Reflections as Signed Two-Sided Operators on the Group Algebra

Introduction

A reflection of the group algebra is a signed two-sided operator of the form $f\mapsto u\,\alpha(f)\,u^{-1}$ that is its own inverse: it is an involution produced by the convolution product, the grade involution and the inverse of a unit. This article reads the reflections of a graded group algebra as operators, identifies the units that carry them, computes the closed subalgebra they fix, and records the two ways the correspondence between reflections and carriers can degenerate — the grade involution inner, or the group algebra commutative. It is the two-sided operator theory of the signed conjugations of which the signed sandwich is the general form.

The article assumes the group, its Haar measure and the modular function from Locally Compact Groups and Haar Measure; the algebra $L^1(G)$ and its norm from The Convolution Algebra $L^1(G)$; the convolution operators and the group algebra as an algebra of operators from Convolution on a Group and The Group Algebra as an Algebra of Operators; the grade involution, the signed and unsigned sandwiches, their composition laws, the coset, the inverse and the signed conjugation from The Signed Sandwich on the Group Algebra, the immediately preceding article; the general reflections of an algebra and their degenerate cases from Reflections as Signed Two-Sided Operators on an Algebra and Reflections as Signed Two-Sided Operators on a Banach Algebra (Topology on Linear Algebras); the topological-group reflections, fixed subgroups and carriers from Reflections as Signed Two-Sided Operators on a Topological Group and Involutive Topological Groups; and the bounded operators, the operator norm and the closed subalgebras from Operator Algebras and Topological Algebras and Banach Algebras. The adjoint of a reflection is The Signed Adjoint of the Reflection on the Group Algebra, in the - * Operator Theory group of this category; the measure algebra is The Involution on the Measure Algebra, later. A reflection in the geometric sense — in a hyperplane, preserving a form — needs a form and belongs to the later categories; a reflection here is only a bounded operator of order two on the group algebra, and no adjoint is used.

Throughout, $G$ is a locally compact Hausdorff group with identity $e$ and left Haar measure $dx$; $\mathcal{A} = L^1(G)$ is the group algebra with convolution $f*g$, norm $\|f\|_1$, centre $Z(\mathcal{A})$ and unit group $\mathcal{A}^\times$; $\alpha$ is a continuous involutive automorphism, $\mathrm{A}f = \alpha(f)$, and $c_t(f) = t*f*t^{-1}$ is the inner automorphism by a unit $t$; the reflector set and the reflection family are

$$ R^\times(\mathcal{A},\alpha) = \{u \in \mathcal{A}^\times : u*\alpha(u) \in Z(\mathcal{A})\}, \qquad \rho_u = c_u\alpha = S_{u,u^{-1}} , \quad \rho_u(f) = u*\alpha(f)*u^{-1} , $$

and $\mathcal{A}^\pm_\rho$ are the $\pm1$-eigenspaces of a reflection $\rho$ when $2$ is invertible.

The Reflector and the Reflection

Definition. A reflector of $\mathcal{A}$ with respect to $\alpha$ is a unit $u$ with $u*\alpha(u) \in Z(\mathcal{A})$, and the reflection determined by a reflector $u$ is $\rho_u = c_u\alpha$. The set of reflections is written $\mathrm{Ref}(\mathcal{A},\alpha)$.

Theorem (the reflection is an involutive automorphism). For every unit $u$ the signed conjugation $\rho_u = c_u\alpha$ is a bounded algebra automorphism of $\mathcal{A}$ with $\rho_u = c_u\alpha = \alpha c_{\alpha(u)}$ and

$$ \rho_u^2 = c_{u*\alpha(u)} , $$

so $\rho_u$ is an involutive automorphism, $\rho_u^2 = \mathrm{id}$, exactly when $u$ is a reflector; conversely every signed conjugation that is an involution has a reflector parameter.

Proof. The signed conjugation is the composite of the automorphisms $c_u$ and $\alpha$, hence an automorphism, and it is bounded because $c_u$, $\alpha$ and $c_{u^{-1}}$ are. The square is $\rho_u^2(f) = u*\alpha(u*\alpha(f)*u^{-1})*u^{-1} = (u*\alpha(u))*f*(u*\alpha(u))^{-1} = c_{u*\alpha(u)}(f)$, using that $\alpha$ is an automorphism and $\alpha^2 = \mathrm{id}$. An inner automorphism is the identity exactly when its element is central, which gives the involution criterion; reading the computation backwards gives the converse. $\square$

Corollary (the identity and the grade involution). When $\mathcal{A}$ has a unit, $1$ is a reflector with $\rho_1 = \alpha$, so the grade involution is always a reflection of the family; and $\rho_u = \mathrm{id}$ exactly when $u$ is a central unit and $\alpha = c_{u^{-1}}$, which for $\alpha = \mathrm{id}$ means exactly $u \in Z(\mathcal{A})^\times$.

