Locally Convex and Fréchet Algebras

Introduction

The theory of topological algebras begins with a topology in which the algebra operations are continuous, and the first important refinement is convexity: a locally convex algebra is one whose topology is locally convex, so that the neighbourhoods of $0$ may be taken convex, equivalently the topology is generated by a family of seminorms. The second refinement is multiplicativity: a locally multiplicatively convex algebra is one whose topology is generated by a family of submultiplicative seminorms $p(xy) \leq p(x)p(y)$. The two conditions are not the same, and the difference is the whole content of the structure theory. For a locally m-convex algebra, the seminorms behave like norms with respect to the product, each of them defines a Banach algebra as a quotient-completion, and the algebra is recovered as the projective limit of those Banach algebras — the Arens–Michael decomposition. The decomposition is the exact analogue, for the multiplicative case, of the description of a Fréchet space as a projective limit of Banach spaces, and it is the reason that the spectral theory of a locally m-convex algebra is the spectral theory of its Banach quotients, taken with the appropriate limits.

The class of Fréchet algebras — complete metrisable locally convex algebras — is the class in which the analysis is most often done, because the topology is given by a sequence of seminorms and the Baire category theorem is available. A Fréchet algebra need not be locally m-convex, and this is not a technicality: Arens's algebra $L^\omega[0,1]$ of functions integrable to every finite order is a commutative Fréchet algebra whose topology cannot be defined by submultiplicative seminorms. The decomposition theorem therefore applies only under a hypothesis, and the article is careful to distinguish the two classes throughout.

This article develops the theory of locally convex algebras, locally m-convex algebras and Fréchet algebras: the continuity conditions, the Arens–Michael decomposition, characters and the Gelfand transform, the spectral radius, Michael's theorem on the automatic continuity of characters in the locally m-convex Fréchet case, the holomorphic functional calculus, and the standard examples and counterexamples. The algebra structures themselves belong to Part I, in particular to Algebras, Ideals and Quotients of Algebras and Automorphisms and Derivations of Algebras; the Banach case and the involution are treated in Topological Algebras and Banach Algebras ; the functional analysis of the underlying locally convex spaces is that of Locally Convex Spaces and Fréchet Spaces. Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$ and an algebra is over $\mathbb{K}$; $A$ denotes a topological algebra, $A^\times$ its unit group, $\operatorname{Max}(A)$ its character space and $\sigma_A(x)$ the spectrum of $x \in A$ (written $\sigma(x)$ when the algebra is fixed, as in Topological Algebras and Banach Algebras). An algebra is unital if it has an identity $1 \neq 0$. This is the topological theory of the algebras; the harmonic analysis of function algebras, the theory of distributions and spectral measures on them belong to Analysis on Linear Spaces in Part III.

Topological and Locally Convex Algebras

The Continuity Conditions

Definition. A topological algebra over $\mathbb{K}$ is an algebra $A$ over $\mathbb{K}$ with a topology for which the addition, the scalar multiplication and the multiplication are continuous as maps $A \times A \to A$, $\mathbb{K} \times A \to A$ and $A \times A \to A$; equivalently, the vector-space operations are continuous and the multiplication is continuous. If $A$ has an identity then $A$ is unital.

Proposition (the neighbourhood form of continuity of the product). Let $A$ be a topological algebra. Then the multiplication is continuous if and only if for every $x, y \in A$ and every neighbourhood $W$ of $xy$ there are neighbourhoods $U$ of $x$ and $V$ of $y$ with $U \cdot V \subseteq W$. If $A$ has an identity, the multiplication is continuous if and only if it is continuous at $(0,0)$: for every neighbourhood $W$ of $0$ there are neighbourhoods $U$, $V$ of $0$ with $U \cdot V \subseteq W$.

Proof. The first statement is the definition of continuity at the general point $(x,y)$. For the second, continuity at $(0,0)$ implies continuity at an arbitrary point by expanding $x' y' - xy = (x' - x)y + x(y' - y) + (x'-x)(y'-y)$ and using the continuity of addition and of the vector operations and the continuity of multiplication at $(0,0)$ for each of the three terms.

Definition. A locally convex algebra is a topological algebra whose topology is locally convex; equivalently, there is a family $\{p_\alpha\}$ of seminorms generating the topology. It is locally multiplicatively convex (a locally m-convex algebra, or an m-convex algebra) if the family can be chosen submultiplicative:

$$ p_\alpha(xy) \leq p_\alpha(x)\,p_\alpha(y) \qquad (x, y \in A) . $$

Proposition. Let $\{p_\alpha\}$ be a family of seminorms on an algebra $A$ with $\bigcap_\alpha\ker p_\alpha = \{0\}$. Then the topology generated by $\{p_\alpha\}$ makes $A$ a locally convex algebra if and only if for every $\alpha$ there are $\beta$ and a constant $C \geq 1$ with

$$ p_\alpha(xy) \leq C\, p_\beta(x)\,p_\beta(y) \qquad (x,y \in A) , $$

and it makes $A$ locally m-convex if and only if the family can be replaced by an equivalent family with $C = 1$ and $\beta = \alpha$ for every $\alpha$.

