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{{efn|An increasing family means that for each $a \in A,$

: $\|a\|_0 \leq \|a\|_1 \leq \cdots \leq \| a \|_n \leq \cdots$ .}} 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 $a, b \in A$ .{{efn|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,

:: $\begin{align}
&{} \| a_k b_k -a b \|_n \\
&= \| a_k b_k - a b_k + a b_k - ab \|_n \\
&\leq \| a_k b_k - a b_k \|_n + \| a b_k - ab \|_n \\
&\leq C_n \biggl( \| a_k - a \|_m \|b_k\|_m + \| a\|_m\| b_k - b \|_m \biggr) \\
&\leq C_n \biggl( \| a_k - a \|_m \|b\|_m + \| a_k -a\|_m \|b_k - b\|_m + \| a\|_m\| b_k - b \|_m \biggr).
\end{align}$ }} Fréchet algebras are also called B₀-algebras.{{sfnm|1a1=Mitiagin|1a2=Rolewicz|1a3=Żelazko|1y=1962|2a1=Żelazko|2y=2001}}

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 $a, b \in A.$ {{efn|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: $p(fg) \le p(f) p(g),$ and the algebra is complete.}} $m$ -convex Fréchet algebras may also be called Fréchet algebras.{{harvnb|Husain|1991}}; {{harvnb|Żelazko|2001}}.

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.{{harvnb|Waelbroeck|1971|loc=Chapter VII, Proposition 1}}; {{harvnb|Palmer|1994|loc= $\S$ 2.9}}.
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.{{sfn|Waelbroeck|1971|loc=Chapter VII, Proposition 2}} 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 $A$ {{efn|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 $(a+ \mu 1)(b + \lambda 1) = ab + \mu b + \lambda a + \mu \lambda 1.$ }} and work with $inv A^+$ , or the set of quasi invertibles{{efn|If $a \in A$ , then $b \in A$ is a quasi-inverse for $a$ if $a + b -ab = 0$ .}} 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 $n \in \N$ .{{sfn|Mitiagin|Rolewicz|Żelazko|1962|loc=Lemma 1.2}} A commutative Fréchet $Q$ -algebra is $m$ -convex,{{sfn|Żelazko|1965|loc=Theorem 13.17}} but there exist examples of non-commutative Fréchet $Q$ -algebras which are not $m$ -convex.{{sfn|Żelazko|1994|pp=283–290}}
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.{{sfn|Michael|1952|loc=Theorem 5.1}} An element of $A$ is invertible if and only if its image in each Banach algebra of the projective limit is invertible.{{efn|If $A$ is non-unital, replace invertible with quasi-invertible.}}{{sfn|Michael|1952|loc=Theorem 5.2}}See also {{harvnb|Palmer|1994|loc=Theorem 2.9.6}}.

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 $\left\| \varphi \right\|_{n} = \left \| \varphi^{(n)} \right \|_{\infty}, \qquad \varphi \in A,$ 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 $\varphi^{(n)}$ .{{efn|To see the completeness, let $\varphi_{k}$ be a Cauchy sequence. Then each derivative $\varphi_{k}^{(l)}$ is a Cauchy sequence in the sup norm on $S^1$ , and hence converges uniformly to a continuous function $\psi_{l}$ on $S^1$ . It suffices to check that $\psi_{l}$ is the $l$ th derivative of $\psi_{0}$ . But, using the fundamental theorem of calculus, and taking the limit inside the integral (using uniform convergence), we have

: $\begin{align}
&{} \psi_{l}(x) - \psi_{l}(x_{0}) \\
= &{} \lim_{k \to \infty}\left ( \varphi_{k}^{(l)}(x) - \varphi_{k}^{(l)}(x_{0})\right) \\
= &{} \lim_{k \to\infty} \int_{x_{0}}^{x} \varphi_{k}^{(l+1)}(t) dt \\
= &{} \int_{x_{0}}^{x} \psi_{l+1}(t) dt.
\end{align}$ }} Then, by the product rule for differentiation, we have $\begin{align}
\| \varphi \psi \|_{n} &= \left \| \sum_{i = 0}^{n} {n \choose i} \varphi^{(i)} \psi^{(n-i)} \right \|_{\infty} \\
&\leq \sum_{i=0}^{n} {n \choose i} \| \varphi \|_{i} \| \psi \|_{n-i} \\
&\leq \sum_{i=0}^{n} {n \choose i} \| \varphi \|'_{n} \| \psi \|'_{n} \\
&= 2^n\| \varphi \|'_{n} \| \psi \|'_{n},
\end{align}$ where ${n \choose i} = \frac{n!}{{i! (n-i)!}},$ 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 $\Complex^\N$ 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 $U^{0} = \{ e \}$ .{{efn|

