ . By the estimate
$\begin{aligned} \sum_{g \in S} ℓ (g)^{d} | φ (g) | \\ \leq \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | + \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) | \\ \leq ‖ φ_{n} - φ ‖_{d} + ‖ φ_{n} ‖_{d}, \end{aligned}$
we obtain $‖ φ ‖_{d} < \infty$ . Since this holds for each $d \in ℕ$ , we have $φ \in A$ and $φ_{n} \to φ$ in the Fréchet topology, so $A$ is complete.
↑
$\begin{aligned} ‖ φ * ψ ‖_{d} \\ \leq \sum_{g \in G} (\sum_{h \in G} ℓ (g)^{d} | φ (h) | | ψ (h^{- 1} g) |) \\ \leq \sum_{g, h \in G} {(ℓ (h) + ℓ (h^{- 1} g))}^{d} | φ (h) | | ψ (h^{- 1} g) | \\ = \sum_{i = 0}^{d} (\binom{d}{i}) (\sum_{g, h \in G} | ℓ^{i} φ (h) | | ℓ^{d - i} ψ (h^{- 1} g) |) \\ = \sum_{i = 0}^{d} (\binom{d}{i}) (\sum_{h \in G} | ℓ^{i} φ (h) |) (\sum_{g \in G} | ℓ^{d - i} ψ (g) |) \\ = \sum_{i = 0}^{d} (\binom{d}{i}) ‖ φ ‖_{i} ‖ ψ ‖_{d - i} \\ \leq 2^{d} ‖ φ ‖'_{d} ‖ ψ ‖'_{d} \end{aligned}$

Citations

↑ Mitiagin, Rolewicz & Żelazko 1962; Żelazko 2001.
↑ Husain 1991; Żelazko 2001.
↑ Waelbroeck 1971, Chapter VII, Proposition 1; Palmer 1994, $§$ 2.9.
↑ Waelbroeck 1971, Chapter VII, Proposition 2.
↑ Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2.
↑ Żelazko 1965, Theorem 13.17.
↑ Żelazko 1994, pp. 283–290.
↑ Michael 1952, Theorem 5.1.
↑ Michael 1952, Theorem 5.2.
↑ See also Palmer 1994, Theorem 2.9.6.
↑ Fragoulopoulou 2005, Example 6.13 (2).
↑ Waelbroeck 1971.
↑ Rudin 1973, 1.8(e).
↑ Michael 1952; Husain 1991.
↑ Fragoulopoulou 2005, Chapter 1.
↑ Michael 1952, $§$ 12, Question 1; Palmer 1994, $§$ 3.1.
↑ Patel, S. R. (2022-06-28). "On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory". arXiv:2006.11134 [math.FA].

Sources

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.

[1] An increasing family means that for each $a \in A,$
$‖ a ‖_{0} \leq ‖ a ‖_{1} \leq \dots \leq ‖ a ‖_{n} \leq \dots$ .

[2] Joint continuity of multiplication means that for every absolutely convex neighborhood $V$ of zero, there is an absolutely convex neighborhood $U$ of zero for which $U^{2} \subseteq V,$ from which the seminorm inequality follows. Conversely,
$\begin{aligned} ‖ a_{k} b_{k} - a b ‖_{n} \\ = ‖ a_{k} b_{k} - a b_{k} + a b_{k} - a b ‖_{n} \\ \leq ‖ a_{k} b_{k} - a b_{k} ‖_{n} + ‖ a b_{k} - a b ‖_{n} \\ \leq C_{n} (‖ a_{k} - a ‖_{m} ‖ b_{k} ‖_{m} + ‖ a ‖_{m} ‖ b_{k} - b ‖_{m}) \\ \leq C_{n} (‖ a_{k} - a ‖_{m} ‖ b ‖_{m} + ‖ a_{k} - a ‖_{m} ‖ b_{k} - b ‖_{m} + ‖ a ‖_{m} ‖ b_{k} - b ‖_{m}) . \end{aligned}$

[4] In other words, an $m$ -convex Fréchet algebra is a topological algebra, in which the topology is given by a countable family of submultiplicative seminorms: $p (f g) \leq p (f) p (g),$ and the algebra is complete.

[8] If $A$ is an algebra over a field $k$ , the unitization $A^{+}$ of $A$ is the direct sum $A \oplus k 1$ , with multiplication defined as $(a + μ 1) (b + λ 1) = a b + μ b + λ a + μ λ 1 .$

[9] If $a \in A$ , then $b \in A$ is a quasi-inverse for $a$ if $a + b - a b = 0$ .

[14] If $A$ is non-unital, replace invertible with quasi-invertible.

[17] To see the completeness, let $φ_{k}$ be a Cauchy sequence. Then each derivative $φ_{k}^{(l)}$ is a Cauchy sequence in the sup norm on $S^{1}$ , and hence converges uniformly to a continuous function $ψ_{l}$ on $S^{1}$ . It suffices to check that $ψ_{l}$ is the $l$ th derivative of $ψ_{0}$ . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have
$\begin{aligned} ψ_{l} (x) - ψ_{l} (x_{0}) \\ = & \lim_{k \to \infty} (φ_{k}^{(l)} (x) - φ_{k}^{(l)} (x_{0})) \\ = & \lim_{k \to \infty} \int_{x_{0}}^{x} φ_{k}^{(l + 1)} (t) d t \\ = & \int_{x_{0}}^{x} ψ_{l + 1} (t) d t . \end{aligned}$

[18] We can replace the generating set $U$ with $U \cup U^{- 1}$ , so that $U = U^{- 1}$ . Then $ℓ$ satisfies the additional property $ℓ (g^{- 1}) = ℓ (g)$ , and is a length function on $G$ .