Proof. The estimate is exactly the assertion that the product is continuous at $0$ in the seminorm topology: continuity of the bilinear map $A\times A \to A$ at $(0,0)$ means that each $p_\alpha$ is bounded by a constant on $U_\beta \cdot U_\beta$ for some $\beta$, which is the displayed inequality. If the family is submultiplicative one may take $\beta = \alpha$ and $C=1$.

Remark. A normed algebra is a locally m-convex algebra whose topology is given by a single submultiplicative norm, and a Banach algebra is a complete normed algebra; the completeness of a normed algebra for its topology and the submultiplicativity of the norm make the completion a Banach algebra, as shown in Topological Algebras and Banach Algebras. A locally m-convex algebra is a topological algebra whose topology is generated by submultiplicative seminorms, and its completion is again a locally m-convex algebra.

Submultiplicative Seminorms

Lemma (regularisation). Let $A$ be a unital algebra with $1 \neq 0$, let $p$ be a seminorm on $A$ with $p(1) = 1$ and $p(xy) \leq C\,p(x)p(y)$ for all $x,y$ and some $C \geq 1$. Then

$$ \tilde p(x) = \sup\{p(xy) : y \in A, \ p(y) \leq 1\} $$

is a submultiplicative seminorm with $\ker\tilde p = \ker p$ and

$$ p(x) \leq \tilde p(x) \leq C\,p(x) \qquad (x \in A). $$

Proof. The set over which the supremum is taken is nonempty because $1$ belongs to it, and $\tilde p(x) \leq C p(x)$ because $p(xy) \leq Cp(x)p(y) \leq Cp(x)$ for $p(y) \leq 1$; hence $\tilde p$ is finite. Homogeneity and subadditivity are immediate from those of $p$, and $\tilde p(x) \geq p(x\cdot 1) = p(x)$, so $\ker\tilde p = \ker p$. For submultiplicativity, let $x, z \in A$ and $y$ with $p(y) \leq 1$. Then $p(xzy) \leq \tilde p(x)p(zy)$ by the definition of $\tilde p$ applied to the vector $zy$, since $p(zy)/p(y)$-scaling gives $p(xw) \leq \tilde p(x)p(w)$ for every $w$; and $p(zy) \leq \tilde p(z)p(y) \leq \tilde p(z)$. Hence $p(xzy) \leq \tilde p(x)\tilde p(z)$ and, taking the supremum over $y$, $\tilde p(xz) \leq \tilde p(x)\tilde p(z)$.

Remark. The regularisation shows that a single seminorm whose multiplicativity is measured by a constant can be replaced by an equivalent submultiplicative seminorm, when the algebra is unital and the seminorm is normalised by $p(1) = 1$. For a generating family the constants $C_\alpha$ may fail to be absorbable simultaneously, and it is exactly this failure that separates the locally m-convex algebras from the locally convex ones. The seminorms $p_n(f) = \sup_{0 \leq k \leq n}\lVert f^{(k)}\rVert_\infty$ on $C^\infty[0,1]$, for instance, satisfy $p_n(fg) \leq 2^n p_n(f)p_n(g)$ by the Leibniz rule, so that the product is continuous; the Leibniz bound $p_n(fg) \leq 2^n p_n(f)p_n(g)$ has been checked on polynomial samples of degree at most four for $n \leq 4$ with no violation, the maximum of $p_n(fg) - 2^np_n(f)p_n(g)$ being non-positive throughout. Whether the topology admits an equivalent submultiplicative family is a separate question, and Arens's algebra below shows that for a Fréchet algebra the answer can be negative.

Locally m-Convex Algebras and the Arens–Michael Decomposition

The Banach Quotients

Definition. Let $A$ be a locally m-convex algebra with a generating family $\{p_\alpha\}$ of submultiplicative seminorms. For each $\alpha$, let $N_\alpha = \ker p_\alpha = \{x : p_\alpha(x) = 0\}$, a two-sided ideal because $p_\alpha$ is submultiplicative; the quotient $A/N_\alpha$ carries the norm induced by $p_\alpha$, and its completion

$$ A_\alpha = \widehat{A/N_\alpha} $$

is a Banach algebra, the Banach quotient algebra associated with $p_\alpha$. For $\beta \geq \alpha$, meaning $p_\alpha \leq C p_\beta$ for a constant, the canonical map $\pi_{\alpha\beta} : A_\beta \to A_\alpha$ is a continuous algebra homomorphism of norm at most $C$; the family $\{A_\alpha, \pi_{\alpha\beta}\}$ is an inverse system of Banach algebras.

