Involutive Topological Linear Algebras

Introduction

An involutive topological linear algebra is a topological algebra together with a continuous involution of it. A topological $R$-algebra is a unital associative $R$-algebra $A$ whose underlying $R$-module is a topological module over a topological ring $R$, commutative with $1 \neq 0$, and whose product $A \times A \to A$ is continuous; when $R$ is a complete valued field $\mathbb{K}$, either $\mathbb{R}$ or $\mathbb{C}$, this is the topological algebra of Topological Algebras and Banach Algebras, whose first section is the owner of the definition, and it is a topological vector space carrying the product when $R$ is a topological field. The involution of the algebra is measured against an involution of the scalars: $\varsigma$ is a continuous involution of $R$, so that $R$ is an involutive topological ring in the sense of Involutive Topological Rings and Fields, and a map $\sigma : A \to A$ is a $\varsigma$-semilinear involution when

$$ \sigma(\lambda a + \mu b) = \varsigma(\lambda)\sigma(a) + \varsigma(\mu)\sigma(b), \qquad \sigma(ab) = \sigma(b)\sigma(a), \qquad \sigma(1) = 1, \qquad \sigma^2 = \mathrm{id}, $$

the ordinary involution of Involutive Linear Algebras being the case $\varsigma = \mathrm{id}$, which is $R$-linear. A topological involution is a $\varsigma$-semilinear involution that is continuous; over $\mathbb{C}$ with the complex conjugation the map is called antilinear. Beside it stands the second order-two map of the algebra case, which the abstract theory separates from the first: an involutive topological automorphism is a continuous algebra automorphism $\alpha$ with $\alpha^2 = \mathrm{id}$. On a commutative algebra the two notions coincide, because an anti-automorphism of a commutative algebra is an automorphism; on a non-commutative one they do not, and every statement below says which map it is about.

Four additions are the substance of the article. Continuity is a hypothesis, and it is free more often than on a ring: it is free for an isometric involution, for every linear involution in finite dimension over a complete valued field, and for an involution that preserves the ideals of a linear topology; it is not free in general, and the $\mathbb{R}$-algebra $\mathbb{R}[x]$ with the $(x)$-adic topology carries the linear involutions $f(x) \mapsto f(b - x)$, of which only $b = 0$ is continuous. The symmetric and the skew parts are closed, being equalizers, and the symmetric part is a closed topological module over the closed fixed subring, closed under the symmetrised product; that it is a subalgebra only when its elements commute pairwise is the failure of the abstract theory, and the topology adds to that failure the closedness of the part. When $2$ is invertible the averaging maps are continuous and the algebra is the topological direct sum $A^+ \oplus A^-$ of the two parts, so the decomposition of the abstract theory survives as a topological splitting of modules and not of algebras. A closed stable ideal has an involutive quotient, and a continuous involution extends to the completion. And in the $\mathrm{C}^*$-case the involution is isometric and even determines the norm, so that the topology there is a consequence of the involutive algebra rather than an extra datum.

Layout and boundaries. The article has seven sections and a closing comparison: the continuous involutions and their two kinds; the involution and the linear topology with the criterion and the completion; the symmetric and the skew part with the closedness and the topological splitting; the ideals, the quotients and the products; the $\mathrm{C}^*$-case; the semilinear case with the fixed field; and the comparison. The abstract theory is Involutive Linear Algebras, whose involution, symmetric and skew elements, involutive automorphisms, $\sigma$-ideals, quotients, products, exchange involution and descent are used without repetition; the topological background is Topological Modules and Vector Spaces for the linear topologies, the neighbourhoods of zero, the quotient topology, the completion $\widehat{V} = \varprojlim_n V/U_n$ and the finite-dimensional proposition, Topological Algebras and Banach Algebras for the topological algebra, the closure of $\{0\}$ as a two-sided ideal, the submultiplicative norm, the Banach algebras, the $\mathrm{C}^*$-algebras, Gelfand duality and the spectral radius, Topological Groups for the uniformities, Involutive Topological Rings and Fields for the continuous ring involution, the closed fixed subring, the $I$-adic criterion and the closed fixed field of index two, and Involutive Topological Linear Spaces for the closedness, the splitting and the completion read on the underlying space. Several boundaries are marked rather than crossed. The norms, the Banach algebras and the Fréchet algebras are Normed and Banach Spaces and Locally Convex and Fréchet Algebras: the norm appears below in the $\mathrm{C}^*$-section alone and is used with the meaning it has there. The non-commutative $\mathrm{C}^*$-theory, the modules over a $\mathrm{C}^*$-algebra, the states, the positivity and the Gelfand–Naimark theorem are Operator Algebras, and the representations, the von Neumann algebras and the modular theory are the rest of that article. The completed tensor products are Topological Tensor Products, and the forms, the adjoint involution a form defines, the orthogonal and the symplectic type and the reduced norm are Hilbert Algebras: no form is written below and no tensor product is topologised. The graded structures of an involutive automorphism are Superalgebras and Graded Structures, the commutator and the symmetrised product are owned by Lie Algebras and Jordan Algebras, and the involution restricted to the group of units is Involutive Groups. Throughout, $R$ is a topological ring, commutative with $1 \neq 0$, $A$ is a unital associative topological $R$-algebra, Hausdorff when a closedness claim is made, $\varsigma$ is a continuous involution of $R$, $\sigma$ is a $\varsigma$-semilinear involution of $A$, $\alpha$ is an involutive topological automorphism, $A^+ = A^\sigma$ is the set of symmetric elements and $A^-$ the set of skew elements, $I$ is a two-sided ideal, $I_0 \supseteq I_1 \supseteq \cdots$ is a fundamental system of two-sided ideals, $\widehat{A} = \varprojlim_n A/I_n$ is the completion, $\iota : A \to \widehat{A}$ is the map into it, $\rho$ is a quotient map, $\pi_\pm$ are the averaging maps, and $2$ is invertible in $R$ whenever the averaging maps are used.

Continuous Involutions

Definition and First Properties

Definition. A topological involution of the topological $R$-algebra $A$, relative to the involution $\varsigma$ of $R$, is a $\varsigma$-semilinear involution $\sigma$ of $A$ that is continuous for the topology. An involutive topological algebra is a pair $(A,\sigma)$ consisting of a topological $R$-algebra and a topological involution of it. A morphism $(A,\sigma) \to (B,\tau)$ is a continuous unital $R$-algebra homomorphism $\varphi : A \to B$ with $\varphi \circ \sigma = \tau \circ \varphi$; the condition is the algebra form of the condition on the underlying spaces, and it forces the two scalar involutions to agree, since $\sigma(\lambda \cdot 1) = \varsigma(\lambda) \cdot 1$ and $\varphi(1) = 1$.

Proposition (continuity at the origin). Let $\sigma$ be a $\varsigma$-semilinear involution of the topological $R$-algebra $A$. Then $\sigma$ is continuous if and only if it is continuous at $0$; a topological involution is a homeomorphism of $A$ with $\sigma^{-1} = \sigma$, it is uniformly continuous for the additive uniformity, and it is determined by its values on any dense subset of $A$.

Proof. $\sigma$ is additive with $\sigma(0) = 0$, so continuity at every point is continuity at $0$; $\sigma^2 = \mathrm{id}$ exhibits $\sigma$ as a bijection whose inverse is itself, so a continuous involution is a homeomorphism. Being additive and continuous it is a continuous homomorphism of the additive topological group, and it carries the base set $E_U$ of the left uniformity into $E_{\sigma(U)}$, so it is uniformly continuous for the left uniformity and likewise for the right one. For the last assertion, a continuous map into a Hausdorff space is determined by its values on a dense set, and $A$ is Hausdorff.

Proposition (the Hausdorff quotient). The closure of $\{0\}$ in a topological $R$-algebra $A$ is a two-sided ideal, by Topological Algebras and Banach Algebras, and a continuous involution $\sigma$ carries it into itself; hence $\sigma$ descends to a topological involution $\bar\sigma$ of the Hausdorff quotient $A/\overline{\{0\}}$, of the same kind, and the quotient map is a morphism.

