Locally Convex Spaces with an Involution

Introduction

On a locally convex space the continuous involutions are exactly those that are bounded with respect to the generating seminorms, and the topology can always be generated by invariant seminorms, those satisfying $p(\theta x) = p(x)$: from any generating family the seminorms $\sup(p, p \circ \theta)$ generate the same topology and are invariant. The involution then acts isometrically on each seminorm, its fixed and negated subspaces are closed and are locally convex for the induced topology, and when $2$ is invertible the space is the topological direct sum of the two summands, each of which is itself a locally convex space with the induced structure. On the dual the transposed involution $\theta'$ is continuous for the weak and the strong topologies, and the annihilator calculus identifies its fixed and negated parts with the annihilators of the summands.

This article develops the locally convex theory of an involution. The topological involution, its continuity criterion, the closedness and the topological splitting are Involutive Topological Linear Spaces; the seminorms, the generating families, the topology of uniform convergence and the strong dual are Locally Convex Spaces; the dual pairs, the weak and the strong topologies and the polar calculus are Duality Theory; the quotient and the completion are Topological Modules and Vector Spaces. The normed, Banach, Fréchet and nuclear cases are the following articles of this group; the forms and the adjoints are Part III. No norm is assumed here.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $\varsigma$ is the involution of $\mathbb{K}$ (the identity, or the complex conjugation), $E$ is a Hausdorff locally convex space over $\mathbb{K}$, $\mathcal{P}$ is a generating family of seminorms, and $\theta$ is a continuous $\varsigma$-semilinear involution of $E$ with $\theta^{2} = \mathrm{id}$; the linear case is $\varsigma = \mathrm{id}$ and is written $T$, and the fixed and negated subspaces are $E^{\theta} = \ker(\theta - \mathrm{id})$ and $E^{-} = \ker(\theta + \mathrm{id})$.

Invariant Seminorms and Continuity

Definition. A seminorm $p$ on $E$ is $\theta$-invariant when $p(\theta x) = p(x)$ for all $x$. A locally convex space with an involution is involutive when it carries a continuous involution, and its topology is $\theta$-admissible when it is generated by $\theta$-invariant seminorms.

Theorem (invariant generation). Let $\theta$ be a continuous involution of $E$ and let $\mathcal{P}$ be any generating family of seminorms. Then the family

$$ \mathcal{P}^{\theta} = \{\, \sup(p, p \circ \theta) : p \in \mathcal{P} \,\} $$

generates the same topology and consists of $\theta$-invariant seminorms; hence every continuous involution admits a $\theta$-admissible topology equal to the given one. In particular a continuous involution is an isometry for each seminorm of some generating family.

Proof. The supremum of two seminorms is a seminorm, and $p \circ \theta$ is a seminorm because $\theta$ is continuous and semilinear; the unit balls satisfy $B_{\sup(p, p\circ\theta)} = B_{p} \cap B_{p\circ\theta}$, so the family generates exactly the topology of $\mathcal{P}$. Invariance is $p \circ \theta^{2} = p$: $\sup(p, p\circ\theta)(\theta x) = \sup(p(\theta x), p(\theta^{2}x)) = \sup(p(\theta x), p(x)) = \sup(p,p\circ\theta)(x)$.

Theorem (continuity criterion). Let $\theta$ be a $\varsigma$-semilinear involution of $E$. Then $\theta$ is continuous if and only if for every seminorm $q$ of some generating family the seminorm $q \circ \theta$ is continuous, that is, is dominated by a finite supremum of generating seminorms. For a $\theta$-invariant generating family the criterion is $q \circ \theta \leq C q$ for a constant $C$.

Proof. Continuity of $\theta$ means that $q \circ \theta$ is continuous for every continuous $q$; conversely, if every $q \circ \theta$ is continuous then for each generating seminorm $q$ the preimage of $[0,1)$ under $q\circ\theta$ is a neighbourhood of $0$, so $\theta$ is continuous at $0$ and hence continuous. A seminorm is continuous exactly when it is bounded by a finite supremum of generating seminorms, by the definition of the topology; on an invariant family the finite supremum reduces to a single seminorm up to a constant, giving the stated form.

Corollary (equivalent invariant norms and seminorms). If $E$ is a normed space and $\theta$ is a bounded involution, then $\lVert x\rVert_{\theta} = \sup(\lVert x\rVert, \lVert\theta x\rVert)$ is an equivalent norm for which $\theta$ is isometric; conversely an isometric involution is bounded. The same statement holds for each seminorm of a generating family of a locally convex space.