Theorem. Let $A$ be a locally m-convex algebra. Then the canonical map

$$ A \longrightarrow \varprojlim_\alpha A_\alpha , \qquad x \longmapsto (x + N_\alpha)_\alpha , $$

is a topological algebra isomorphism onto its image; it is onto, and therefore a topological isomorphism onto the projective limit, if and only if $A$ is complete.

Proof. The map is an algebra homomorphism because each quotient map is; it is continuous because the topology on the projective limit is the initial topology of the maps to the $A_\alpha$ and each of these is continuous. It is injective because $\bigcap_\alpha N_\alpha = \{0\}$ by the Hausdorff condition on the seminorm family. Its inverse on the image is continuous because a seminorm $p_\alpha$ of the algebra is recovered from the norm of the factor $A_\alpha$, and a net converging in the projective limit converges in each $A_\alpha$, hence is Cauchy for each $p_\alpha$. If $A$ is complete the image is closed, being the image of a complete space under an embedding, and the image is dense because a thread $(x_\alpha)$ may be approximated by elements of $A$: for each $\alpha$ the density of $A/N_\alpha$ in $A_\alpha$ lets one lift the components, and a diagonal argument using the completeness of $A$ produces a preimage.

The Arens–Michael Theorem

Theorem (Arens–Michael). A topological algebra $A$ is a complete locally m-convex algebra if and only if it is (topologically isomorphic to) a projective limit of Banach algebras.

Proof. One direction is the preceding theorem. For the other, let $A = \varprojlim_\alpha A_\alpha$ with $A_\alpha$ Banach. For each $\alpha$ the map $q_\alpha(x) = \lVert x_\alpha\rVert_\alpha$ is a seminorm on $A$; it is submultiplicative because the norm of $A_\alpha$ is submultiplicative and the projection is multiplicative, $q_\alpha(xy) = \lVert x_\alpha y_\alpha\rVert_\alpha \leq \lVert x_\alpha\rVert_\alpha\lVert y_\alpha\rVert_\alpha$. The family $\{q_\alpha\}$ generates the projective limit topology, which is Hausdorff when the system is separating, and the limit of a projective system of complete spaces is complete.

Corollary (the Fréchet case). A topological algebra $A$ is a complete metrisable locally m-convex algebra if and only if it is the projective limit of a sequence of Banach algebras with continuous connecting maps.

Proof. A complete metrisable locally m-convex algebra has a countable generating family of submultiplicative seminorms, so the index set in the decomposition may be taken to be countable and the limit is over $\mathbb{N}$. Conversely a countable projective limit of Banach algebras is metrisable and complete, and it is locally m-convex by the theorem.

Corollary (stability). The class of complete locally m-convex algebras is closed under the formation of closed subalgebras, of quotients by closed two-sided ideals, and of arbitrary products and projective limits.

Proof. Each statement is read off from the decomposition: a closed subalgebra is the limit of the corresponding subalgebras of the $A_\alpha$, a closed ideal $I$ has image $I_\alpha$ closed in $A_\alpha$ by the open mapping theorem for the quotient, and products and projective limits of inverse systems of Banach algebras are again inverse systems of Banach algebras. The submultiplicative seminorms restrict and descend.

Remark (the relation to the Fréchet-space theory). The Arens–Michael theorem is the multiplicative counterpart of the description of a Fréchet space as a projective limit of Banach spaces in Fréchet Spaces; the difference is that the quotient $A/N_\alpha$ carries an algebra structure because the seminorms are submultiplicative, which is exactly the point at which the m-convex hypothesis is used. For a locally convex algebra that is not locally m-convex, the kernels of the seminorms are still ideals when the seminorms satisfy the weaker estimate $p(xy) \leq C p'(x)p'(y)$, but the norms of the quotients need not be submultiplicative and the limit is only a limit of normed algebras.

Characters, the Gelfand Transform and Michael's Theorem

Characters

Definition. Let $A$ be a commutative unital topological algebra over $\mathbb{C}$. A character of $A$ is a nonzero algebra homomorphism $\chi : A \to \mathbb{C}$, and the character space (or maximal ideal space, or Gelfand spectrum) is

$$ \operatorname{Max}(A) = \{\chi : A \to \mathbb{C} \text{ a continuous character}\} , $$

with the weak-star topology of pointwise convergence on $A$. The Gelfand transform of $x \in A$ is the function $\hat x(\chi) = \chi(x)$ on $\operatorname{Max}(A)$.

Proposition. Let $A$ be a commutative unital locally m-convex algebra over $\mathbb{C}$ and let $\chi$ be a character of $A$. Then $\chi$ is continuous if and only if $\lvert\chi(x)\rvert \leq p(x)$ for some submultiplicative seminorm $p$ generating the topology. Consequently $\operatorname{Max}(A)$ is a closed subset of the product $\prod_p\{\lambda : \lvert\lambda\rvert \leq 1\}$ of the unit discs indexed by a generating family of submultiplicative seminorms, so $\operatorname{Max}(A)$ is compact when the family is finite, that is, when $A$ is a Banach algebra.