Proof. $\sigma(\overline{\{0\}}) \subseteq \overline{\sigma(\{0\})} = \overline{\{0\}}$ because $\sigma$ is continuous and $\sigma(0) = 0$, and the reverse inclusion follows from $\sigma^2 = \mathrm{id}$; the ideal is two-sided and closed by the cited proposition, so the quotient is a Hausdorff topological $R$-algebra and $\sigma$ induces a map of it, which is continuous because $\sigma$ and the quotient map are.

Theorem (the topological opposite algebra). Give the opposite algebra $A^{\mathrm{op}}$ the topology of $A$. Then $A^{\mathrm{op}}$ is a topological $R$-algebra, and the topological involutions of $A$ are exactly the topological $R$-algebra isomorphisms $\sigma : A \to A^{\mathrm{op}}$ of order two. In particular a topological $R$-algebra carries a continuous involution if and only if it is isomorphic to its opposite algebra by an order-two topological isomorphism, and the involution is the certificate of that isomorphism.

Proof. The product of $A^{\mathrm{op}}$ is the composite of the product of $A$ with the swap $A \times A \to A \times A$, which is a homeomorphism, so the product of $A^{\mathrm{op}}$ is continuous and $A^{\mathrm{op}}$ is a topological $R$-algebra with the same addition, scalars and topology. An involution of $A$ is an $R$-algebra isomorphism $A \to A^{\mathrm{op}}$ of order two by Involutive Linear Algebras, and it is continuous as a map $A \to A$ exactly when it is continuous as a map $A \to A^{\mathrm{op}}$, the two spaces having the same topology.

The Two Kinds, and What the Topology Sees

Remark. The distinction of Involutive Linear Algebras between the involution and the involutive automorphism survives the topology unchanged, and it decides what the fixed set is. The fixed set $A^\alpha$ of an involutive topological automorphism is a closed subalgebra, because $\alpha$ is multiplicative and continuous; the fixed set $A^+$ of a topological involution is closed, being the fixed set of a continuous map, but it is a subalgebra exactly when its elements commute pairwise, and it is closed under the symmetrised product $x \bullet y = \tfrac12(xy + yx)$ in general. The topology adds closedness to both and nothing to either, and the algebraic failure is the failure of the abstract theory, now with a topological face. The grading $A = A^+ \oplus A^-$ of the automorphism case is a grading of algebras and the decomposition of the involution case is not; the graded theory is Superalgebras and Graded Structures.

Remark (continuity is not free, and what makes it free). The identity is continuous, every map is continuous for the discrete and the indiscrete topology, an involution that preserves the ideals of a linear topology is continuous, every linear involution in finite dimension over a complete valued field is continuous, and the involution of a $\mathrm{C}^*$-algebra is isometric and hence continuous; each of these is proved below or in the article that owns it. It fails for the family $f(x) \mapsto f(b - x)$ with $b \neq 0$ on $\mathbb{R}[x]$ with the $(x)$-adic topology, and that failure is the reason the criterion of the next section is stated at all.

Examples

Example (the matrix algebras, verdict: two continuous involutions of different kinds). Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, let $A = M_n(\mathbb{K})$ with the topology of entrywise convergence, which is a topological algebra by Topological Algebras and Banach Algebras, and let $n \geq 2$. The transpose $X \mapsto X^{\mathsf{T}}$ is a $\mathbb{K}$-linear map with $(XY)^{\mathsf{T}} = Y^{\mathsf{T}}X^{\mathsf{T}}$ and $(X^{\mathsf{T}})^{\mathsf{T}} = X$, so it is a topological involution, of the kind $\varsigma = \mathrm{id}$: it is a continuous $\mathbb{K}$-linear involution, and it is an anti-automorphism and not an automorphism for $n \geq 2$, since with $E_{12}$ and $E_{21}$ the two matrix units one has $(E_{12}E_{21})^{\mathsf{T}} = E_{11}$ while $E_{12}^{\mathsf{T}}E_{21}^{\mathsf{T}} = E_{22}$. Over $\mathbb{C}$ the adjoint $X \mapsto X^* = \bar{X}^{\mathsf{T}}$ satisfies $X^* = \bar{X}^{\mathsf{T}}$, so $(XY)^* = Y^*X^*$ and $(X^*)^* = X$ and $(\lambda X)^* = \bar\lambda X^*$: it is a continuous $\varsigma$-semilinear involution for the conjugation of $\mathbb{C}$, that is an antilinear one, and it is not linear; the same pair of matrix units shows that it is not an automorphism. Its symmetric elements are the Hermitian matrices and those of the transpose are the symmetric matrices, so the two involutions are different maps on the same algebra. Both are continuous for the entrywise topology because the topology is finite-dimensional.

Example (the three fields and the complex numbers, verdict: a continuous linear involution over $\mathbb{R}$ and a continuous antilinear one over $\mathbb{C}$). The conjugation of $\mathbb{C}$ is an additive map of order two with $\overline{\lambda a} = \bar\lambda \bar a$ and $\overline{ab} = \bar a \bar b$. Read on the $\mathbb{R}$-algebra $\mathbb{C}$ with its usual topology, whose scalar involution is the identity of $\mathbb{R}$, it is a continuous $\mathbb{R}$-linear involution, that is a topological involution of the $\mathbb{R}$-algebra $\mathbb{C}$, with $A^+ = \mathbb{R}$ and $A^- = \mathbb{R}i$; read on the $\mathbb{C}$-algebra $\mathbb{C}$, whose scalars are $\mathbb{C}$ with the conjugation, the same map is a continuous $\varsigma$-semilinear involution and not $\mathbb{C}$-linear. The two readings are the two rows of the same map in the two categories, and the reader is meant to fix the base field before calling the map linear or antilinear.

Example (the function algebra, verdict: a continuous antilinear isometric involution). Let $X$ be a compact Hausdorff space and let $A = C(X,\mathbb{C})$ with pointwise operations and the topology of uniform convergence, which is a commutative unital topological algebra. The map $\sigma(f) = \bar f$ is an involution of the $\mathbb{C}$-algebra $C(X,\mathbb{C})$ relative to the conjugation of the scalars: it is additive, it satisfies $\sigma(\lambda f) = \bar\lambda\sigma(f)$ and $\sigma(fg) = \sigma(f)\sigma(g)$, its square is the identity, and it is continuous. Its verdict is that of a continuous $\varsigma$-semilinear involution which is an automorphism, the algebra being commutative; its fixed algebra is $C(X,\mathbb{R})$. The $\mathrm{C}^*$-structure of this algebra, its sup norm and its Gelfand duality are Topological Algebras and Banach Algebras.

The Involution and the Linear Topology

The Criterion

Proposition (linear topologies of algebras). Let the topology of $A$ be linear, with the two-sided ideals $I_0 \supseteq I_1 \supseteq \cdots$ a fundamental system of neighbourhoods of $0$. Then the product of $A$ is continuous, so $A$ is a topological $R$-algebra, each $I_n$ is open and closed, and each quotient $A/I_n$ is a Hausdorff topological $R$-algebra.

Proof. For $a, b \in A$ and $n$, the elements $a + I_n$ and $b + I_n$ are neighbourhoods and $(a + I_n)(b + I_n) \subseteq ab + aI_n + I_nb + I_nI_n \subseteq ab + I_n$, because $I_n$ is a two-sided ideal; hence the product is continuous at $(a,b)$. A subgroup that is a neighbourhood of $0$ is open, and an open subgroup is closed, so $I_n$ is closed and the quotient is Hausdorff; the quotient of a topological algebra by a closed two-sided ideal is a topological algebra.

Theorem (criterion). Let the topology of $A$ be linear with the two-sided ideals $I_0 \supseteq I_1 \supseteq \cdots$ a fundamental system of neighbourhoods of $0$, and let $\sigma$ be a $\varsigma$-semilinear involution of $A$. Then $\sigma$ is continuous if and only if

$$ \text{for every } n \text{ there is } m \text{ with } \sigma(I_m) \subseteq I_n . $$

Proof. If $\sigma$ is continuous then $\sigma^{-1}(I_n)$ is a neighbourhood of $0$, so it contains some $I_m$, which says $\sigma(I_m) \subseteq I_n$. Conversely suppose the condition holds and let $\Omega$ be a neighbourhood of $0$; some $I_n$ is contained in $\Omega$, and for that $n$ the inclusion $\sigma(I_m) \subseteq I_n \subseteq \Omega$ shows that $\sigma^{-1}(\Omega)$ contains the neighbourhood $I_m$ of $0$, so $\sigma$ is continuous at $0$ and hence continuous.