Proof. The supremum is an equivalent norm because $\theta$ is bounded, and it is invariant by the computation of the theorem; the converse is immediate.

The Fixed and Negated Subspaces

Proposition (closedness and local convexity). The fixed subspace $E^{\theta}$ and the negated subspace $E^{-}$ are closed subspaces of $E$, and each is a locally convex space for the induced topology; the induced topology on $E^{\theta}$ is generated by the restrictions of the $\theta$-invariant seminorms of $E$.

Proof. The subspaces are the kernels of the continuous maps $\mathrm{id} - \theta$ and $\mathrm{id} + \theta$, hence closed; a subspace of a locally convex space with the induced topology is locally convex, generated by the restrictions of the seminorms. The invariance of the restricted seminorms is inherited.

Theorem (topological splitting). Suppose $2$ is invertible in $\mathbb{K}$. Then the averaging maps

$$ \pi_{\pm} = \tfrac{1}{2}(\mathrm{id} \pm \theta) $$

are continuous $\theta$-invariant projections with $\mathrm{im}\,\pi_{+} = E^{\theta}$, $\mathrm{im}\,\pi_{-} = E^{-}$, and

$$ E = E^{\theta} \oplus E^{-} $$

as a topological direct sum; the projection onto $E^{\theta}$ along $E^{-}$ is $\pi_{+}$, and the direct sum is locally convex for the product topology.

Proof. This is the topological splitting of Involutive Topological Linear Spaces: the maps are continuous linear (or semilinear over the fixed field) idempotents with the stated images, their sum is the identity and their product vanishes, so they are the projections of a topological direct sum. The local convexity of the product is Locally Convex Spaces.

Proposition (the fixed subspace over the fixed field). In the antilinear case $\varsigma \neq \mathrm{id}$, the fixed subspace $E^{\theta}$ is a real vector space with the induced topology; it is a locally convex space over the real field, generated by the restrictions of the $\theta$-invariant seminorms, and $\dim_{\mathbb{R}}$ of a finite-dimensional $E^{\theta}$ equals $\dim_{\mathbb{C}} E$ in the finite-dimensional case.

Proof. For $\lambda$ real and $x$ fixed, $\theta(\lambda x) = \varsigma(\lambda)\theta(x) = \lambda x$, so $E^{\theta}$ is closed under real scalars; the induced topology and the seminorm description are the proposition above, and the dimension statement is the real structure of Involutive Topological Linear Spaces.

The Transposed Involution and the Dual

Definition. The transposed involution of $\theta$ is the map $\theta'$ on the continuous dual $E'$ defined by

$$ \theta'(\varphi) = \varphi \circ \theta , $$

so that $\langle \theta x, \varphi\rangle = \varsigma\bigl(\langle x, \theta'\varphi\rangle\bigr)$ under the dual pairing; in the linear case $\theta'$ is linear and in the antilinear case it is again $\varsigma$-semilinear.

Proposition (the transposed involution is continuous). The map $\theta'$ is an involution of $E'$ of the same kind as $\theta$, and it is continuous for the weak-star topology $\sigma(E', E)$, for the weak topology $\sigma(E', E'')$ and for the strong topology $\beta(E', E)$; on the weak-star dual it is a homeomorphism, and its fixed and negated parts are the annihilators

$$ (E')^{\theta'} = (E^{-})^{\circ}, \qquad (E')^{-} = (E^{\theta})^{\circ} . $$

Proof. $\theta'$ is the transpose of the involution $\theta$, so it is an involution and is weak-star continuous by The Dual Operator and the Weak Topology; continuity for the strong topology is the strong continuity of the transpose of The Dual Operator and the Weak Topology, and the weak-star homeomorphism statement is the same place. A functional is fixed under $\theta'$ exactly when $\varphi(\theta x) = \varphi(x)$ for all $x$; on $E^{-}$ this reads $\varphi(-x) = \varphi(x)$, so $\varphi$ vanishes there, and on $E^{\theta}$ it is automatic; hence the fixed part is the annihilator of $E^{-}$, and the negated part is the annihilator of $E^{\theta}$.

Corollary (the involution and the bipolar theorem). The annihilators of the two summands are the two summands of the dual, $(E^{\theta})^{\circ} = (E')^{-}$ and $(E^{-})^{\circ} = (E')^{\theta'}$, and the bipolar theorem of Duality Theory recovers the summands, $\bigl((E^{\theta})^{\circ}\bigr)^{\circ} = E^{\theta}$ and $\bigl((E^{-})^{\circ}\bigr)^{\circ} = E^{-}$, the summands being weakly closed. In the reflexive case the canonical identification $E'' = E$ exchanges the fixed and negated parts of $E$ with the corresponding parts of $E''$.

