Operators on a Topological Group

Introduction

A topological group $G$ carries two structures at once, and its operators must respect both: an operator on $G$ is a continuous map, and the two-sided operators built from the multiplication are the ones by which $G$ acts on itself. This article lays down the operator layer of the category — the monoid of continuous self-maps, the group of homeomorphisms it contains, the subgroup generated by the two families of translations, and the group of continuous automorphisms — and it fixes the conventions, the symbols and the natural pairing that the later groups of the category use.

The article assumes the topological group and its homogeneity from Topological Groups, in particular that the translations are homeomorphisms, the local base at the identity, the left and right uniformities and the completion; it assumes the abstract translations, their composition laws, their commutation and the description of the subgroup they generate from Left and Right Multiplication in a Group; and it assumes the symmetric group, the automorphism group of a structure and the correspondence between actions and permutation representations from Transformation Groups. No involution of the group, no signed operator and no adjoint is used here: they belong to the later groups of the category, and the present article supplies them with the continuous translations and the natural pairing. Nothing analytic and nothing geometric is used, and no measure occurs; the Haar integral is named once, at the boundary, and never used.

Throughout, $G$ is a topological group, Hausdorff when a closedness statement is made, with identity $e$ and inversion $\iota(g) = g^{-1}$; $L_a$, $R_b$ are the translations, $\operatorname{Ad}_g$ is the inner automorphism, $\operatorname{Homeo}(G)$ is the group of homeomorphisms of $G$ onto itself, and $\operatorname{Aut}(G)$, $\operatorname{Aut}_c(G)$, $\operatorname{Inn}(G)$ are the automorphisms, the continuous automorphisms and the inner automorphisms.

The Continuous Operators

The Operator Layer

Definition. A continuous operator on $G$ is a continuous map $G \to G$. The continuous operators form a monoid $\operatorname{Map}_c(G)$ under composition, with identity the identity map, and the invertible elements of this monoid are exactly the homeomorphisms of $G$ onto itself, which form the group $\operatorname{Homeo}(G)$.

Proof. The composite of two continuous maps is continuous, and composition is associative with identity $\mathrm{id}$; a continuous map with a continuous inverse is by definition a homeomorphism, and a homeomorphism is continuous and invertible in the monoid.

The passage from the monoid to its group of units is the first structural statement of the operator layer: an operator is invertible exactly when it is a homeomorphism, which is the topological form of the algebraic statement that a bijective operator is a permutation.

Definition. A continuous automorphism of $G$ is a group automorphism that is continuous; the continuous automorphisms form a subgroup $\operatorname{Aut}_c(G)$ of $\operatorname{Aut}(G)$ and a subgroup of $\operatorname{Homeo}(G)$. The inner automorphisms $\operatorname{Ad}_g(x) = gxg^{-1}$ form a normal subgroup $\operatorname{Inn}(G) \subseteq \operatorname{Aut}_c(G)$.

Proposition. $\operatorname{Aut}_c(G) = \operatorname{Aut}(G) \cap \operatorname{Homeo}(G)$, and $\operatorname{Inn}(G) \subseteq \operatorname{Aut}_c(G)$ with the assignment $g \mapsto \operatorname{Ad}_g$ a continuous homomorphism $G \to \operatorname{Aut}_c(G)$ when the latter carries the topology of uniform convergence on compacta.

Proof. An automorphism that is a homeomorphism has a continuous inverse, so it is a continuous automorphism; conversely a continuous automorphism is a continuous bijection and its inverse is continuous, by Topological Groups. The inner automorphism $\operatorname{Ad}_g = L_g R_{g^{-1}}$ is a homeomorphism and an automorphism, and $\operatorname{Ad}_{gh} = \operatorname{Ad}_g \operatorname{Ad}_h$; continuity in $g$ is the continuity of the evaluation map $(g,x) \mapsto gxg^{-1}$ on $G\times G$ together with the definition of the function-space topology.

The Translations

The two families of translations are the elementary operators, and in the topological setting they are elementary in a second sense: they are continuous for free.

Theorem (translations are homeomorphisms). For every $g \in G$ the left translation $L_g$ and the right translation $R_g$ are homeomorphisms, the inversion $\iota$ is a homeomorphism, and $\operatorname{Ad}_g$ is a homeomorphism and a continuous automorphism.

Proof. Multiplication is continuous, so $L_g$ is continuous, and it has the continuous inverse $L_{g^{-1}}$; likewise $R_g$ has inverse $R_{g^{-1}}$. Inversion is continuous by the axioms and is its own inverse. Finally $\operatorname{Ad}_g = L_g \circ R_{g^{-1}}$ is a composite of homeomorphisms and is an automorphism, hence a continuous automorphism.