Corollary ($\sigma$-stable filtrations). If $\sigma(I_n) \subseteq I_n$ for every $n$ then $\sigma$ is continuous. Conversely, if $\sigma$ is continuous then

$$ J_n = I_n \cap \sigma(I_n) $$

is a fundamental system of two-sided ideals cofinal with $(I_n)$, it defines the same topology, and $\sigma(J_n) = J_n$ for every $n$; so a continuous involution may always be read on a filtration that it preserves.

Proof. The first assertion is the criterion with $m = n$. For the second, $\sigma(I_n)$ is a two-sided ideal, the image of a two-sided ideal under an anti-automorphism, so $J_n$ is a two-sided ideal; it is contained in $I_n$, and given $I_n$ the continuity of $\sigma$ supplies $m$ with $\sigma(I_m) \subseteq I_n$, whence $J_m \subseteq I_m \cap I_n = I_n$ for $m \geq n$, so the family $(J_n)$ is cofinal with $(I_n)$ and defines the same topology. Finally $\sigma(J_n) = \sigma(I_n) \cap \sigma(\sigma(I_n)) = \sigma(I_n) \cap I_n = J_n$.

Corollary ($I$-adic topologies). Let $I$ be a two-sided ideal of $A$ and give $A$ the $I$-adic topology with the neighbourhoods $I^n$. Then a $\varsigma$-semilinear involution $\sigma$ is continuous if and only if

$$ \sigma(I)^m \subseteq I \qquad \text{for some } m \geq 1, $$

and in particular $\sigma$ is continuous whenever $\sigma(I) = I$. For a linear involution the condition $\sigma(I)^m \subseteq I$ is automatic when $\sigma(I) \subseteq I$, which for an involution is equivalent to $\sigma(I) = I$. When the $I$-adic and the $\sigma(I)$-adic topologies coincide, $\sigma$ is a homeomorphism for them.

Proof. The criterion with $I_n = I^n$ gives $\sigma(I^m) \subseteq I$ for some $m$, and $\sigma(I^m) = \sigma(I)^m$ because $\sigma$ reverses products; conversely $\sigma(I)^m \subseteq I$ gives $\sigma(I)^{mn} \subseteq I^n$ for every $n$, that is $\sigma(I^{mn}) \subseteq I^n$, and the criterion applies. The case $\sigma(I) = I$ is $m = 1$, and for an involution $\sigma(I) \subseteq I$ implies $I = \sigma(\sigma(I)) \subseteq \sigma(I)$. The last statement is the change-of-ideal argument of Involutive Topological Rings and Fields, applied to the two filtrations.

Remark (the difference from the module case). Involutive Topological Linear Spaces proves that every linear involution of an $I$-adic topological module is continuous, $I$ being an ideal of the scalars; that argument computes $\sigma(I^nV) \subseteq I^nV$ and is unavailable here, because the interesting $I$ is an ideal of the algebra itself, and then $\sigma(I^n) = \sigma(I)^n$ and the criterion is a condition on $\sigma(I)$, not a tautology. The linear case is therefore free for the topology generated by the ideals of the scalars and not for the topology generated by the ideals of the algebra, and the example below is the witness.

The Completion

Theorem (extension to the completion). Let $A$ be a Hausdorff topological $R$-algebra whose topology is linear with the two-sided ideals $I_0 \supseteq I_1 \supseteq \cdots$, let $\widehat{A} = \varprojlim_n A/I_n$ be the completion, and let $\sigma$ be a continuous $\varsigma$-semilinear involution of $A$. Replacing $(I_n)$ by the cofinal family $J_n = I_n \cap \sigma(I_n)$ if necessary, $\sigma$ descends to each quotient $A/I_n$, the family of the induced involutions is compatible with the projections, and it defines a continuous $\varsigma$-semilinear involution $\widehat{\sigma}$ of $\widehat{A}$ with

$$ \widehat{\sigma} \circ \iota = \iota \circ \sigma , $$

the unique continuous extension of $\sigma$. The pair $(\widehat{A},\widehat{\sigma})$ is an involutive topological $R$-algebra, and if $2$ is invertible in $R$ then

$$ (\widehat{A})^{\widehat{\sigma}} = \overline{\iota(A^+)} , $$

the closure being taken in $\widehat{A}$.

Proof. By the corollary above the family $J_n$ is cofinal, consists of two-sided ideals and is $\sigma$-stable, so we may suppose $\sigma(I_n) = I_n$. The involution descends to each quotient $A/I_n$ and commutes with the projections, so the family defines a map $\widehat{\sigma}$ of the inverse limit with $\widehat{\sigma}^2 = \mathrm{id}$; it is anti-multiplicative, being the limit of the anti-multiplicative maps of the quotients, so it is an involution of the algebra $\widehat{A}$, and it is $\varsigma$-semilinear for the same reason. The product of the inverse limit is continuous and makes $\widehat{A}$ a topological $R$-algebra, because the product of two compatible families is the compatible family of the products, and the composites with the projections are continuous; $\widehat{\sigma}$ is continuous because it carries the kernel of the projection $\widehat{A} \to A/I_n$ into itself, and that kernel is a basic neighbourhood of $0$ in $\widehat{A}$. For the fixed set, let $a \in \widehat{A}$ be fixed. The image of $\iota$ is dense, so $a$ is a limit of $\iota(a_k)$ with $a_k \in A$; applying $\widehat{\sigma}$ and using $\widehat{\sigma}\iota = \iota\sigma$ gives $a = \lim \iota(\sigma(a_k))$ as well, hence $\iota(a_k - \sigma(a_k)) \to 0$ and, multiplying by $\tfrac12 \in R$ which exists by hypothesis,

$$ \iota\!\left(\tfrac12(a_k + \sigma(a_k))\right) = \tfrac12\bigl(\iota(a_k) + \iota(\sigma(a_k))\bigr) \longrightarrow \tfrac12(a + a) = a, $$

with $\tfrac12(a_k + \sigma(a_k)) \in A^+$ for every $k$; so $a$ lies in the closure of $\iota(A^+)$. The reverse inclusion holds because $\widehat{\sigma}$ is continuous and $\widehat{A}$ is Hausdorff, so its fixed set is closed.

Corollary (the completed algebra of a stable filtration). Let $A$ be an involutive topological $R$-algebra whose topology is given by the $\sigma$-stable ideals $I_n$, so that $\sigma$ is continuous by the criterion. Then $\widehat{A}$ is an involutive topological $R$-algebra, $\widehat{\sigma}$ is the unique continuous extension of $\sigma$, the completion is complete and Hausdorff, and when $2$ is invertible the completion of the fixed subalgebra is the closure of its image in the fixed subalgebra of the completion. The completion of $A$ is the same whether the topology is read through $(I_n)$ or through any cofinal family, and in particular through $(J_n)$.

Proof. The completion is complete and Hausdorff by Topological Modules and Vector Spaces; the uniqueness of $\widehat{\sigma}$ is that of a continuous extension to a dense subset of a Hausdorff space; the rest is the theorem.

Examples

Example (the $(x)$-adic algebra $\mathbb{R}[x]$, verdict: a linear involution, continuous exactly for $b = 0$). Let $A = \mathbb{R}[x]$ be the polynomial algebra over the discrete field $\mathbb{R}$, with the $(x)$-adic topology, whose neighbourhoods of $0$ are the ideals $(x)^n$ of the polynomials divisible by $x^n$; these are two-sided ideals, so $A$ is a topological $\mathbb{R}$-algebra by the proposition above, and the topology is Hausdorff because only the zero polynomial is divisible by every $x^n$. For $b \in \mathbb{R}$ let $\sigma_b$ be the substitution $\sigma_b(f)(x) = f(b - x)$. Then $\sigma_b$ is an $\mathbb{R}$-linear map of order two, because $x \mapsto b - x$ is an involution of the polynomial algebra and the constants are fixed, so it is a linear involution of the algebra, and it is an automorphism as well. Its verdict is that of an $\mathbb{R}$-linear topological involution for $b = 0$ and of a discontinuous $\mathbb{R}$-linear involution for $b \neq 0$. Indeed $\sigma_b((x)) = (b - x)$ is the ideal generated by $b - x$, and $(b - x)^m$ has constant term $b^m$, which is nonzero for $b \neq 0$, so $(b - x)^m \not\subseteq (x)$ for every $m$ and the $I$-adic criterion fails; for $b = 0$ the map is $f(x) \mapsto f(-x)$, the ideal $(x)$ is preserved, and $\sigma_0$ is continuous. So exactly one member of the family is continuous and the whole rest of it is not, although all the members are the same kind of map and all are automorphisms of the same algebra. The family is the family of the ring involutions of Involutive Topological Rings and Fields and of the linear involutions of Involutive Topological Linear Spaces, read here as algebra involutions of the topological algebra $\mathbb{R}[x]$; the constant term of $(b-x)^m$ is $b^m$, while the coefficient of $x^m$ is $(-1)^m$, and it is the constant term that decides the criterion.

