The Signed Adjoint of the Reflection on a Topological Vector Space

Introduction

A reflection of a topological vector space is a continuous linear involution $r$ whose fixed hyperplane is closed, and read as a signed two-sided operator it carries the grade involution $\alpha_{r}(x) = rxr^{-1}$ that it defines. Its signed inner sandwich acts trivially, $\Theta^{\alpha_{r}}_{r,r^{-1}} = \mathrm{id}$, so the reflection is self-adjoint and unitary for its own signed structure; for an arbitrary grade involution $\alpha$ the adjoint of the signed inner sandwich of $r$ is $\Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)}$, the sandwich by the image of $r$, and the sandwich is unitary always while being self-adjoint exactly when $\alpha(r) = r$, the reflection being fixed by the grade involution. The degenerate cases, in which $r = \mathrm{id}$ or the grade involution moves $r$, are recorded and their collapse described.

This article develops the adjoint of a reflection read as a signed operator. The reflection and its fixed hyperplane are Reflections as Signed Two-Sided Operators on a Topological Vector Space; the grade involution attached to an involution is The Signed Sandwich on a Topological Vector Space; the adjoint computation of the signed family is The Signed Adjoint Sandwich on a Topological Vector Space, of which this is the case of the inner sandwich; the trace form is The Adjoint of the Left Multiplication on a Topological Vector Space. The pairing-based self-adjointness of a reflection — the orthogonal reflection — belongs to Part III. The forms are Part III.

Throughout, $E$ is a Hausdorff locally convex algebra over $\mathbb{K}$ with a unit and jointly continuous multiplication, $\tau$ is a continuous trace and $\langle x, y\rangle = \tau(xy)$ the trace form of the category, assumed non-degenerate, ${}^{\dagger}$ is the trace-adjoint, $r \in E$ is a reflection, that is an involution $r^{2} = 1$ with $r \neq 1$ and with the fixed hyperplane of the action closed, $\alpha_{r}(x) = rxr^{-1}$ is the grade involution it defines, $\Theta^{\alpha}_{a,b}(x) = a\alpha(x)b$ is the signed sandwich, and $\alpha$ is a trace-preserving grade involution.

The Inner Sandwich of a Reflection

Theorem (the reflection is fixed by its own signed inner sandwich). For a reflection $r$ whose inner sandwich is $\Theta^{\alpha_{r}}_{r,r^{-1}}$ one has

$$ \Theta^{\alpha_{r}}_{r,r^{-1}} = \mathrm{id} , $$

so the reflection is fixed by its own signed sandwich; its trace-adjoint is itself,

$$ (\Theta^{\alpha_{r}}_{r,r^{-1}})^{\dagger} = \Theta^{\alpha_{r}}_{r^{-1},r} = \mathrm{id} , $$

and the reflection is both self-adjoint and unitary for its own signed structure.

Proof. $\Theta^{\alpha_{r}}_{r,r^{-1}}(x) = r\alpha_{r}(x)r^{-1} = r(rxr^{-1})r^{-1} = r^{2}xr^{-2} = x$, using $r^{2} = 1$ and $r^{-1} = r$. The adjoint is $\Theta^{\alpha_{r}}_{\alpha_{r}(r^{-1}),\alpha_{r}(r)} = \Theta^{\alpha_{r}}_{r^{-1},r}$ by The Signed Adjoint Sandwich on a Topological Vector Space, and $\alpha_{r}(r) = rr r^{-1} = r$, so $\Theta^{\alpha_{r}}_{r^{-1},r} = \Theta^{\alpha_{r}}_{r,r^{-1}} = \mathrm{id}$; unitarity is the previous article with $u = \alpha_{r}(r^{-1})\alpha_{r}(r) = r^{-1}r = 1$.

The Adjoint for an Arbitrary Grade Involution

Theorem (the adjoint of the signed inner sandwich). For an arbitrary trace-preserving grade involution $\alpha$ the signed inner sandwich of the reflection has the trace-adjoint

$$ (\Theta^{\alpha}_{r,r^{-1}})^{\dagger} = \Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)} , $$

the signed sandwich by the image of $r$ under the grade involution; the sandwich is therefore self-adjoint exactly when $\Theta^{\alpha}_{r,r^{-1}} = \Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)}$, that is exactly when

$$ \alpha(r) = r , $$

and it is unitary for every $\alpha$.