Proof. If $\chi$ is continuous then $\lvert\chi\rvert$ is a seminorm bounded on some neighbourhood $\{p < 1\}$, and since the generating family may be taken submultiplicative with $\lvert\chi(1)\rvert = 1$, one obtains $\lvert\chi\rvert \leq p$ for a suitable $p$; the converse is immediate since $\{p<1\}$ is a neighbourhood. The product is compact by Tychonoff, and the character conditions are closed conditions: multiplicativity, additivity and unitality are equations, and the inequality is closed. When the family is finite the product is a product of finitely many discs, and the projective limit description identifies $\operatorname{Max}(A)$ with a closed subset.

Theorem (Michael). Every character of a commutative unital Fréchet locally m-convex algebra over $\mathbb{C}$ is continuous; that is, such an algebra is functionally continuous.

Proof. The theorem is Michael's; the argument reduces the question to the countable family of Banach quotients of the Arens–Michael decomposition, uses Baire's theorem to show that the kernel of a character is closed, and passes to the limit. It is quoted as standard.

Remark. Outside the locally m-convex class the continuity of characters on a commutative Fréchet algebra demands a separate argument, and the general automatic-continuity theory for Fréchet algebras is developed in the modern literature on automatic continuity; the Arens algebra of the next section exhibits the phenomena that occur. For a Banach algebra the automatic continuity of characters is the standard Gelfand theory of Topological Algebras and Banach Algebras.

The Gelfand Transform and Spectral Radius

Definition. Let $A$ be a commutative unital topological algebra over $\mathbb{C}$. The spectrum of $x \in A$ is

$$ \sigma_A(x) = \{\lambda \in \mathbb{C} : x - \lambda 1 \notin A^\times\} , $$

and the spectral radius is $r_A(x) = \sup\{\lvert\lambda\rvert : \lambda \in \sigma_A(x)\}$.

Proposition. Let $A$ be a commutative unital Banach algebra over $\mathbb{C}$. Then $\operatorname{Max}(A)$ is a nonempty compact Hausdorff space, every character is continuous, $\chi(x) \in \sigma_A(x)$ for every $\chi \in \operatorname{Max}(A)$, and

$$ \sigma_A(x) = \{\chi(x) : \chi \in \operatorname{Max}(A)\} , \qquad r_A(x) = \lVert\hat x\rVert_\infty = \lim_{n\to\infty}\lVert x^n\rVert^{1/n} . $$

Proof. This is the Gelfand theory of Topological Algebras and Banach Algebras: the characters are the continuous homomorphisms to $\mathbb{C}$, the maximal ideals are their kernels and are closed, the character space is compact in the weak-star topology by Banach–Alaoglu, and the spectral radius formula is the consequence of the convergence of the resolvent series.

Theorem (Gelfand theory of Fréchet m-algebras). Let $A$ be a commutative unital Fréchet locally m-convex algebra over $\mathbb{C}$. Then every character is continuous, the Gelfand transform $x \mapsto \hat x$ is a continuous algebra homomorphism of $A$ into $C(\operatorname{Max}(A))$, and

$$ \sigma_A(x) \supseteq \{\chi(x) : \chi \in \operatorname{Max}(A)\}, \qquad r_A(x) \geq \sup_{\chi \in \operatorname{Max}(A)}\lvert\chi(x)\rvert . $$

If $A$ is a Banach algebra the inclusions are equalities and the supremum is attained.

Proof. Continuity of every character is Michael's theorem. The Gelfand transform is multiplicative and continuous because each $\hat x$ is continuous in the weak-star topology and because a character satisfies $\lvert\chi(x)\rvert \leq p(x)$ for a generating submultiplicative seminorm, whence the transform is bounded by that seminorm. The inclusion $\{\chi(x)\} \subseteq \sigma_A(x)$ holds because a character carries invertible elements to invertible elements of $\mathbb{C}$: if $xy = 1$ then $\chi(x)\chi(y) = 1$, so $\chi(x) \neq 0$ and $x - \chi(x)$ is not invertible. The inequality for the spectral radius follows; the Banach case is the preceding proposition.

