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^[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 $m \ge n$ for each $n$ such that $\| a b \|_n \leq C_n \| a \|_m \|b \|_m$ for all $a, b \in A$.^[2] Fréchet algebras are also called B₀-algebras (Mitiagin et al. 1962) (Żelazko 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 $a, b \in A$.^[3] $m$-convex Fréchet algebras may also be called Fréchet algebras (Husain 1991) (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

1. Continuity of multiplication. Multiplication is separately continuous if $a_k b \rightarrow ab$ and $ba_k \rightarrow ba$ for every $a, b \in A$ and sequence $a_k \rightarrow a$ converging in the Fréchet topology of $A$. Multiplication is jointly continuous if $a_k \rightarrow a$ and $b_k \rightarrow b$ imply $a_k b_k \rightarrow ab$. 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 (Waelbroeck 1971, Chapter VII, Proposition 1), (Palmer 1994, $\S$ 2.9).
2. Group of invertible elements. If $invA$ is the set of invertible elements of $A$, then the inverse map $u \mapsto u^{-1}$, $invA \rightarrow invA$ is continuous if and only if $invA$ is a $G_\delta$ (Waelbroeck 1971, 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 $A$^[4] and work with $inv A^+$, or the set of quasi invertibles^[5] may take the place of $inv A$.)
3. 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 $m \in \mathbb N$ 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 $n \in \mathbb N$ (Mitiagin et al. 1962, Lemma 1.2). A commutative Fréchet $Q$-algebra is $m$-convex (Żelazko 1965, Theorem 13.17). But there exist examples of non-commutative Fréchet $Q$-algebras which are not $m$-convex (Żelazko 1994).

4. 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 (Michael 1952, Theorem 5.1). An element of $A$ is invertible if and only if it's image in each Banach algebra of the projective limit is invertible (Michael 1952, Theorem 5.2).^[6] See also (Palmer 1994, Theorem 2.9.6).

Examples

1. Zero multiplication. If $E$ is any Fréchet space, we can make a Fréchet algebra structure by setting $e * f = 0$ for all $e, f \in E$.
2. 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

   $\| \varphi \|_{n} = \| \varphi^{(n)} \|_{\infty}, \qquad \varphi \in A,$ where $\| \varphi^{(n)} \|_{\infty} = \sup_{x \in {S^1}} |\varphi^{(n)}(x)|$ denotes the supremum of the absolute value of the $n$th derivative $\varphi^{(n)}$.^[7] Then, by the product rule for differentiation, we have

   $\begin{align} \| \varphi \psi \|_{n} &=& \biggl\| \sum_{i = 0}^{n} {n \choose i} \varphi^{(i)} \psi^{(n-i)} \biggr\|_{\infty} & \leq & \sum_{i =0}^{n} {n \choose i} \| \varphi \|_{i} \| \psi \|_{n-i} \\ & \leq & \sum_{i =0}^{n} {n \choose i} \| \varphi \|_{n}^{\prime} \| \psi \|_{n}^{\prime} & = & 2^n \| \varphi \|_{n}^{\prime} \| \psi \|_{n}^{\prime}, \end{align}$ where ${n \choose i}$ denotes the binomial coefficient ${n! \over{i! (n-i)!}}$, and $\| \cdot \|_{n}^{\prime} = \max_{k \leq n} \| \cdot \|_{k}$. The primed seminorms are submultiplicative after re-scaling by $C_n=2^n$.

3. 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 $\| \varphi \|_n = \max_{k\leq n} |\varphi(k)|$. With pointwise multiplication, $\C^\N$ 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 $A$.
4. Equipped with the topology of uniform convergence on compact sets, and pointwise multiplication, $C(\mathbb C)$, the algebra of all continuous functions on the complex plane $\mathbb C$, or to the algebra $Hol(\mathbb C)$ of holomorphic functions on $\mathbb C$.
5. 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 the union of all products $\bigcup_{n=0}^{\infty} U^n$ equals $G$. Without loss of generality, we may also assume that the identity element $e$ of $G$ is contained in $U$. Define a function $\ell \colon G \rightarrow [0, \infty)$ 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 $U^{0} = \{ e \}$.^[8] Let $A$ be the $\C$-vector space

   $S(G) = \biggr\{ \varphi\colon G \rightarrow {\mathbb C} \,\,\biggl|\,\, \| \varphi \|_{d} < \infty, \quad d = 0,1, 2, \dots \biggr\},$ where the seminorms $\| \cdot \|_{d}$ are defined by

   $\| \varphi \|_{d} = \| \ell^d \varphi \|_{1} = \sum_{g \in G} \ell(g)^d |\varphi(g)|.$^[9]

   $A$ is an $m$-convex Fréchet algebra for the convolution multiplication

   $\varphi {*} \psi (g) = \sum_{h \in G} \varphi(h) \psi(h^{-1}g),$^[10] $A$ is unital because $G$ is discrete, and $A$ is commutative if and only if $G$ is Abelian.