Proof. $1*\alpha(1) = 1$ is central and $\rho_1 = c_1\alpha = \alpha$; $\rho_u = \mathrm{id}$ is $c_u\alpha = \mathrm{id}$, that is $\alpha = c_{u^{-1}}$. $\square$

Proposition (the reflection is an operator of the sandwich family). The reflection is the signed sandwich whose second factor is the inverse of the first, $\rho_u = S_{u,u^{-1}}$; conversely the signed sandwich is recovered from the reflections and the one-sided operators by

$$ S_{a,b} = \rho_a\,R_{\alpha(b)*\alpha(a)} \qquad (a \in \mathcal{A}^\times), $$

so the reflections together with the right convolutions generate the whole signed family.

Proof. $S_{u,u^{-1}}(f) = u*\alpha(f)*u^{-1} = \rho_u(f)$. For the reconstruction, $\rho_aR_{\alpha(b)*\alpha(a)}(f) = \rho_a(f*\alpha(b)*\alpha(a)) = a*\alpha(f)*\alpha(\alpha(b)*\alpha(a))*a^{-1} = a*\alpha(f)*(b*a)*a^{-1} = a*\alpha(f)*b = S_{a,b}(f)$, using $\alpha^2 = \mathrm{id}$ and the associativity of convolution. $\square$

The Eigenvalue Decomposition

Proposition (fixed subalgebra and negated part). Let $u$ be a reflector and $\rho = \rho_u$. When $2$ is invertible,

$$ \mathcal{A} = \mathcal{A}^+_\rho\oplus\mathcal{A}^-_\rho, \qquad \mathcal{A}^+_\rho = \{f : \rho(f) = f\}, \qquad \mathcal{A}^-_\rho = \{f : \rho(f) = -f\}, $$

the fixed set $\mathcal{A}^+_\rho$ is a closed subalgebra, the negated part $\mathcal{A}^-_\rho$ is a closed subspace, and the multiplication table of the grading holds:

$$ \mathcal{A}^+_\rho\mathcal{A}^+_\rho\subseteq\mathcal{A}^+_\rho, \quad \mathcal{A}^+_\rho\mathcal{A}^-_\rho\subseteq\mathcal{A}^-_\rho, \quad \mathcal{A}^-_\rho\mathcal{A}^+_\rho\subseteq\mathcal{A}^-_\rho, \quad \mathcal{A}^-_\rho\mathcal{A}^-_\rho\subseteq\mathcal{A}^+_\rho . $$

Proof. $\rho$ is a continuous involutive automorphism, so its $\pm1$-eigenspaces decompose $\mathcal{A}$ by the averaging $f = \tfrac12(f+\rho(f)) + \tfrac12(f-\rho(f))$; the multiplication table is the multiplicativity of $\rho$, $\rho(fg) = \rho(f)\rho(g)$. The fixed set is the kernel of the continuous map $\rho - \mathrm{id}$ and the negated part the kernel of $\rho + \mathrm{id}$, hence both closed. $\square$

Corollary (determination by the fixed subalgebra). A reflection is determined by its fixed subalgebra and by its negated part; two reflections with the same fixed subalgebra are equal.

Proof. An automorphism of order two is determined by its action on the $\pm1$-eigenspaces, and the two eigenspaces determine each other by the direct sum. $\square$

Remark (the topology adds closedness). The algebraic content of the decomposition is that of any involutive automorphism; the topology adds that the two parts are closed, because $\rho$ is continuous, and that the averaging maps are continuous. No norm, no form and no measure enters the algebra of the reflection beyond that.

The Correspondence

Theorem (the kernel of the parametrisation). The assignment $u\mapsto\rho_u$ is a surjection of the reflectors onto the reflections, and

$$ \rho_u = \rho_v \quad\Longleftrightarrow\quad v^{-1}*u \in Z(\mathcal{A})^\times , $$

so the reflections are parametrised by the reflectors modulo the central units, $\mathrm{Ref}(\mathcal{A},\alpha)\cong R^\times(\mathcal{A},\alpha)/Z(\mathcal{A})^\times$.