Example (the completed polynomial algebra, verdict: the two involutions extend and the fixed algebras are the closures). The $(x)$-adic completion of $\mathbb{R}[x]$ is the algebra $\mathbb{R}[[x]]$ of formal power series, the ideal $(x)$ being preserved by $\sigma_0$, so $\sigma_0$ extends to the substitution $x \mapsto -x$ on $\mathbb{R}[[x]]$, whose fixed algebra is the algebra $\mathbb{R}[[x^2]]$ of the even power series, and $\mathbb{R}[[x^2]]$ is the closure of $\mathbb{R}[x^2]$, the fixed algebra of the polynomials. The discontinuous $\sigma_b$ with $b \neq 0$ does not extend to the completion at all, exactly because it is not continuous, and this is the sharper statement of the same fact. Verdict: a continuous linear involution of the completed topological algebra, with the fixed algebra the closure of the fixed algebra of the dense subalgebra.

Example (the complex polynomials, verdict: a continuous antilinear involution and its completed fixed algebra). On $A = \mathbb{C}[x]$ with the $(x)$-adic topology, the map $\theta(f)(x) = \overline{f(-x)}$ is additive, satisfies $\theta(\lambda f) = \bar\lambda\,\theta(f)$ and $\theta(fg) = \theta(f)\theta(g)$ because the algebra is commutative, and has $\theta^2 = \mathrm{id}$: it is a $\varsigma$-semilinear involution of the $\mathbb{C}$-algebra $\mathbb{C}[x]$ for the conjugation of the scalars, that is an antilinear one. It preserves every ideal $(x)^n$, so it is continuous, and its fixed algebra consists of the polynomials whose even coefficients are real and whose odd coefficients are purely imaginary. On the completion $\mathbb{C}[[x]]$ the same substitution and conjugation define $\widehat{\theta}$, whose fixed algebra is the algebra of the series with real even coefficients and purely imaginary odd coefficients, the closure of the fixed algebra of the polynomials, in agreement with the theorem. Verdict: a continuous antilinear involution, extending to the completion, with the two fixed algebras as described.

The Symmetric and the Skew Part

Closedness

Theorem (closedness of the two parts). Let $\sigma$ be a topological involution of the Hausdorff topological $R$-algebra $A$. Then the symmetric part $A^+$ and the skew part $A^-$ are closed in $A$, and both are closed $R^\varsigma$-submodules of $A$, hence topological $R^\varsigma$-modules for the subspace topology. If $\sigma$ is an involutive automorphism, or if $A$ is commutative, then $A^+$ is a closed subalgebra; in general, with $2$ invertible in $R$, the set $A^+$ is closed under the symmetrised product $x \bullet y = \tfrac12(xy + yx)$ and $A^-$ is closed under the commutator $[x,y] = xy - yx$; both products are continuous, so $A^-$ is a closed topological Lie algebra and, when $2$ is invertible, $A^+$ a closed topological Jordan algebra.

Proof. $A^+$ is the equalizer of the continuous maps $\sigma$ and $\mathrm{id}_A$, and $A^-$ is the equalizer of $\sigma$ and $-\mathrm{id}_A$; the equalizer of two continuous maps into a Hausdorff space is closed, being the preimage of the diagonal. If $x \in A^+$ and $r \in R^\varsigma$ then $\sigma(rx) = \varsigma(r)\sigma(x) = rx$, so $A^+$ is an $R^\varsigma$-submodule, and likewise $A^-$; the ring $R^\varsigma$ is closed in $R$ by Involutive Topological Rings and Fields, so it is a topological ring in the subspace topology and the two parts are topological $R^\varsigma$-modules. If $\sigma$ is multiplicative its fixed set is a subalgebra, and if $A$ is commutative every anti-automorphism is an automorphism. For $x,y \in A^+$ one has $\sigma(xy) = \sigma(y)\sigma(x) = yx$, so $\sigma(x \bullet y) = \tfrac12(\sigma(xy) + \sigma(yx)) = \tfrac12(yx + xy) = x \bullet y$ and $x \bullet y \in A^+$; for $x,y \in A^-$ one has $\sigma(xy) = yx$, so $\sigma([x,y]) = \sigma(xy) - \sigma(yx) = yx - xy = -[x,y]$ and $[x,y] \in A^-$. The two products are continuous, being composites of the continuous product of $A$ with addition and with the fixed scalar $\tfrac12$ for the symmetrised one, so the two parts are topological algebras for their structures.

Corollary (a dense fixed part forces the trivial involution). If $A^+$ is dense in the Hausdorff algebra $A$ then $\sigma = \mathrm{id}_A$. In particular a continuous involution is determined by its values on any dense subset.

Proof. A dense closed subset is the whole space; $A^+ = A$ says $\sigma = \mathrm{id}$.

The Topological Splitting

Theorem (the averaging maps). Let $\sigma$ be a topological involution of $A$ and suppose $2$ is invertible in $R$. Then the averaging maps

$$ \pi_+ = \tfrac12(\mathrm{id}_A + \sigma), \qquad \pi_- = \tfrac12(\mathrm{id}_A - \sigma), $$

are continuous idempotent $R^\varsigma$-linear endomorphisms of the $R^\varsigma$-module $A$ with $\pi_+ + \pi_- = \mathrm{id}_A$, $\pi_+\pi_- = \pi_-\pi_+ = 0$, images $A^+$ and $A^-$ and kernels $A^-$ and $A^+$. Consequently

$$ A \cong A^+ \times A^- $$

as topological $R^\varsigma$-modules, $A$ is the topological direct sum of its symmetric and its skew part, and $A^+$ is a retract of $A$. The projection $\pi_+$ is multiplicative, that is an algebra homomorphism, exactly when $\sigma$ is multiplicative, that is for an involutive automorphism or for a commutative $A$; otherwise the splitting is a splitting of modules and not of algebras.

Proof. The maps are composites of the continuous operations of the module with the fixed scalar $1/2$, hence continuous, and $R^\varsigma$-linear because $\pi_\pm(rx) = \tfrac12(rx \pm \varsigma(r)\sigma(x)) = r\pi_\pm(x)$ for $r \in R^\varsigma$. They are idempotent with the stated sums, products and kernels, directly from $\sigma^2 = \mathrm{id}$ and $2 \cdot \tfrac12 = 1$; the image of $\pi_+$ is $A^+$ and the image of $\pi_-$ is $A^-$. Hence $x \mapsto (\pi_+x, \pi_-x)$ is a bijection $A \to A^+ \times A^-$ with continuous components and continuous inverse the sum, so it is a homeomorphism of topological $R^\varsigma$-modules, and the second component of $\pi_+$ being zero exhibits $A^+$ as a retract. For multiplicativity, $\pi_+(xy) = \pi_+(x)\pi_+(y)$ for all $x,y$ is equivalent to $\sigma(xy) = \sigma(x)\sigma(y)$ for all $x,y$.

Remark. The theorem is the topological refinement of the decomposition of Involutive Linear Algebras, where the symmetric part is a Jordan algebra, the skew part a Lie algebra and the splitting an identity of spaces; topologically the dimensions are replaced by the closedness of the two parts and the continuity of the projections, and the two parts are closed by the theorem above whether or not $2$ is invertible, while the splitting itself needs $2$, exactly as in the abstract theory.