Corollary (homogeneity). For any $g, h \in G$ the homeomorphism $L_{hg^{-1}}$ carries $g$ to $h$; hence a local property of $G$ at one point holds at every point, and the operator layer acts transitively on $G$ by homeomorphisms.

Proof. $L_{hg^{-1}}(g) = hg^{-1}g = h$. This is the homogeneity of Topological Groups, restated as a statement about the translation operators.

The Group Generated by the Translations

Theorem. Let $L(G) = \{L_g\}$ and $R(G) = \{R_g\}$. Each is a subgroup of $\operatorname{Homeo}(G)$ isomorphic to $G$, the two commute elementwise, and the subgroup $L(G)R(G)$ that they generate is the image of

$$ \pi : G\times G \longrightarrow \operatorname{Homeo}(G), \qquad \pi(a,b) = L_aR_b , $$

with kernel the anti-diagonal copy of the centre, $\ker\pi = \{(z,z^{-1}) : z \in Z(G)\}$. Hence $L(G)R(G) \cong (G\times G)/Z(G)$.

Proof. The composition laws $L_aL_b = L_{ab}$ and $R_bR_c = R_{cb}$, the commutation $L_aR_b = R_bL_a$ and the kernel computation are the abstract statements of Left and Right Multiplication in a Group; the only new ingredient is that each of these operators is a homeomorphism, by the theorem above, so the subgroup lies in $\operatorname{Homeo}(G)$ and not merely in $\operatorname{Sym}(G)$.

Corollary. $L(G) \cap R(G) = L(Z(G)) = R(Z(G))$, the subgroup of the central translations, which is the image of $Z(G)$ and is a normal subgroup of $L(G)R(G)$.

Proof. If $L_a = R_b$ then $ax = xb$ for all $x$, so $a = b$ and $a \in Z(G)$; conversely a central $z$ gives $L_z = R_z$. Normality follows because $L(G)$ and $R(G)$ commute.

The Action by Translation

Definition. The action by translation of $G$ on itself is the map

$$ G \times G \longrightarrow G, \qquad (g, x) \mapsto gx , $$

whose associated permutation representation is $g \mapsto L_g : G \to \operatorname{Homeo}(G) \subseteq \operatorname{Sym}(G)$.

Theorem. The action by translation is continuous, free and transitive: it is continuous as a map $G\times G\to G$; the only $g$ with $gx = x$ for some $x$ is $g = e$; and for any $x, y$ there is a unique $g$ with $gx = y$, namely $g = yx^{-1}$. Consequently the map $g \mapsto L_g$ is an isomorphism of $G$ onto $L(G)$ whose inverse is $L_g \mapsto g = L_g(e)$.

Proof. Continuity of the action is the continuity of multiplication in $G\times G$. Freeness: $gx = x$ gives $g = e$ on multiplying by $x^{-1}$. Transitivity: $g = yx^{-1}$ works and is unique. The map $g\mapsto L_g$ is an injective homomorphism because $L_g = L_h$ forces $g = ge = L_g(e) = L_h(e) = h$.

The action is not merely transitive but sharply transitive, and this is the topological content of the statement that the group is recovered from its left regular image: the orbit map at $e$ is a homeomorphism, so the abstract isomorphism $G \cong L(G)$ is a homeomorphism when $L(G)$ carries the compact-open topology and the two-sided action is continuous.

Corollary (the two-sided action). The map $G\times G \to \operatorname{Homeo}(G)$, $(a,b) \mapsto L_aR_b$, is a homomorphism with the kernel above, and it is the topological form of the two-sided action of The Action of a Group on Itself; the group $L(G)R(G)$ is closed in $\operatorname{Homeo}(G)$ for the compact-open topology when $G$ is locally compact.

Proof. The homomorphism statement is the theorem above. For closedness, $\operatorname{Homeo}(G)$ with the compact-open topology is a topological group when $G$ is locally compact Hausdorff, the map $(a,b)\mapsto L_aR_b$ is continuous, and it is a homeomorphism onto its image, which is therefore closed in the open-compact topology; the point-set statements are those of the function-space topology of Part II.

The Natural Pairing

The later groups of the category build adjoints, and an adjoint is taken with respect to a pairing. This article fixes the pairing once.

Definition. Let $k$ be a field and $k[G]$ the group algebra, with basis $G$ and finitely supported elements; write $C(G)$ for the $k$-algebra of continuous functions $G \to k$, carrying the topology of uniform convergence on compacta. The natural pairing of the category is

$$ \langle\cdot,\cdot\rangle : C(G) \times k[G] \longrightarrow k, \qquad \langle f, u\rangle = \sum_{g \in G} f(g)\,u_g , $$

the sum being finite because $u$ has finite support.