Proof. If $\rho_u = \rho_v$ then $u*\alpha(f)*u^{-1} = v*\alpha(f)*v^{-1}$ for all $f$, so $(v^{-1}*u)*\alpha(f) = \alpha(f)*(v^{-1}*u)$ for all $f$; since $\alpha$ is onto, $v^{-1}*u$ commutes with every element and lies in $Z(\mathcal{A})^\times$. Conversely a central unit is absorbed by the inner conjugation, and every reflector is in the image by definition. $\square$

Corollary (the reflector as the correcting element). For a reflector $u$, $\rho_u(u) = \alpha(u)$, and $\rho_u$ is the composite of the inner automorphism by $u$ with the twist; the reflector is the element whose inner automorphism corrects $\alpha$ to the desired reflection.

Proof. $u*\alpha(u)$ is central, so $u$ commutes with it and $\rho_u(u) = u*\alpha(u)*u^{-1} = \alpha(u)$. $\square$

Proposition (the coset of the inner automorphisms). The reflections relative to $\alpha$ are exactly the involutive automorphisms lying in the coset $\mathrm{Inn}(\mathcal{A})\alpha$ of the inner automorphism group, and the map $[u]\mapsto[c_u]$ sends the reflectors to the classes of the reflections in $\mathrm{Out}(\mathcal{A}) = \mathrm{Aut}(\mathcal{A})/\mathrm{Inn}(\mathcal{A})$. If every automorphism of $\mathcal{A}$ is inner then every involutive automorphism is a reflection for a suitable $\alpha$.

Proof. The reflection is $c_u\alpha$ by the corollary, so it lies in the coset; conversely an element of the coset is $c_u\alpha$, and it is an involution exactly when $u$ is a reflector. The map $[u]\mapsto[c_u]$ is the standard isomorphism $\mathcal{A}^\times/Z(\mathcal{A})^\times\to\mathrm{Inn}(\mathcal{A})$. $\square$

The Degenerate Cases

Theorem (the inner grade involution). Suppose $\alpha = c_z$ for a unit $z$. Then

$$ \rho_u = c_{u*z}, \qquad \mathrm{Ref}(\mathcal{A},\alpha) = \{\text{involutive inner automorphisms}\}, $$

so the reflections are exactly the inner automorphisms of $\mathcal{A}$ of order two, with reflector condition $u*z*u*z^{-1} \in Z(\mathcal{A})$.

Proof. $\rho_u = c_u\alpha = c_uc_z = c_{u*z}$, an inner automorphism; it is an involution exactly when $(u*z)*f*(u*z)^{-1} = f$ for all $f$, that is when $u*z\in Z(\mathcal{A})$. $\square$

Theorem (the commutative case). If $G$ is abelian then every unit is a reflector and $\rho_u = \alpha$ for every unit $u$; the reflection family collapses to the two-element group $\{\mathrm{id},\alpha\}$ and the correspondence $u\mapsto\rho_u$ is constant on the units.

Proof. In a commutative algebra every element is central, so every unit is a reflector and $\rho_u = c_u\alpha = \alpha$; the reflections are therefore only $\alpha$, together with $\mathrm{id}$ when $\alpha = \mathrm{id}$. $\square$

Remark (the failure of the correspondence). The map $u\mapsto\rho_u$ is neither injective nor, when $\alpha$ is not inner, surjective onto all involutive automorphisms: only the signed conjugations are reflections, and the involutive automorphisms outside the coset $\mathrm{Inn}(\mathcal{A})\alpha$ are not. The parametrisation is also not a group homomorphism in general, because the product of two reflections is an unsigned sandwich rather than a reflection; each reflection is individually of order two, and the reflections are a coset of the inner automorphisms, not a subgroup.

The Group Instance

Example (the discrete group). Let $G$ be discrete and $\alpha = \alpha_\theta$ induced by an involutive automorphism of $G$. The reflectors among the point masses are the elements $h$ with $h\theta(h) \in Z(G)$, and

$$ \rho_{\delta_h}(\delta_g) = \delta_{h\theta(g)h^{-1}} , $$

so that the restriction of $\rho_{\delta_h}$ to the point masses is exactly the signed conjugation $\rho_h$ of Reflections as Signed Two-Sided Operators on a Topological Group; the fixed subalgebra $\mathcal{A}^+_\rho$ is the linear span of the fixed subgroup $\{g : \alpha(g) = h^{-1}gh\}$ together with the topological closedness inherited from $\ell^1(G)$.