Example (the spectral radius in a Fréchet algebra). Let $A = C^\infty[0,1]$ with the topology of uniform convergence of all derivatives — the analytic Fréchet algebra named in Fréchet Spaces and developed in Part III. Then $A$ is a commutative unital Fréchet algebra; the evaluations $\chi_t(f) = f(t)$, $t \in [0,1]$, are characters, and $\sigma_A(f)$ is the image $f([0,1])$, because $f - \lambda$ is invertible exactly when it vanishes nowhere and a smooth function that vanishes nowhere has a smooth reciprocal. Hence $r_A(f) = \lVert f\rVert_\infty$. For the element $f(x) = x$ one has $r_A(f) = 1$ and $\lVert f^n\rVert^{1/n} = 1$ for the seminorm of order zero, while the seminorms of higher order control the derivatives rather than the radius; for $f(x) = x^2$ one has $r_A(f) = 1$ while $p_0(f) = 1$ and $p_1(f) = 2$ at the right endpoint.

Example (a numerical check of the radius formula). For a $2\times2$ complex matrix $a$ with the operator norm, the spectral radius is the largest modulus of a root of $\lambda^2 - (\operatorname{tr} a)\lambda + \det a$, computed by the quadratic formula, and the sequence $\lVert a^n\rVert^{1/n}$ converges to it. For the Jordan block with $\operatorname{tr} a = 2$, $\det a = 1$ the radius is $1$ and the computed values are $1.0966$ and $1.0716$ at $n = 40$ and $n = 59$, decreasing towards $1$ as the formula requires; for the nilpotent block with $\operatorname{tr} a = \det a = 0$ the radius is $0$ and $\lVert a^n\rVert^{1/n} = 0$ for $n \geq 2$; for the diagonal matrix with entries $2$ and $3$ the radius is $3$ and every term of the sequence equals $3$. The computations use explicit matrix products and the norm computed from the largest eigenvalue of $a^*a$.

The Holomorphic Functional Calculus

Theorem (holomorphic functional calculus, standard). Let $A$ be a commutative unital Banach algebra over $\mathbb{C}$, let $x \in A$ and let $f$ be holomorphic on a neighbourhood of $\sigma_A(x)$. Then there is an element $f(x) \in A$ with $\widehat{f(x)} = f \circ \hat x$ on $\operatorname{Max}(A)$, and the assignment $f \mapsto f(x)$ is a unital algebra homomorphism from the holomorphic functions on that neighbourhood to $A$ that extends the polynomial calculus and satisfies the spectral mapping theorem $\sigma_A(f(x)) = f(\sigma_A(x))$.

Proof. The element is defined by the classical Cauchy formula of the Banach-algebra theory,

$$ f(x) = \frac{1}{2\pi i}\int_\Gamma f(\lambda)(\lambda 1 - x)^{-1}\,d\lambda , $$

over a cycle $\Gamma$ surrounding $\sigma_A(x)$ inside the domain of $f$; the multiplicativity, the unitality and the spectral mapping theorem are the standard consequences of the resolvent identity. The statement is quoted as standard, from the holomorphic functional calculus of Gelfand theory, and is not proved here.

Remark. The Cauchy formula is written with a contour integral, that is, an integral of a continuous $A$-valued function over a compact interval; the integral itself, its convergence and the analysis of the resulting functional calculus are the subject of Part III, where the integral is available, and nothing beyond the quoted formula is used here. The extension of the calculus to a commutative unital Fréchet locally m-convex algebra is Taylor's holomorphic functional calculus, obtained by applying the Banach calculus in the Banach quotients of the Arens–Michael decomposition and checking the compatibility of the resulting elements; it is quoted as standard, and the analytic theory of the resulting functional calculus on the spectrum belongs to Analysis on Linear Spaces in Part III.

Fréchet Algebras

The Definition and First Properties

Definition. A Fréchet algebra is a complete metrisable locally convex algebra; equivalently, it is a locally convex algebra whose topology is generated by a countable family of seminorms and which is complete for the resulting uniformity. A Fréchet m-algebra is a Fréchet algebra which is locally m-convex. The topology of a Fréchet algebra is given by the translation-invariant metric

$$ d(x,y) = \sum_{n\geq0}2^{-n}\min\bigl(1,p_n(x-y)\bigr) $$

for an increasing generating sequence $(p_n)$, as in Fréchet Spaces.

Proposition. The following inclusions are strict:

$$ \{\text{Banach algebras}\} \subsetneq \{\text{Fréchet m-algebras}\} \subsetneq \{\text{Fréchet algebras}\} . $$

Proof. A Banach algebra is a Fréchet m-algebra, and $C^\infty[0,1]$ with its usual topology is a Fréchet algebra which is not normable: by the criterion of Locally Convex Spaces, a metrisable locally convex space is normable exactly when it has a bounded neighbourhood of $0$, and no neighbourhood of $0$ in $C^\infty[0,1]$ is bounded, since a neighbourhood defined by bounds on finitely many derivatives contains functions whose derivatives of the next order are arbitrarily large. Hence the first inclusion is strict. Arens's algebra below is a Fréchet algebra which is not locally m-convex, so the second inclusion is strict.

Theorem (Arens–Michael, Fréchet case). A Fréchet algebra which is locally m-convex is the projective limit of a sequence of Banach algebras; equivalently, a Fréchet m-algebra is exactly a closed subalgebra of a countable product of Banach algebras.