Example (the splitting of the polynomial algebra, verdict: closed parts and a continuous non-multiplicative projection). On $A = \mathbb{R}[x]$ with the $(x)$-adic topology and the continuous involution $\sigma_0(f)(x) = f(-x)$, the symmetric part is the algebra $\mathbb{R}[x^2]$ of the even polynomials and the skew part is the module $x\mathbb{R}[x^2]$ of the odd ones; both are closed in the $(x)$-adic topology, the direct sum $\mathbb{R}[x^2] \oplus x\mathbb{R}[x^2]$ is all of $\mathbb{R}[x]$, and the projection $\pi_+$ is continuous. The projection is not multiplicative and not an algebra homomorphism: $\pi_+(x) = 0$, so $\pi_+(x)\pi_+(x) = 0$, while $\pi_+(x^2) = x^2$. On $A = \mathbb{C}[x]$ with the antilinear $\theta(f)(x) = \overline{f(-x)}$ the symmetric part is the real subspace $\mathbb{R}[x^2] \oplus ix\mathbb{R}[x^2]$, whose elements have real even and purely imaginary odd coefficients, the skew part is $i\mathbb{R}[x^2] \oplus x\mathbb{R}[x^2]$, and $A$ is the topological direct sum of the two over the fixed field $\mathbb{R}$ of the conjugation. Verdict: in the first case a continuous linear involution whose symmetric part is a closed subalgebra, in the second a continuous antilinear involution whose symmetric part is a closed real subspace and not a complex one.

Ideals, Quotients and Products

Definition. A subset $B \subseteq A$ is $\sigma$-stable when $\sigma(B) = B$; an ideal that is $\sigma$-stable is a $\sigma$-ideal in the sense of Involutive Linear Algebras, and an ideal is $\sigma$-stable exactly when $\sigma(I) \subseteq I$, because $\sigma$ is an involution.

Proposition (closures of stable sets). Let $\sigma$ be a continuous involution of $A$ and let $B \subseteq A$ be $\sigma$-stable. Then the closure $\overline{B}$ is $\sigma$-stable; in particular the closure of a $\sigma$-stable two-sided ideal is a closed $\sigma$-stable two-sided ideal, and it is the smallest closed one containing $I$.

Proof. $\sigma(\overline{B}) \subseteq \overline{\sigma(B)} = \overline{B}$ by continuity, and applying $\sigma$ again with $\sigma^2 = \mathrm{id}$ gives equality. A closed $\sigma$-stable ideal containing $I$ contains $\overline{I}$ because it is closed, and $\overline{I}$ is an ideal and is $\sigma$-stable.

Theorem (the quotient carries the involution). Let $I$ be a closed two-sided $\sigma$-stable ideal of $A$. Then $A/I$ is a Hausdorff topological $R$-algebra, the induced map $\bar\sigma(\rho(a)) = \rho(\sigma(a))$ is a topological involution of $A/I$ of the same kind as $\sigma$, and

$$ \rho(A^+) \subseteq (A/I)^{\bar\sigma} , $$

with equality when $2$ is invertible in $R$. The quotient symmetric part can be strictly larger when $2$ is not invertible.

Proof. The quotient of a topological algebra by a closed two-sided ideal is Hausdorff and topological, the multiplication of the quotient being continuous because the quotient map is open, by Topological Modules and Vector Spaces. The map $\bar\sigma$ is well defined and of order two because $\sigma$ preserves $I$, and it is continuous because $\rho$ and $\sigma$ are. An element of $\rho(A^+)$ has a symmetric representative and is symmetric, whence the inclusion. If $2$ is invertible and $\rho(a)$ is symmetric then $\sigma(a) - a \in I$; hence $\tfrac12(a - \sigma(a)) \in A^- \cap I$, because $I$ is a submodule and $2$ is invertible in $R$, while $\tfrac12(a + \sigma(a)) \in A^+$; adding the two gives $a \in A^+ + I$, so $\rho(a) \in \rho(A^+)$.

Corollary (kernels and images of morphisms). The kernel of a continuous morphism $(A,\sigma) \to (B,\tau)$ of involutive topological algebras is a closed two-sided $\sigma$-stable ideal, so the image of the morphism with the quotient topology carries the induced involution and the morphism factors through it.

Proof. A continuous homomorphism of topological algebras has closed kernel, the target being Hausdorff, and the kernel is two-sided; it is $\sigma$-stable because for $x \in \ker\varphi$ one has $\varphi(\sigma(x)) = \tau(\varphi(x)) = \tau(0) = 0$, so that $\sigma(x) \in \ker\varphi$.

Example (strictness in characteristic two, verdict: a continuous linear involution whose quotient symmetric part is strictly larger). Let $K = \mathbb{F}_2$, let $A = K[x,y]$ with the $(x + y)$-adic topology, and let $\sigma$ be the swap $\sigma(x) = y$, $\sigma(y) = x$, extended to the algebra. Then $\sigma$ is a $K$-linear involution and an automorphism of the commutative algebra, the ideal $I = (x + y)$ is two-sided and $\sigma$-stable, and $I$ is a neighbourhood of $0$, hence open and closed; so $\sigma$ is continuous and the theorem applies with $2 = 0$ not invertible. The quotient $A/I$ is the polynomial algebra $K[x]$ in the class of $x$, because $y \equiv x$, and the induced involution is the identity of $K[x]$, since $\bar\sigma(x) = \bar y = \bar x$: every class is symmetric. The image of $A^+$ is the image of the symmetric polynomials, which are the polynomials in the elementary symmetric functions $x + y$ and $xy$; modulo $I$ the first is $0$ and the second is $x^2$, so the image is the subalgebra $K[x^2]$ of the even polynomials, of index two in $K[x]$. The class of $x$ is symmetric and is not in the image, so

$$ \rho(A^+) \subsetneq (A/I)^{\bar\sigma} , $$

and the strictness occurs exactly because $2$ is not invertible in $K$. This is the algebra form of the strictness computed for a module in Involutive Topological Linear Spaces and for a ring in Involutive Topological Rings and Fields, and the algebra brings the symmetric polynomials into the computation in place of the fixed subring.

Theorem (products and the exchange). Let $A$ and $B$ be involutive topological $R$-algebras with involutions $\sigma$ and $\tau$. Then $A \times B$ with the product algebra structure and the product topology is an involutive topological $R$-algebra with the involution $\sigma \times \tau$, which is continuous, and

$$ (A \times B)^{\sigma \times \tau} = A^+ \times B^+ , \qquad (A \times B)^- = A^- \times B^- . $$

On $A \times A^{\mathrm{op}}$, with the product $(a_1,a_2)(b_1,b_2) = (a_1b_1, b_2a_2)$ and the product topology, the swap $\sigma(a_1,a_2) = (a_2,a_1)$ is a topological involution and an automorphism of the algebra, its symmetric part is the diagonal, which is homeomorphic to $A$, and the diagonal is a subalgebra exactly when $A$ is commutative. On $A \times A$ the swap is an involutive topological automorphism with the diagonal as a closed subalgebra.

Proof. The product of the topological algebras is a topological algebra, the product topology is the product of the two, and $\sigma \times \tau$ is continuous because its two components are; it is an anti-automorphism of order two, because the two factors are and the products are taken componentwise, and the symmetric part is computed componentwise. For the exchange, the product of $A \times A^{\mathrm{op}}$ is continuous and the swap is a homeomorphism of the space and an anti-automorphism, $A$ being identified with $(A^{\mathrm{op}})^{\mathrm{op}}$; the product $(a,a)(b,b) = (ab,ba)$ lies on the diagonal exactly when $ab = ba$, so the diagonal is a subalgebra exactly when $A$ is commutative, though it is always the symmetric part of the swap and always closed. On $A \times A$ the swap is multiplicative, hence an involutive automorphism, and its fixed subalgebra is the diagonal.

Remark (the tensor product). The tensor product of two involutive algebras carries the involution $\sigma \otimes \tau$, whose symmetric part is $A^+ \otimes B^+ \oplus A^- \otimes B^-$ and whose skew part is $A^+ \otimes B^- \oplus A^- \otimes B^+$, by Involutive Linear Algebras; the topology that makes the tensor product an involutive topological algebra is the subject of Topological Tensor Products, and no topology on it is fixed here. In the finite-dimensional case over a complete valued field there is only one linear topology, so $\sigma \otimes \tau$ is continuous there as soon as $\sigma$ and $\tau$ are, and that is the only case used below.