Example (the sign character and the two-element group). Let $\alpha = \alpha_\varepsilon$ for a sign character and let $G$ be abelian. Then $\varepsilon$ is the only nontrivial reflection, the family is $\{\mathrm{id},\alpha_\varepsilon\}$, and the reflector condition is $h^2 \in Z(G) = G$, automatically satisfied by every unit: the correspondence is constant, in accordance with the commutative degeneracy. On a finite abelian group the two reflections are the identity and the sign, which is the group-algebra reading of the Fourier sign symmetry.

Remark (the group-level dictionary). The reflector condition $u*\alpha(u)\in Z(\mathcal{A})$, the square $\rho_u^2 = c_{u*\alpha(u)}$, the parametrisation by $Z(\mathcal{A})^\times$ and the two degenerate cases are the group-algebra form of the dictionary of Involutive Topological Groups and Reflections as Signed Two-Sided Operators on a Topological Group; on the point masses of a discrete group the latter theory is recovered exactly, and on a non-discrete group only the integrable reflections survive.

Summary

A reflection of the group algebra $\mathcal{A} = L^1(G)$ with respect to the invertible twist $\alpha$ is the signed conjugation $\rho_u(f) = u*\alpha(f)*u^{-1} = S_{u,u^{-1}}$, a bounded algebra automorphism with $\rho_u = c_u\alpha = \alpha c_{\alpha(u)}$ and square $c_{u*\alpha(u)}$; it is an involutive automorphism exactly when $u$ is a reflector, $u*\alpha(u)\in Z(\mathcal{A})$. An involutive reflection splits $\mathcal{A} = \mathcal{A}^+_\rho\oplus\mathcal{A}^-_\rho$ into a closed fixed subalgebra and a closed negated part with the multiplication table of a grading, and it is determined by its fixed subalgebra. The map $u\mapsto\rho_u$ is a surjection of the reflectors onto the reflections with kernel the central units, so the reflections are parametrised by $R^\times(\mathcal{A},\alpha)/Z(\mathcal{A})^\times$, and they are exactly the involutive automorphisms in the coset $\mathrm{Inn}(\mathcal{A})\alpha$, mapping to $\mathrm{Out}(\mathcal{A})$ by the class of the inner part. When $\alpha$ is inner every reflection is an inner automorphism, $c_{u*z}$; when $G$ is abelian every unit is a reflector, every reflection equals $\alpha$, and the family collapses to $\{\mathrm{id},\alpha\}$. The correspondence is neither injective nor, for a non-inner $\alpha$, surjective onto all involutive automorphisms, and the reflections form a coset of the inner automorphisms rather than a subgroup. On a discrete group the point masses reproduce the signed conjugations of the topological-group theory; the adjoint and the measure algebra are the - * Operator Theory group and The Involution on the Measure Algebra.

Summary of Notation

Symbol	Meaning
$\mathcal{A} = L^1(G)$, $Z(\mathcal{A})$, $\mathcal{A}^\times$	Group algebra, centre, unit group
$\alpha$, $\mathrm{A}f = \alpha(f)$, $c_t(f) = tft^{-1}$	Grade involution; inner automorphism
$\rho_u = c_u\alpha = S_{u,u^{-1}}$	Reflection by the unit $u$, $\rho_u(f) = u\alpha(f)u^{-1}$
$R^\times(\mathcal{A},\alpha)$	Reflectors: units with $u*\alpha(u)\in Z(\mathcal{A})$
$\mathrm{Ref}(\mathcal{A},\alpha)\cong R^\times(\mathcal{A},\alpha)/Z(\mathcal{A})^\times$	Reflections and their parametrisation
$\rho_u^2 = c_{u*\alpha(u)}$	Square; involution iff $u$ is a reflector
$\mathcal{A}^+_\rho$, $\mathcal{A}^-_\rho$	Fixed subalgebra and negated part of $\rho$
$\rho_1 = \alpha$	The grade involution is a reflection
$\mathrm{Inn}(\mathcal{A})\alpha$, $\mathrm{Out}(\mathcal{A})$	The coset of the reflections; the outer quotient
$\rho_{\delta_h}(\delta_g) = \delta_{h\theta(g)h^{-1}}$	Group instance on a discrete group