[19] To see that $A$ is Fréchet space, let $φ_{n}$ be a Cauchy sequence. Then for each $g \in G$ , $φ_{n} (g)$ is a Cauchy sequence in $ℂ$ . Define $φ (g)$ to be the limit. Then
$\begin{aligned} \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | \\ \leq \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ_{m} (g) | + \sum_{g \in S} ℓ (g)^{d} | φ_{m} (g) - φ (g) | \\ \leq ‖ φ_{n} - φ_{m} ‖_{d} + \sum_{g \in S} ℓ (g)^{d} | φ_{m} (g) - φ (g) |, \end{aligned}$
where the sum ranges over any finite subset $S$ of $G$ . Let $ϵ > 0$ , and let $K_{ϵ} > 0$ be such that $‖ φ_{n} - φ_{m} ‖_{d} < ϵ$ for $m, n \geq K_{ϵ}$ . By letting $m$ run, we have
$\sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | < ϵ$
for $n \geq K_{ϵ}$ . Summing over all of $G$ , we therefore have ${‖ φ_{n} - φ ‖}_{d} < ϵ$ for $n \geq K_{ϵ}$ . By the estimate
$\begin{aligned} \sum_{g \in S} ℓ (g)^{d} | φ (g) | \\ \leq \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) - φ (g) | + \sum_{g \in S} ℓ (g)^{d} | φ_{n} (g) | \\ \leq ‖ φ_{n} - φ ‖_{d} + ‖ φ_{n} ‖_{d}, \end{aligned}$
we obtain $‖ φ ‖_{d} < \infty$ . Since this holds for each $d \in ℕ$ , we have $φ \in A$ and $φ_{n} \to φ$ in the Fréchet topology, so $A$ is complete.

[20] 
$\begin{aligned} ‖ φ * ψ ‖_{d} \\ \leq \sum_{g \in G} (\sum_{h \in G} ℓ (g)^{d} | φ (h) | | ψ (h^{- 1} g) |) \\ \leq \sum_{g, h \in G} {(ℓ (h) + ℓ (h^{- 1} g))}^{d} | φ (h) | | ψ (h^{- 1} g) | \\ = \sum_{i = 0}^{d} (\binom{d}{i}) (\sum_{g, h \in G} | ℓ^{i} φ (h) | | ℓ^{d - i} ψ (h^{- 1} g) |) \\ = \sum_{i = 0}^{d} (\binom{d}{i}) (\sum_{h \in G} | ℓ^{i} φ (h) |) (\sum_{g \in G} | ℓ^{d - i} ψ (g) |) \\ = \sum_{i = 0}^{d} (\binom{d}{i}) ‖ φ ‖_{i} ‖ ψ ‖_{d - i} \\ \leq 2^{d} ‖ φ ‖'_{d} ‖ ψ ‖'_{d} \end{aligned}$

[FOOTNOTEMitiaginRolewiczŻelazko1962Żelazko2001-3] Mitiagin, Rolewicz & Żelazko 1962; Żelazko 2001.

[5] Husain 1991; Żelazko 2001.

[6] Waelbroeck 1971, Chapter VII, Proposition 1; Palmer 1994, $§$ 2.9.

[FOOTNOTEWaelbroeck1971Chapter_VII,_Proposition_2-7] Waelbroeck 1971, Chapter VII, Proposition 2.

[FOOTNOTEMitiaginRolewiczŻelazko1962Lemma_1.2-10] Mitiagin, Rolewicz & Żelazko 1962, Lemma 1.2.

[FOOTNOTEŻelazko1965Theorem_13.17-11] Żelazko 1965, Theorem 13.17.

[FOOTNOTEŻelazko1994283–290-12] Żelazko 1994, pp. 283–290.

[FOOTNOTEMichael1952Theorem_5.1-13] Michael 1952, Theorem 5.1.

[FOOTNOTEMichael1952Theorem_5.2-15] Michael 1952, Theorem 5.2.

[16] See also Palmer 1994, Theorem 2.9.6.

[FOOTNOTEFragoulopoulou2005Example_6.13_(2)-21] Fragoulopoulou 2005, Example 6.13 (2).

[FOOTNOTEWaelbroeck1971-22] Waelbroeck 1971.

[FOOTNOTERudin19731.8(e)-23] Rudin 1973, 1.8(e).

[FOOTNOTEMichael1952Husain1991-24] Michael 1952; Husain 1991.

[FOOTNOTEFragoulopoulou2005Chapter_1-25] Fragoulopoulou 2005, Chapter 1.

[26] Michael 1952, $§$ 12, Question 1; Palmer 1994, $§$ 3.1.

[27] Patel, S. R. (2022-06-28). "On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory". arXiv:2006.11134 [math.FA].

[lower-alpha 1]

[lower-alpha 2]

[1]

[lower-alpha 3]

[2]

[3]

[4]

[lower-alpha 4]

[lower-alpha 5]

[5]

[6]

[7]

[8]

[lower-alpha 6]

[9]

[10]

[lower-alpha 7]

[lower-alpha 8]

[lower-alpha 9]

[lower-alpha 10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

Fréchet algebra

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Properties

Examples

Generalizations

Michael's Conjecture

Notes

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