Proof. The annihilator identifications are the proposition above; the bipolar theorem applies because the summands are closed subspaces, hence weakly closed, and their bipolars in the appropriate dual are their weak closures by Duality Theory. The reflexive statement is the identification $E'' = E$ and the naturality of the transpose.

Quotients, Products and Completion

Proposition (quotients). Let $W \subseteq E$ be a closed $\theta$-stable subspace. Then the quotient $E/W$ carries the involution induced by $\theta$, of the same kind, continuous for the quotient topology; the fixed subspace of the quotient contains the image of $E^{\theta}$ with equality over a field of characteristic not two, and the quotient is locally convex.

Proof. This is the quotient statement of Involutive Topological Linear Spaces, together with the fact that a quotient of a locally convex space by a closed subspace is locally convex, from Locally Convex Spaces.

Proposition (products and completion). A product of involutive locally convex spaces over the same $\varsigma$ carries the componentwise involution, continuous, of the same kind in every factor; a continuous involution of a Hausdorff locally convex space extends uniquely to a continuous involution of the same kind of the completion, and when $2$ is invertible the fixed subspace of the completion is the closure of the fixed subspace of the space.

Proof. The product statement is the product example of Involutive Topological Linear Spaces; the completion statement is its completion theorem, and the completion of a locally convex space is locally convex by Locally Convex Spaces.

Examples

Example (the antilinear conjugation of a complex space). On a complex locally convex space $E$ the conjugation $\theta(x) = \bar x$ of the complexification, or the componentwise conjugation of $\mathbb{C}^n$, is antilinear with fixed subspace a real locally convex space; the invariant seminorms are the real ones, and the splitting $E = E^{\theta} \oplus iE^{\theta}$ is the real structure.

Example (the transposition involution). On $E = \mathbb{C}$ with the usual topology the conjugation is continuous and the invariant seminorm is the modulus; on $\mathbb{C}^n$ with the product topology the componentwise conjugation is continuous with fixed subspace $\mathbb{R}^n$, and the quotient by $\mathbb{R}^n$ is the imaginary subspace.

Example (the sparse topology). On $K[x]$ with the sparse linear topology the map $f(x) \mapsto f(1 - x)$ is a linear involution that is not continuous, so the pair is not an involutive locally convex space; this shows that the criterion above is a genuine restriction and not a formality.

Summary

A locally convex space with a continuous involution $\theta$ can always be described by $\theta$-invariant seminorms, since the assignment $p \mapsto \sup(p, p\circ\theta)$ replaces any generating family by an invariant one with the same topology; consequently a continuous involution is an isometry for some admissible generating family, and it is continuous exactly when it is bounded with respect to the seminorms. The fixed and negated subspaces $E^{\theta}$ and $E^{-}$ are closed locally convex subspaces, and when $2$ is invertible the averaging maps $\pi_{\pm} = \frac12(\mathrm{id}\pm\theta)$ are continuous projections exhibiting $E$ as the topological direct sum $E^{\theta}\oplus E^{-}$; in the antilinear case the fixed subspace is a real locally convex space. The transposed involution $\theta'$ on the dual is continuous for the weak-star, weak and strong topologies, and its fixed and negated parts are the annihilators of $E^{-}$ and $E^{\theta}$; quotients by closed stable subspaces and products carry the induced involution, and a continuous involution extends to the completion with the fixed subspace of the completion the closure of the fixed subspace. The conjugation of a complex space, the transposition involution and the discontinuous sparse involution are the standard examples.

Summary of Notation

Symbol	Meaning
$\mathbb{K}$, $\varsigma$	Field and its involution (identity or conjugation)
$E$	Hausdorff locally convex space over $\mathbb{K}$
$\mathcal{P}$, $p$	Generating family of seminorms
$\theta$, $T$	Continuous $\varsigma$-semilinear involution; the linear case
$E^{\theta}$, $E^{-}$	Fixed and negated subspaces, $\ker(\theta \mp \mathrm{id})$
$\sup(p, p\circ\theta)$	Invariant seminorm generating the same topology
$\lVert\cdot\rVert_{\theta}$	Equivalent invariant norm
$\pi_{\pm} = \frac12(\mathrm{id}\pm\theta)$	Averaging projections
$\theta'$, $(E^{-})^{\circ}$	Transposed involution and the annihilator calculus
$\widehat{E}$	Completion, carrying the extended involution