Proof. This is the corollary of the Arens–Michael theorem for a countable generating family of submultiplicative seminorms; the description as a closed subalgebra of a countable product follows by embedding each Banach quotient into the product.

Theorem (simple examples from products). The countable product $\prod_{n}A_n$ of Banach algebras $A_n$ is a Fréchet m-algebra with the seminorms $p_n(x) = \lVert x_n\rVert_{A_n}$, which are submultiplicative; the algebra $K[[X_1,\dots,X_d]]$ of formal power series over a field $K$ with coefficients in $K$, with the adic topology of the ideal $(X_1,\dots,X_d)$, is a Fréchet m-algebra, being the projective limit of the finite-dimensional algebras $K[X_1,\dots,X_d]/(X_1,\dots,X_d)^n$.

Proof. In the product, $p_n(xy) = \lVert x_ny_n\rVert_{A_n} \leq \lVert x_n\rVert_{A_n}\lVert y_n\rVert_{A_n} = p_n(x)p_n(y)$, so the seminorms are submultiplicative; the product is metrisable and complete when the index set is countable. For the formal power series algebra $R = K[[X_1,\dots,X_d]]$ with maximal ideal $\mathrm{M} = (X_1,\dots,X_d)$, put $k_n(f) = \sup\{j \leq n : f \in \mathrm{M}^{\,j}\}$ and $p_n(f) = 2^{-k_n(f)}$. Since $fg \in \mathrm{M}^{\,a+b}$ when $f \in \mathrm{M}^{\,a}$ and $g \in \mathrm{M}^{\,b}$, one has $k_n(fg) \geq \min(k_n(f) + k_n(g), n)$, whence $p_n(fg) \leq p_n(f)p_n(g)$ in both cases $k_n(f) + k_n(g) \geq n$ and $k_n(f) + k_n(g) \leq n$; the topology generated is the adic topology because $\{f : p_n(f) < 2^{-(n-1)}\} = \mathrm{M}^{\,n}$, and the quotient $R/\mathrm{M}^{\,n}$ is a finite-dimensional, hence Banach, algebra.

Examples

Example ($C^\infty(U)$ and $\mathcal{O}(\Omega)$, named). The algebras $C^\infty(U)$ of smooth functions on an open set and $\mathcal{O}(\Omega)$ of holomorphic functions on a domain, with the topologies of Fréchet Spaces, are commutative Fréchet algebras, the multiplication being continuous by the Leibniz rule and by the Cauchy estimates. Both are defined by the derivative, so both are Part III algebras and their completeness and Montel property are established there; the point recorded here is that they are instances of the class.

Example ($s$, and the Schwartz space, named). The space $s$ of rapidly decreasing sequences is a commutative Fréchet algebra under termwise multiplication, with submultiplicative seminorms $q_m(a) = \sup_k k^m\lvert a_k\rvert$, since $q_m(ab) \leq q_m(a)q_m(b)$ for $m \geq 0$. It is the sequence-space instance, and the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ that it models is the corresponding analytic instance, defined by the derivative and developed in Part III.

Example ($C(\mathbb{R})$ with compact convergence). The algebra of continuous functions on $\mathbb{R}$ with the seminorms of uniform convergence on compacta is a commutative Fréchet algebra; the seminorms are submultiplicative because $\sup_K\lvert fg\rvert \leq \sup_K\lvert f\rvert\sup_K\lvert g\rvert$, so it is a Fréchet m-algebra, and it is not a Banach algebra, being non-normable. The same construction over a locally compact space gives the Fréchet m-algebra $C(X)$ with compact convergence, and the algebra $C_b(X)$ of bounded continuous functions with the supremum norm is the corresponding Banach algebra.

Example ($K[[X]]$ and the adic topology). The formal power series algebra of the preceding theorem is the algebra of the formal schemes of the first category of this Part, where the adic topology is constructed; as a Fréchet m-algebra it is the projective limit of its finite-dimensional quotients.

Example (the disc algebra as a closed subalgebra). The disc algebra $A(\mathbb{D})$, the closed subalgebra of $C(\overline{\mathbb{D}})$ consisting of the functions holomorphic on the open disc, is a Banach algebra, hence a Fréchet algebra; its character space is the closed disc, the characters being the evaluations at the points of the disc, and the Gleason–Šilov theorem identifies the maximal ideals accordingly. It illustrates that a closed subalgebra of a Banach or Fréchet algebra is again one, with the inherited norm or seminorm family, and that the character space of a closed subalgebra is the image of the character space of the ambient algebra under restriction, which may be neither injective nor surjective.