The $\mathrm{C}^*$-Case: When the Involution Determines the Topology

In this section $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, $A$ is a $\mathbb{K}$-algebra with a submultiplicative norm $\|\cdot\|$, and the notions of a normed algebra, a Banach algebra and a $\mathrm{C}^*$-algebra, with the $\mathrm{C}^*$-identity $\|a^*a\| = \|a\|^2$, are those of Topological Algebras and Banach Algebras; the spectral radius is $r(a)$, and Gelfand duality and the spectral radius formula are that article's. The non-commutative theory — the Gelfand–Naimark theorem, the states, the positivity, the Gelfand–Naimark–Segal construction and the von Neumann algebras — is Operator Algebras and is not used.

Proposition (the involution of a $\mathrm{C}^*$-algebra is isometric). Let $A$ be a $\mathrm{C}^*$-algebra. Then $\|a^*\| = \|a\|$ for every $a \in A$. Hence the involution of a $\mathrm{C}^*$-algebra is isometric, therefore continuous: over $\mathbb{R}$ it is a topological involution of the real algebra $A$, and over $\mathbb{C}$ it is a topological $\varsigma$-semilinear involution for the conjugation of the scalars, that is an antilinear one. In the complex case the self-adjoint part $A^+ = \{a : a^* = a\}$ is a closed real subspace and $A = A^+ \oplus iA^+$ as real spaces, by the splitting theorem over the fixed field $\mathbb{R}$, whose hypothesis that $2$ be invertible holds there.

Proof. The $\mathrm{C}^*$-identity gives $\|a\|^2 = \|a^*a\| \leq \|a^*\|\,\|a\|$, whence $\|a\| \leq \|a^*\|$ for $a \neq 0$ and trivially for $a = 0$; applying the same inequality to $a^*$, whose adjoint is $a$, gives $\|a^*\| \leq \|a\|$, so the two are equal. An isometry is continuous, and a continuous involution is a topological involution by the first section; the rest is the closedness theorem and the splitting theorem applied to the conjugation of $\mathbb{C}$ over the fixed field $\mathbb{R}$.

Theorem (the norm is determined by the involution). Let $A$ be a unital $\mathrm{C}^*$-algebra. Then

$$ \|a\|^2 = \|a^*a\| = r(a^*a) = \sup\{|\lambda| : \lambda \in \sigma(a^*a)\} , $$

so the norm of $a$ is computed from the -algebra structure of $A$ alone. Consequently a unital -algebra carries at most one norm making it a $\mathrm{C}^*$-algebra, and the topology of a $\mathrm{C}^*$-algebra is determined by its involutive algebra structure: on a $\mathrm{C}^*$-algebra the involution determines the topology it is continuous for, and "$\mathrm{C}^*$-algebra" is a property of a *-algebra and not a structure added to one.

Proof. For a normal element $h$ of $A$ the subalgebra $C^*(h,1)$ is a commutative unital $\mathrm{C}^*$-algebra, and Gelfand duality of Topological Algebras and Banach Algebras identifies it with $C(\operatorname{Max}(C^*(h,1)))$ by an isometric -isomorphism carrying $h$ to a function whose supremum modulus is $r(h)$; hence $\|h\| = r(h)$ for normal $h$. The element $a^*a$ is self-adjoint, hence normal, so $\|a\|^2 = \|a^*a\| = r(a^*a)$ by the $\mathrm{C}^*$-identity. The spectrum $\sigma(a^*a)$ is the set of scalars $\lambda$ for which $a^*a - \lambda 1$ is not invertible, and invertibility is defined by the -algebra structure, so $\sigma(a^*a)$, and with it $r(a^*a)$, does not depend on the norm. Hence two $\mathrm{C}^*$-norms on the same unital *-algebra agree, and the identity displayed computes each of them from the algebra.

Theorem (automatic continuity of $*$-homomorphisms). Let $\varphi : A \to B$ be a unital $*$-homomorphism of unital $\mathrm{C}^*$-algebras. Then $\varphi$ is contractive,

$$ \|\varphi(a)\| \leq \|a\| \qquad \text{for every } a \in A , $$

so it is continuous with $\|\varphi\| \leq 1$, and a unital $*$-isomorphism of $\mathrm{C}^*$-algebras is isometric, hence a homeomorphism.

Proof. If $a - \lambda 1$ is invertible with inverse $b$ then $\varphi(b)$ is an inverse of $\varphi(a) - \lambda 1 = \varphi(a - \lambda 1)$ in $B$, so $\sigma(\varphi(a)) \subseteq \sigma(a)$ and $r(\varphi(a)) \leq r(a)$. Since $\varphi$ is a *-homomorphism, $\varphi(a^*a) = \varphi(a)^*\varphi(a)$ is self-adjoint, hence normal, and the theorem above applied in $B$ and in $A$ gives

$$ \|\varphi(a)\|^2 = r\bigl(\varphi(a)^*\varphi(a)\bigr) = r\bigl(\varphi(a^*a)\bigr) \leq r(a^*a) = \|a\|^2 . $$

A unital -isomorphism has a unital -homomorphism as inverse, so both inequalities hold and it is isometric.

Remark. The two theorems are the answer this article gives to its own question in the normed case. On a $\mathrm{C}^*$-algebra the involution is continuous not by hypothesis but by the $\mathrm{C}^*$-identity, and the topology itself is a function of the involution; and the morphisms of the category are continuous automatically, so continuity is free a second time, on the maps rather than on the involution. That the continuity of an involution of a topological algebra is a genuine hypothesis is what the $(x)$-adic family of the previous section shows, and the $\mathrm{C}^*$-identity is exactly the hypothesis that removes it.

Examples

Example (the matrix algebras as $\mathrm{C}^*$-algebras, verdict: an antilinear isometric involution and a linear isometric one). On $A = M_n(\mathbb{C})$ with the operator norm, the largest singular value, the adjoint $X \mapsto X^*$ and the transpose $X \mapsto X^{\mathsf{T}}$ are both isometric: $\|X^*\| = \|X\| = \|X^{\mathsf{T}}\|$ and $\|X^*X\| = \|X\|^2$, which is the $\mathrm{C}^*$-identity. Their verdicts are the same as before, and the norm adds one thing: the adjoint is antilinear and isometric, the transpose is linear and isometric, and the two are different maps whenever $n \geq 2$, the symmetric part of the first being the Hermitian matrices and that of the second the symmetric ones. The norm of the theorem is computed from the involution, $\|X\| = \sqrt{r(X^*X)}$, and $M_n(\mathbb{C})$ carries exactly one $\mathrm{C}^*$-norm; the norm is $*$-invariant and submultiplicative, and both facts are Topological Algebras and Banach Algebras.

Example (the function algebra as a $\mathrm{C}^*$-algebra, verdict: an antilinear isometric involution). On $A = C(X,\mathbb{C})$ with the sup norm, which is a commutative unital $\mathrm{C}^*$-algebra, the involution $\sigma(f) = \bar f$ of the example above is isometric, $\|\bar f\|_\infty = \|f\|_\infty$, and the $\mathrm{C}^*$-identity $\|\bar f f\|_\infty = \|f\|_\infty^2$ holds. Its verdict: a continuous $\varsigma$-semilinear isometric involution, an automorphism because the algebra is commutative, with fixed algebra $C(X,\mathbb{R})$; the theorem on the norm says that this sup norm is the only norm on $C(X,\mathbb{C})$ making it a $\mathrm{C}^*$-algebra, and the theorem on morphisms says that a unital *-homomorphism $C(X,\mathbb{C}) \to M_n(\mathbb{C})$, that is a representation, is automatically contractive.