Proposition. The natural pairing is bilinear, it separates points of each side, and it is separately continuous for the topology of $C(G)$ and the discrete topology of $k[G]$.

Proof. Bilinearity is the bilinearity of the sum. For separation: if $\langle f, u\rangle = 0$ for all $f \in C(G)$ then evaluating at the continuous functions vanishing off each finite support gives $u = 0$; if $\langle f, u\rangle = 0$ for all $u$ then taking $u = \delta_g$, the basis element at $g$, gives $f(g) = 0$ for every $g$. Separate continuity: for fixed $f$ the functional $u \mapsto \langle f, u\rangle$ depends on finitely many coordinates, hence is continuous for the discrete topology; for fixed $u$ the functional $f \mapsto \langle f,u\rangle$ is a finite linear combination of evaluations, each continuous in the topology of uniform convergence on compacta.

Proposition (the translations are adjoint to their inverses). On $C(G)$ let $L_a$ act by $(L_af)(x) = f(a^{-1}x)$ and $R_b$ by $(R_bf)(x) = f(xb)$; on $k[G]$ let $L_a$, $R_b$ act by left and right multiplication. Then

$$ \langle L_a f, u\rangle = \langle f, L_{a^{-1}}u\rangle , \qquad \langle R_b f, u\rangle = \langle f, R_{b^{-1}}u\rangle . $$

Proof. $\langle L_af, u\rangle = \sum_x f(a^{-1}x)u_x = \sum_y f(y)u_{ay}$, on reindexing $y = a^{-1}x$, and $(a^{-1}u)_y = u_{ay}$, so the first identity is the definition of the pairing. Similarly $\langle R_bf, u\rangle = \sum_x f(xb)u_x = \sum_y f(y)u_{yb^{-1}}$, on reindexing $y = xb$, and $(ub^{-1})_y = u_{yb}$, which gives the second identity.

The two identities are the algebraic content of the statement that the inversion carries the left regular operators to their adjoints. They are what The Adjoint of the Left Multiplication on a Topological Group and The Group Inversion as an Adjoint formalise, and the twisted pairing built from a continuous involution is The Signed Adjoint of the Left Multiplication on a Topological Group.

Summary

A continuous operator on a topological group is a continuous self-map, the invertible ones are exactly the homeomorphisms, and the continuous automorphisms are the automorphisms that are homeomorphisms. The translations $L_a$, $R_b$ and the inversion are homeomorphisms for free, and the inner automorphisms $\operatorname{Ad}_g = L_gR_{g^{-1}}$ are continuous automorphisms; homogeneity is the transitivity of the translation action by homeomorphisms. Each of the two translation families is a subgroup of $\operatorname{Homeo}(G)$ isomorphic to the abstract group, the two commute, their intersection is the central translations, and the group they generate is the image of $(a,b)\mapsto L_aR_b$ with kernel the anti-diagonal copy of the centre, so that $L(G)R(G) \cong (G\times G)/Z(G)$. The action by translation is continuous, free and sharply transitive, so $g \mapsto L_g$ is an isomorphism of $G$ onto $L(G)$ and, at the level of the function-space topology, a homeomorphism onto it. Finally the category carries a natural pairing between the continuous functions and the group algebra, and with respect to it the left and right translations are adjoint to their inverses, which is the form the inversion takes on the operator layer.

Summary of Notation

Symbol	Meaning
$G$, $e$, $\iota$	a topological group, its identity, and the inversion $\iota(g) = g^{-1}$
$\operatorname{Map}_c(G)$	the monoid of continuous self-maps of $G$
$\operatorname{Homeo}(G)$	the group of homeomorphisms of $G$, the units of $\operatorname{Map}_c(G)$
$\operatorname{Aut}(G)$, $\operatorname{Aut}_c(G)$	all automorphisms, and the continuous ones
$\operatorname{Inn}(G)$, $\operatorname{Ad}_g$	the inner automorphisms and $\operatorname{Ad}_g(x) = gxg^{-1}$
$L_a$, $R_b$	left translation $x \mapsto ax$ and right translation $x \mapsto xb$, both homeomorphisms
$L(G)$, $R(G)$	the translation subgroups, each $\cong G$
$\pi(a,b) = L_aR_b$	the parametrisation of the generated subgroup
$\ker\pi = \{(z,z^{-1}) : z \in Z(G)\}$	the anti-diagonal copy of the centre
$Z(G)$	the centre, whose translations are the intersection $L(G)\cap R(G)$
$k[G]$, $C(G)$	the group algebra of finitely supported functions, and the continuous functions
$\langle f, u\rangle = \sum_g f(g)u_g$	the natural pairing of the category
$\langle L_af,u\rangle = \langle f, L_{a^{-1}}u\rangle$	the translations are adjoint to their inverses