We can replace the generating set $U$ with $U \cup U^{-1}$ , so that $U=U^{-1}$ . Then $\ell$ satisfies the additional property $\ell(g^{-1})=\ell(g)$ , and is a length function on $G$ .}} 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 $\| \varphi \|_{d} = \| \ell^d \varphi \|_{1} =\sum_{g \in G} \ell(g)^d |\varphi(g)|.$ {{efn|

To see that $A$ is Fréchet space, let $\varphi_{n}$ be a Cauchy sequence. Then for each $g \in G$ , $\varphi_{n}(g)$ is a Cauchy sequence in $\Complex$ . Define $\varphi(g)$ to be the limit. Then

: $\begin{align}
&\sum_{g\in S} \ell(g)^d | \varphi_{n}(g) - \varphi(g)| \\
& \leq \sum_{g\in S} \ell(g)^d | \varphi_{n}(g) - \varphi_{m}(g)|+ \sum_{g\in S} \ell(g)^d | \varphi_{m}(g) - \varphi(g) | \\
& \leq \| \varphi_n - \varphi_m \|_d + \sum_{g\in S} \ell(g)^d |\varphi_{m}(g) - \varphi(g) |,
\end{align}$

where the sum ranges over any finite subset $S$ of $G$ . Let $\epsilon >0$ , and let $K_{\epsilon}> 0$ be such that $\| \varphi_n - \varphi_m \|_{d} < \epsilon$ for $m, n \geq K_{\epsilon}$ . By letting $m$ run, we have

: $\sum_{g \in S} \ell(g)^d | \varphi_{n}(g) - \varphi(g)| < \epsilon$

for $n \geq K_{\epsilon}$ . Summing over all of $G$ , we therefore have $\left\| \varphi_n - \varphi \right\|_d < \epsilon$ for $n \geq K_{\epsilon}$ . By the estimate

: $\begin{align}
&{}\sum_{g\in S} \ell(g)^d | \varphi(g) | \\
&{} \leq \sum_{g\in S} \ell(g)^d | \varphi_{n}(g) - \varphi(g)|+ \sum_{g\in S} \ell(g)^d | \varphi_{n}(g) | \\
&{} \leq \| \varphi_n - \varphi\|_d + \| \varphi_n \|_{d},
\end{align}$

we obtain $\| \varphi \|_{d} < \infty$ . Since this holds for each $d \in \N$ , we have $\varphi \in A$ and $\varphi_n \to \varphi$ in the Fréchet topology, so $A$ is complete.}} $A$ is an $m$ -convex Fréchet algebra for the convolution multiplication $\varphi * \psi (g) = \sum_{h \in G} \varphi(h) \psi(h^{-1}g),$ {{efn|

: $\begin{align}
& \| \varphi * \psi \|_{d} \\
& \leq \sum_{g \in G}\left ( \sum_{h \in G} \ell(g)^d |\varphi(h)| \left| \psi(h^{-1}g) \right| \right ) \\
&\leq \sum_{g, h \in G} \left (\ell(h) + \ell \left (h^{-1}g \right ) \right )^d |\varphi(h)| \left| \psi(h^{-1}g)\right| \\
&= \sum_{i=0}^{d} {d \choose i} \left (\sum_{g, h \in G} \left |\ell^i \varphi(h) \right | \left |\ell^{d-i} \psi(h^{-1}g) \right | \right ) \\
&= \sum_{i=0}^{d} {d \choose i} \left (\sum_{h \in G} \left |\ell^i \varphi(h) \right |\right)\left ( \sum_{g\in G} \left |\ell^{d-i} \psi(g) \right | \right) \\
&= \sum_{i=0}^{d} {d \choose i} \| \varphi \|_{i} \| \psi \|_{d-i} \\
&\leq 2^d \| \varphi \|'_{d} \| \psi \|'_{d}
\end{align}$ }} $A$ 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 $[0,1]$ .{{sfn|Fragoulopoulou|2005|loc=Example 6.13 (2)}}

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{{sfn|Waelbroeck|1971}} or an F-space.{{sfn|Rudin|1973|loc=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).{{sfnm|Michael|1952|Husain|1991}} A complete LMC algebra is called an Arens-Michael algebra.{{sfn|Fragoulopoulou|2005|loc=Chapter 1}}

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.{{harvnb|Michael|1952|loc= $\S$ 12, Question 1}}; {{harvnb|Palmer|1994|loc= $\S$ 3.1}} This conjecture is perhaps the most famous open problem in the theory of topological algebras.

Notes

Citations

Sources

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