The Semilinear Case and the Fixed Field

Theorem (the fixed field and the linear structure over it). Let $F$ be a topological field, let $\varsigma$ be a continuous involution of $F$, let $A$ be a Hausdorff topological $F$-algebra and let $\sigma$ be a topological $\varsigma$-semilinear involution of $A$. Then the fixed field $F^\varsigma$ is a closed subfield of $F$, a topological field in the subspace topology and of index two in $F$ when $\varsigma \neq \mathrm{id}$, and $\sigma$ is $F^\varsigma$-linear; consequently $A$ is a topological $F^\varsigma$-algebra, the two parts $A^+$ and $A^-$ are closed topological $F^\varsigma$-submodules of $A$, the skew part is a closed topological Lie algebra over $F^\varsigma$, and when the characteristic of $F$ is not two the symmetric part is a closed topological Jordan algebra over $F^\varsigma$ and the algebra is the topological direct sum

$$ A = A^+ \oplus A^- $$

of $F^\varsigma$-modules. The symmetric part is a subalgebra over $F^\varsigma$ exactly when its elements commute pairwise.

Proof. That $F^\varsigma$ is a closed subfield of index two and a topological field is the theorem of Involutive Topological Rings and Fields; restricting the scalars of $A$ from $F$ to the closed subfield $F^\varsigma$ makes $A$ a topological $F^\varsigma$-algebra, the product being $F^\varsigma$-bilinear and continuous. A $\varsigma$-semilinear map is $F^\varsigma$-linear, because $\varsigma(r) = r$ for $r \in F^\varsigma$, so $\sigma$ is $F^\varsigma$-linear and the closedness theorem, the splitting theorem and the Jordan and Lie statements apply with $R$ replaced by $F^\varsigma$. The last assertion is the algebraic criterion of Involutive Linear Algebras.

Theorem (finite dimension over a complete valued field). Let $F$ be a complete valued field, let $A$ be a finite-dimensional $F$-algebra with a topology making it a topological $F$-algebra, and let $\sigma$ be a $\varsigma$-semilinear involution of $A$. Then $\sigma$ is continuous if and only if the involution $\varsigma$ of the scalars is continuous. In particular every linear involution in finite dimension is continuous, and the semilinear ones are continuous exactly as far as the scalar involution is.

Proof. Suppose $\varsigma$ continuous. The fixed field $F^\varsigma$ is closed in the complete field $F$, hence complete in the induced valuation, and $A$ is finite-dimensional over $F^\varsigma$, because $F$ is of degree two over its fixed field; the map $\sigma$ is $F^\varsigma$-linear, by the theorem above, so it is continuous by the finite-dimensional proposition of Topological Modules and Vector Spaces, which states that every linear map of a finite-dimensional space over a complete valued field is continuous. Conversely, suppose $\sigma$ continuous and let $\lambda_k \to \lambda$ in $F$. Then $\lambda_k \cdot 1 \to \lambda \cdot 1$ in $A$, so $\sigma(\lambda_k \cdot 1) = \varsigma(\lambda_k)\cdot 1$ converges to $\sigma(\lambda \cdot 1) = \varsigma(\lambda)\cdot 1$; the subspace $F \cdot 1$ is one-dimensional over $F$, and a linear isomorphism of finite-dimensional spaces over a complete valued field is a homeomorphism, again by the cited proposition, so $\varsigma(\lambda_k) \to \varsigma(\lambda)$ and $\varsigma$ is continuous.

Example (the completeness is needed, verdict: a discontinuous semilinear involution in dimension one). Let $F = \mathbb{Q}(\sqrt{2})$ with the topology inherited from $\mathbb{R}$, a valued field which is not complete, and let $A = F$ as a one-dimensional $F$-algebra. The conjugation $\varsigma(a + b\sqrt{2}) = a - b\sqrt{2}$ is an involution of the field, and it is discontinuous, as Involutive Topological Rings and Fields records: the rationals converge to $\sqrt{2}$ in the induced topology and are fixed, so the image of the limit would have to be both $\sqrt{2}$ and $-\sqrt{2}$. Read on the algebra $A$, the map $\varsigma$ is the $\varsigma$-semilinear involution $\sigma = \varsigma$, since $\varsigma(\lambda a) = \varsigma(\lambda)\varsigma(a)$, and its verdict is that of a discontinuous semilinear involution of a one-dimensional algebra: finite dimension does not force continuity, because the field of scalars is not complete. The theorem above is sharp: the completeness of the field, equivalently of the closed fixed field of the twisting involution, is what carries the proof, and the example shows that it cannot be dropped.

Remark (extension of scalars and descent). Let $K/F$ be a quadratic extension with nontrivial automorphism $\varsigma$, let $t$ be a $\varsigma$-skew element with $K = F \oplus Ft$ when $2 \neq 0$, and let $B$ be an $F$-algebra with an involution $\tau$. The map $\tau \otimes \varsigma$ on the $K$-algebra $B \otimes_F K$ is a $\varsigma$-semilinear involution of the second kind with symmetric part $(B^+ \otimes 1) \oplus (B^- \otimes t)$, by Involutive Linear Algebras, and in the commutative case it descends $B \otimes_F K$ to $B$ by the fixed algebra $B \otimes 1$. Topologically, $B \otimes_F K$ carries many linear topologies in general and the one compatible with the involution is the subject of Topological Tensor Products; when $K/F$ is finite over a complete valued field there is only one linear topology up to equivalence, and then $\tau \otimes \varsigma$ is continuous as soon as $\tau$ and $\varsigma$ are, by the finite-dimensional theorem. The complex case $F = \mathbb{R}$, $K = \mathbb{C}$, $B = M_n(\mathbb{R})$ with the transpose gives $B \otimes_{\mathbb{R}} \mathbb{C} = M_n(\mathbb{C})$ with the conjugate transpose, whose symmetric part is $\mathrm{Sym}_n(\mathbb{R}) \oplus i\,\mathrm{Skew}_n(\mathbb{R})$, the Hermitian matrices, of real dimension $n(n+1)/2 + n(n-1)/2 = n^2$.

Remark (the boundary to the $p$-adic fields). The topological fields with a continuous involution that come after $\mathbb{R}$ and $\mathbb{C}$ are the $p$-adic fields and their finite extensions, where the order-two automorphisms of a quadratic extension give the standard examples; they are Absolute Values, Valuations and Completions and Local Fields, and the finite-dimensional theorem above covers the algebras over them as soon as the twisting involution is continuous, without any valuation being used here. The quaternion conjugation and the involutions of the composition algebras are Division Algebras.

What the Topology Adds

The comparison is between the involutive algebra of Involutive Linear Algebras and its topological form. The new datum is continuity, and its consequences are the following.

• Continuity is a hypothesis, and it is free more often than on a ring. It is automatic for the identity, for every linear involution in finite dimension over a complete valued field, for every involution that preserves the ideals of the filtration, and for the involution of a $\mathrm{C}^*$-algebra; it is not free for the semilinear involutions over a field whose fixed field fails to be complete, where the twisting involution itself may be discontinuous, as the conjugation of $\mathbb{Q}(\sqrt{2})$ with the topology from $\mathbb{R}$ shows, nor for the linear ones over a linear topology that is not generated by stable ideals, as $f(x) \mapsto f(1 - x)$ on $\mathbb{R}[x]$ with the $(x)$-adic topology shows. Abstractly there is no such question, and the criterion $\sigma(I_m) \subseteq I_n$ is the exact measure of the restriction.
• The two parts are closed. The symmetric and the skew part are closed, being equalizers, and they are closed topological modules over the closed fixed subring $R^\varsigma$ of the scalars; abstractly they are only subspaces. A dense symmetric part forces the involution to be the identity, and the symmetric part of a discontinuous involution may be dense and not closed, as $\mathbb{R}[x - x^2]$ for $\sigma_1$ shows.
• The splitting is topological and only of modules. When $2$ is invertible the averaging maps are continuous projectors, $A$ is the topological direct sum $A^+ \oplus A^-$ over $R^\varsigma$, and $A^+$ is a retract; the projection is an algebra homomorphism only for an involutive automorphism. So the topology refines the decomposition of the abstract theory into a splitting of the underlying modules, and it does not repair the failure of the symmetric part to be a subalgebra: it only makes that part closed.
• The algebra keeps its two products and gains their continuity. The skew part is a closed topological Lie algebra under the commutator, and the symmetric part, when $2$ is invertible, a closed topological Jordan algebra under the symmetrised product, over the fixed subring or the fixed field; the algebraic theory of the two is Lie Algebras and Jordan Algebras.
• Ideals, quotients and completions behave by continuity alone. The closure of a stable subalgebra is stable; the quotient by a closed stable ideal is involutive, and over a field of characteristic not two the symmetric part of the quotient is exactly the image of the symmetric part, while in characteristic two it can be strictly larger, as the swap on $\mathbb{F}_2[x,y]$ modulo $(x+y)$ shows; the involution extends uniquely to the completion and the symmetric part of the completion is the closure of the symmetric part of the algebra.
• In the $\mathrm{C}^*$-case the involution is the structure and the topology is its shadow. The involution is isometric, the norm is recovered as $\|a\| = \sqrt{r(a^*a)}$ from the -algebra, so a -algebra carries at most one $\mathrm{C}^*$-norm, and a unital *-homomorphism of $\mathrm{C}^*$-algebras is contractive: continuity is free on the involutions and on the morphisms at once. This is the sharpest form of the comparison, since the abstract theory has no topology to determine and the general topological algebra may have many.
• What is not here. The norms and the Banach and Fréchet algebras are Normed and Banach Spaces and Locally Convex and Fréchet Algebras; the non-commutative $\mathrm{C}^*$-theory, the states, the positivity, the Gelfand–Naimark theorem, the $\mathrm{C}^*$-modules, the representation theory and the von Neumann algebras are Operator Algebras; the completed tensor products are Topological Tensor Products; the forms, the adjoints and the involutions a form induces are Hilbert Algebras; the graded structures are Superalgebras and Graded Structures; and nothing here integrates, so no measure is used.