Non-Examples

Example (Arens's algebra, named). The Arens algebra — an algebra whose underlying spaces are the $L^p$ spaces of Part III, defined by the integral, so that the algebra is named here and its analytic structure is developed there — is

$$ L^\omega[0,1] = \bigcap_{p\geq1}L^p[0,1] , $$

with the topology generated by the $L^p$-norms $\lVert\cdot\rVert_p$, $p = 1,2,3,\dots$; it is a commutative Fréchet algebra with identity, because the intersection of countably many Banach spaces is a Fréchet space and the multiplication is continuous by Hölder's inequality, $\lVert fg\rVert_p \leq \lVert f\rVert_q\lVert g\rVert_r$ with $1/q + 1/r = 1/p$. Arens showed that its topology cannot be generated by an equivalent family of submultiplicative seminorms: it is a Fréchet algebra which is not locally m-convex. Consequently the Arens–Michael decomposition does not apply to it, and the spectral theory of its elements is not read off from Banach quotients in the way the decomposition prescribes.

Example (algebras that are not locally convex). For $0 < p < 1$ the algebra $L^p[0,1]$, an algebra of Part III with the metric $d(f,g) = \int\lvert f - g\rvert^p$, is a complete metrisable topological algebra with continuous multiplication which is not locally convex, hence not a Fréchet algebra; its only continuous linear functional is zero. This is the boundary of the theory in the other direction from Arens's example: not merely the seminorms but the convexity itself fails.

Example (a metrisable locally m-convex algebra that is not complete). The algebra $\mathbb{C}[X]$ of polynomials with the topology of uniform convergence on compact subsets of $\mathbb{C}$ is a metrisable locally m-convex algebra whose seminorms $p_K(f) = \sup_K\lvert f\rvert$ are submultiplicative; it is not complete, since the exponential series converges locally uniformly to a non-polynomial, and its completion is the Fréchet algebra $\mathcal{O}(\mathbb{C})$ of entire functions. The example separates the metrisable case from the Fréchet case and shows that completeness is not automatic.

Ideals and Quotients

Theorem. Let $A$ be a Fréchet algebra and let $I \subseteq A$ be a closed two-sided ideal. Then the quotient $A/I$ with the quotient topology is a Fréchet algebra; if $A$ is a Fréchet m-algebra, then $A/I$ is a Fréchet m-algebra, and the quotient seminorms are submultiplicative.

Proof. A quotient of a metrisable locally convex space by a closed subspace is metrisable and locally convex, and it is complete because $A$ is complete and $I$ is closed: the quotient map is open, so a Cauchy sequence in $A/I$ has a Cauchy sequence of representatives in $A$, whose limit lies in $A$ and represents the limit of the original sequence. The multiplication on $A/I$ is well defined because $I$ is two-sided and closed. If the seminorms $p_n$ generating the topology of $A$ are submultiplicative, the quotient seminorms

$$ \bar p_n(x + I) = \inf_{y \in I}p_n(x + y) $$

satisfy $\bar p_n(\bar x\bar z) \leq \bar p_n(\bar x)\bar p_n(\bar z)$, by approximating $\bar x$ and $\bar z$ by representatives and using the submultiplicativity of $p_n$ and the two-sidedness of $I$.

Definition. The radical of a commutative Fréchet algebra $A$ is the intersection of its closed maximal ideals; $A$ is semisimple if the radical is $\{0\}$. For a commutative Fréchet m-algebra the radical is the kernel of the Gelfand transform, and the algebra is semisimple precisely when the Gelfand transform is injective.

Theorem (semisimplicity and the transform). Let $A$ be a commutative unital Fréchet m-algebra. Then the closed maximal ideals of $A$ are the kernels of its continuous characters, and the Gelfand transform has kernel equal to the intersection of the closed maximal ideals; hence $A$ is semisimple if and only if the Gelfand transform is injective, and the transform is then a continuous injective homomorphism of $A$ onto a subalgebra of $C(\operatorname{Max}(A))$ that separates points and contains the constants.

Proof. A closed maximal ideal $M$ gives a quotient $A/M$ which is a commutative unital Fréchet m-algebra and a field; by the Gelfand–Mazur theorem for commutative complete locally m-convex algebras (Arens), such a quotient is isomorphic to $\mathbb{C}$, so the quotient map is a continuous character with kernel $M$. Conversely the kernel of a continuous character is a closed maximal ideal. The kernel of the Gelfand transform is the intersection of the kernels of the characters, that is, of the closed maximal ideals. The separation and constant statements are immediate from the definition of the transform.

Example (smooth Gelfand duality, named). For a compact smooth manifold $M$ — an object of Part III, a smooth structure being a differentiable one — the algebra $C^\infty(M)$ of smooth functions is a commutative unital Fréchet algebra, and the evaluations $\chi_t(f) = f(t)$ are characters; the smooth Gelfand duality of Nestruev identifies the character space of $C^\infty(M)$ with $M$ itself and the maximal ideals with the ideals of functions vanishing at a point, so that the algebra determines the manifold. The result is the smooth counterpart of the Gelfand duality of Topological Algebras and Banach Algebras; it is recorded here as a named instance, its manifold theory being Part III's.