6. Non $m$-convex Fréchet algebras. The Aren's algebra $A=L^\omega[0,1]= \bigcup_{p \geq 1} L^p[0,1]$ is an example of a commutative non-$m$-convex Fréchet algebra with discontinuous inversion. The topology is given by $L^p$

   $\| f \|_p = \biggl( \int_0^1 | f(t) |^p dt \biggr)^{1/p}, \qquad f \in A,$ and multiplication is given by convolution of functions with respect to Lebesgue measure on $[0,1]$ (Fragoulopoulou 2005, 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 (Waelbroeck 1971) or an F-space (Rudin 1973, 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) (Michael 1952) (Husain 1991). A complete LMC algebra is called an Arens-Michael algebra (Fragoulopoulou 2005, Chapter 1).

Open problems

Perhaps the most famous, still open problem of the theory of topological algebras is whether all linear multiplicative functionals on an $m$-convex Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture (Michael 1952, $\S$ (Palmer 1994, $\S$.

Notes

1. An increasing family means that for each $a \in A$, $\|a\|_0 \leq \|a\|_1 \leq \cdots \leq \| a \|_n \leq \cdots$.
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,

   $\begin{align} \| a_k b_k\, -\, a b \|_n &=& \| a_k b_k\, -\, a b_k\, +\, a b_k\, - \, ab \|_n \\ &\leq & \| a_k b_k \,-\, a b_k \|_n\, +\, \| a b_k\, -\, ab \|_n \\ &\leq & C_n \biggl( \| a_k - a \|_m \|b_k\|_m \, + \, \| a\|_m\| b_k - b \|_m \biggr) \\ &\leq & C_n \biggl( \| a_k - a \|_m \|b\|_m \, +\, \| a_k -a\|_m \|b_k - b\|_m \, +\, \| a\|_m\| b_k - b \|_m \biggr). \end{align}$

3. 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) ≤ p(f)p(g), and the algebra is complete.
4. 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 $(a+ \mu 1)(b + \lambda 1) = ab + \mu b + \lambda a + \mu \lambda 1.$
5. If $a \in A$, then $b \in A$ is a quasi-inverse for $a$ if $a + b -ab = 0$.
6. If $A$ is non-unital, replace invertible with quasi-invertible.
7. 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 $\psi_{l}(x) - \psi_{l}(x_{0}) = \lim_{k \rightarrow \infty}\biggl( \varphi_{k}^{(l)}(x) - \varphi_{k}^{(l)}(x_ {0}) \biggr)= \lim_{k \rightarrow \infty} \int_{x_{0}}^{x} \varphi_{k}^{(l+1)}(t) dt = \int_{x_{0}}^{x} \psi_{l+1}(t) dt.$
8. 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 $G$.
9. 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 $\mathbb C$. Define $\varphi(g)$ to be the limit. Then

   $\begin{align} \sum_{g\in S} \ell(g)^d | \varphi_{n}(g) - \varphi(g)| & \leq & \sum_{g\in S} \ell(g)^d | \varphi_{n}(g) - \varphi_{m}(g)| \,\, &+& \sum_{g\in S} \ell(g)^d | \varphi_{m}(g) - \varphi(g) | \\ & \leq & \qquad \| \varphi_n - \varphi_m \|_d \qquad \qquad &+& \sum_{g\in S} \ell(g)^d |\varphi_{m}(g) - \varphi(g) |, \end{align}$ 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 $\| \varphi_n - \varphi \|_d < \epsilon$ for $n \geq K_{\epsilon}$. By the estimate

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

   we obtain $\| \varphi \|_{d} < \infty$. Since this holds for each $d \in \mathbb N$, we have $\varphi \in A$ and $\varphi_n \rightarrow \varphi$ in the Fréchet topology, so $A$ is complete.

10. $\begin{align} \| \varphi {*} \psi \|_{d} & \leq& \sum_{g \in G}\biggl( \sum_{h \in G} \ell(g)^d |\varphi(h)|\, | \psi(h^{-1}g)| \biggr) \\ &\leq & \sum_{g, h \in G} (\ell(h) + \ell(h^{-1}g))^d |\varphi(h)|\, | \psi(h^{-1}g)| \\ & = & \sum_{i = 0}^{d} {d\choose i} \biggl(\sum_{g, h \in G} |\ell^i \varphi(h)|\, |\ell^{d-i} \psi(h^{-1}g)| \biggr) \\ & = & \sum_{i = 0}^{d} {d \choose i} \biggl(\sum_{h \in G} |\ell^i \varphi(h)|\biggr)\biggl( \sum_{g\in G} |\ell^{d-i} \psi(g)| \biggr) \\ &=& \sum_{i=0}^{d} {d \choose i} \| \varphi \|_{i} \| \psi \|_{d-i} \\ &\leq& 2^d \| \varphi \|_{d}^{\prime} \| \psi \|_{d}^{\prime}. \end{align}$

References

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