Summary

An involutive topological linear algebra is a unital associative topological $R$-algebra $A$ over a topological ring $R$, commutative with $1 \neq 0$, with a continuous $\varsigma$-semilinear involution $\sigma$, where $\varsigma$ is a continuous involution of $R$; the involution is an anti-automorphism of order two, the case $\varsigma = \mathrm{id}$ is the linear one of Involutive Linear Algebras, and over $\mathbb{C}$ with the conjugation it is the antilinear case. Beside the involution stands the involutive topological automorphism, an automorphism of order two, whose fixed set is a closed subalgebra, while the fixed set of an involution is closed but is a subalgebra only when its elements commute pairwise. A topological involution is continuous exactly when it is continuous at $0$, it is then a homeomorphism and uniformly continuous, it is determined on a dense subset, it descends to the Hausdorff quotient by the closure of $\{0\}$, and it is the certificate of a topological isomorphism of $A$ with its opposite algebra of order two.

For a linear topology with the two-sided ideals $I_n$, an involution is continuous exactly when for every $n$ there is $m$ with $\sigma(I_m) \subseteq I_n$; if $\sigma$ is continuous the family $J_n = I_n \cap \sigma(I_n)$ is a cofinal $\sigma$-stable system of ideals defining the same topology; for the $I$-adic topology of a two-sided ideal $I$ the criterion is $\sigma(I)^m \subseteq I$ for some $m$, which holds for $m = 1$ when $\sigma(I) = I$, and a linear involution with $\sigma(I) \subseteq I$ satisfies it because $\sigma$ is an involution. A continuous involution of a Hausdorff algebra with a linear topology descends to the quotients by a stable filtration and extends uniquely to a continuous involution of the same kind of the completion $\widehat{A} = \varprojlim_n A/I_n$, whose symmetric part is the closure of $\iota(A^+)$ when $2$ is invertible. The linear involutions $\sigma_b(f)(x) = f(b - x)$ of $\mathbb{R}[x]$ with the $(x)$-adic topology are continuous exactly for $b = 0$, because $(b-x)^m$ has the nonzero constant term $b^m$ for $b \neq 0$, and the fixed algebra $\mathbb{R}[x - x^2]$ of $\sigma_1$ is dense and not closed; the antilinear $\theta(f)(x) = \overline{f(-x)}$ of $\mathbb{C}[x]$ with the $(x)$-adic topology is continuous, and its extension to $\mathbb{C}[[x]]$ has the series with real even and purely imaginary odd coefficients as its symmetric part.

The symmetric part $A^+$ and the skew part $A^-$ of a topological involution of a Hausdorff algebra are closed, $A^+$ is a closed $R^\varsigma$-submodule and, when $2$ is invertible, a closed topological Jordan algebra under the symmetrised product, and $A^-$ is a closed topological Lie algebra under the commutator, and a dense $A^+$ forces $\sigma = \mathrm{id}$. When $2$ is invertible the averaging maps $\pi_\pm = \tfrac12(\mathrm{id} \pm \sigma)$ are continuous idempotent $R^\varsigma$-linear endomorphisms and $A$ is the topological direct sum $A^+ \oplus A^-$ over $R^\varsigma$, the projection being multiplicative only for an involutive automorphism. The closure of a $\sigma$-stable set is $\sigma$-stable; the quotient by a closed two-sided $\sigma$-stable ideal is a Hausdorff involutive topological algebra whose symmetric part contains the image of $A^+$, with equality when $2$ is invertible and strictness otherwise, the swap on $\mathbb{F}_2[x,y]$ modulo $(x + y)$ giving image $\mathbb{F}_2[x^2]$ strictly inside the symmetric part $\mathbb{F}_2[x]$ of the quotient; the kernel of a continuous morphism is a closed stable ideal; the product algebra carries $\sigma \times \tau$ and the exchange involution on $A \times A^{\mathrm{op}}$ is continuous with the diagonal, a subalgebra exactly when $A$ is commutative, as its symmetric part.

In the $\mathrm{C}^*$-case the involution $a \mapsto a^*$ is isometric, $\|a^*\| = \|a\|$, hence continuous, and the norm is recovered from the -algebra as $\|a\|^2 = \|a^*a\| = r(a^*a)$, so a unital -algebra carries at most one $\mathrm{C}^*$-norm and the topology is a consequence of the involutive structure; a unital -homomorphism of unital $\mathrm{C}^*$-algebras is contractive, because $\sigma(\varphi(a)) \subseteq \sigma(a)$ gives $r(\varphi(a)) \leq r(a)$ and $\|\varphi(a)\|^2 = r(\varphi(a^*a)) \leq r(a^*a) = \|a\|^2$, so a unital -isomorphism is isometric. Over a topological field with the continuous involution $\varsigma$ the fixed field $F^\varsigma$ is a closed subfield of index two and a $\varsigma$-semilinear involution is $F^\varsigma$-linear, so the whole linear theory applies over $F^\varsigma$; over a complete valued field a semilinear involution of a finite-dimensional algebra is continuous exactly when $\varsigma$ is, and the completeness cannot be dropped, the conjugation of $\mathbb{Q}(\sqrt{2})$ with the topology from $\mathbb{R}$ being a discontinuous semilinear involution of a one-dimensional algebra.

Summary of Notation

Further Reading

• Nicolas Bourbaki, General Topology, Chapters 1–4 (Springer, 1995), for the uniform spaces, the uniform continuity of continuous homomorphisms and the uniform completion.
• Nicolas Bourbaki, Topological Vector Spaces, Chapters 1–5 (Springer, 1987), for the topological vector spaces, the linear topologies, the quotients and the completions.
• Nicolas Bourbaki, Commutative Algebra, Chapters 1–7 (Springer, 1998), for the $I$-adic topologies, the filtrations by ideals and the completions of algebras.
• Charles E. Rickart, General Theory of Banach Algebras (Van Nostrand, 1960), for the involutions of Banach algebras, the continuity questions and the role of the $\mathrm{C}^*$-identity.
• Theodore W. Palmer, Banach Algebras and the General Theory of *-Algebras (Cambridge University Press, 1994), for the general theory of involutive Banach algebras and the topological algebras with involution.
• Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I (Academic Press, 1983), for the $\mathrm{C}^*$-algebras, the isometry of the involution, the uniqueness of the $\mathrm{C}^*$-norm and the contractivity of $*$-homomorphisms.
• Serge Lang, Algebra (Springer, third edition, 2002), for the semilinear maps, the descent from a quadratic extension and the automorphisms of $\mathbb{C}$.
• Nathan Jacobson, Structure of Rings (American Mathematical Society, 1956), for the symmetric and the skew elements of a ring with involution and the structures they carry.
• Seth Warner, Topological Fields (North-Holland, 1989), for the topological fields, the continuity of an order-two automorphism and the fixed field of a continuous involution.