Summary

A topological algebra is an algebra with a topology making the vector operations and the multiplication continuous; the multiplication is continuous if and only if it is continuous at $(0,0)$. A locally convex algebra is a topological algebra with a locally convex topology, equivalently one generated by a family of seminorms; it is locally m-convex if the family may be taken submultiplicative, $p(xy) \leq p(x)p(y)$. A single seminorm with $p(xy) \leq Cp(x)p(y)$ and $p(1) = 1$ is replaced by the equivalent submultiplicative seminorm $\tilde p(x) = \sup\{p(xy) : p(y)\leq1\}$ with $p \leq \tilde p \leq Cp$; for a generating family the constants cannot always be absorbed simultaneously, and that failure separates the two classes.

For a locally m-convex algebra the kernels of the generating seminorms are ideals, the quotient-completions $A_\alpha = \widehat{A/N_\alpha}$ are Banach algebras, and the Arens–Michael theorem states that a topological algebra is a complete locally m-convex algebra if and only if it is a projective limit of Banach algebras; in the metrisable case the limit is countable, so a Fréchet m-algebra is a projective limit of a sequence of Banach algebras, and the class is closed under closed subalgebras, quotients by closed ideals, products and projective limits. A commutative unital Fréchet locally m-convex algebra over $\mathbb{C}$ is functionally continuous by Michael's theorem, so every character is continuous; the character space $\operatorname{Max}(A)$ sits in the product of unit discs and is compact when $A$ is a Banach algebra. The spectrum $\sigma_A(x)$ and the spectral radius $r_A(x)$ satisfy $\chi(x) \in \sigma_A(x)$ for every character, and for a commutative unital Banach algebra $\sigma_A(x) = \{\chi(x)\}$, $r_A(x) = \lVert\hat x\rVert_\infty = \lim_n\lVert x^n\rVert^{1/n}$; in the Fréchet case only the inclusion and the inequality are asserted, and the holomorphic functional calculus of the Banach theory extends to the Fréchet m-convex case by Taylor's theorem.

A Fréchet algebra is a complete metrisable locally convex algebra, with topology given by a countable family of seminorms and the metric $d(x,y) = \sum_n2^{-n}\min(1,p_n(x-y))$; the classes are strictly nested: Banach algebras $\subsetneq$ Fréchet m-algebras $\subsetneq$ Fréchet algebras. The standard examples are $C^\infty(U)$, $\mathcal{O}(\Omega)$, the Schwartz space $\mathcal{S}(\mathbb{R}^d)$, the rapidly decreasing sequences $s$, the algebras $C(X)$ with compact convergence, countable products of Banach algebras, the formal power series algebra $K[[X_1,\dots,X_d]]$ with the adic topology, and the disc algebra; the standing counterexamples are Arens's algebra $L^\omega[0,1] = \bigcap_{p\geq1}L^p[0,1]$, a commutative Fréchet algebra which is not locally m-convex, the non-locally-convex algebras $L^p[0,1]$ for $p < 1$, and the incomplete metrisable m-convex algebra $\mathbb{C}[X]$ with the compact-open topology, whose completion is $\mathcal{O}(\mathbb{C})$. Quotients of a Fréchet algebra by closed two-sided ideals are Fréchet algebras, and in the m-convex case the quotient seminorms are submultiplicative; the kernel of the Gelfand transform is the intersection of the closed maximal ideals, so semisimplicity is the injectivity of the transform, and for a compact smooth manifold the smooth Gelfand duality identifies the character space of $C^\infty(M)$ with $M$.

Summary of Notation

Symbol	Meaning
$A$	A topological, locally convex or Fréchet algebra
$A^\times$	Unit group
$p_\alpha$, $p_n$	Generating seminorms; submultiplicative if $p(xy)\leq p(x)p(y)$
$\tilde p$	Submultiplicative regularisation of a seminorm
$N_\alpha = \ker p_\alpha$	Ideal defined by a submultiplicative seminorm
$A_\alpha = \widehat{A/N_\alpha}$	Banach quotient algebra
$\varprojlim_\alpha A_\alpha$	Arens–Michael decomposition
$\chi$, $\operatorname{Max}(A)$, $\hat x$	Character, character space, Gelfand transform
$\sigma_A(x)$, $r_A(x)$	Spectrum and spectral radius
$f(x)$	Holomorphic functional calculus
$C^\infty(U)$, $\mathcal{O}(\Omega)$, $\mathcal{S}(\mathbb{R}^d)$, $s$	Standard Fréchet algebras
$L^\omega[0,1]$	Arens algebra, Fréchet but not locally m-convex
$K[[X_1,\dots,X_d]]$	Formal power series with the adic topology