Proof. The adjoint formula is the general computation with $a = r$, $b = r^{-1}$, using $\alpha(r^{-1}) = \alpha(r)^{-1}$. The two sandwiches have invertible parameters $r$ and $\alpha(r)$, so they are equal exactly when the parameters are equal, $r = \alpha(r)^{-1}$ and $r^{-1} = \alpha(r)$, which for $r^{-1} = r$ is the single condition $\alpha(r) = r$. Unitarity follows from the previous article, $u = \alpha(r^{-1})\alpha(r) = \alpha(r)^{-1}\alpha(r) = 1$ and $w = rr^{-1} = 1$, both central involutions.

Corollary (self-adjointness and the grade involution). The signed inner sandwich of a reflection is self-adjoint precisely when the reflection is fixed by the grade involution, $\alpha(r) = r$; in particular it is self-adjoint for $\alpha = \mathrm{id}$ and for $\alpha = \alpha_{r}$, and it fails to be self-adjoint for any grade involution that moves $r$. The sandwich is unitary in all these cases, so unitarity is the stable property and self-adjointness the exceptional one.

Proof. The theorem gives the criterion; the two cases $\alpha = \mathrm{id}$ and $\alpha = \alpha_{r}$ satisfy it, and a grade involution with $\alpha(r) \neq r$ violates it.

Degenerate Cases

Proposition (the trivial reflection). If $r = 1$ then the inner sandwich is the identity, $\Theta^{\alpha}_{1,1} = \alpha$, and its adjoint is $\Theta^{\alpha}_{1,1} = \alpha$; the reflection is the identity operator, the grade involution coincides with the identity of the algebra, and the whole structure collapses to the algebra with its trace form.

Proof. $\Theta^{\alpha}_{1,1}(x) = \alpha(x)$, and the adjoint of $\alpha$ is $\alpha$ by the self-adjointness of the trace-preserving grade involution; every statement above specialises to this case.

Proposition (the reflection moved by the grade involution). If $\alpha(r) \neq r$ then the signed inner sandwich is unitary but not self-adjoint; its adjoint is the sandwich $\Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)}$, which differs from $\Theta^{\alpha}_{r,r^{-1}}$, and the defect is given by $\alpha(r)r^{-1}$, the element by which the grade involution moves the reflection. When $\alpha(r) = r^{-1} = r$ the defect is trivial and the reflection is fixed.

Proof. The adjoint is the theorem, and the defect $\alpha(r)r^{-1}$ is trivial exactly when $\alpha(r) = r$; the statement is the contrapositive of the self-adjointness criterion.

Proposition (the failure in characteristic two). In characteristic two the involution $r$ has $r + 1$ nilpotent, the fixed hyperplane is the kernel of $x \mapsto (r+1)x$, and the reflection is unipotent; the signed inner sandwich is still the identity, but the decomposition of the space into the fixed and the negated parts fails, and the reflection is not diagonalisable, so the two-sided reading is the only available one.

Proof. In characteristic two $r - 1 = r + 1$ and $(r+1)^{2} = r^{2} + 1 = 0$, so $r + 1$ is nilpotent; the fixed set is the kernel, which is not a complement of the image, and the grade involution $\alpha_{r}$ is still an involution but the sum decomposition $E = E^{+}\oplus E^{-}$ is unavailable.

Examples

Example (the reflection of a linear space). For $E = \mathrm{End}_F(V)$ with $\operatorname{tr}$ and $r$ an involution of type $(n-1,1)$ the grade involution is $\alpha_{r}(X) = rXr$ and the inner sandwich is the identity; the reflection is self-adjoint and unitary for the trace form, in agreement with the linear-space computation.

Example (the reflection of a Hermitian space). For a reflection that is self-adjoint for a form, the grade involution $\alpha_{r}$ is the conjugation by $r$ and the inner sandwich is the identity, so the reflection is self-adjoint for the trace form; the pairing-based self-adjointness with respect to the form is a different operator, and the two coincide only when the form is the trace form, which is Part III.

Example (a grade involution moving the reflection). On a graded algebra with grade involution $\alpha$ and a reflection $r$ of odd part, $\alpha(r) = -r \neq r$, so the signed inner sandwich is unitary but not self-adjoint, and its adjoint is the sandwich by $-r$; the defect is $\alpha(r)r^{-1} = -r\cdot r = -1$, the central sign of the odd part.

Summary

A reflection $r$ read as a signed two-sided operator defines the grade involution $\alpha_{r}(x) = rxr^{-1}$ and has inner sandwich $\Theta^{\alpha_{r}}_{r,r^{-1}} = \mathrm{id}$, so it is self-adjoint and unitary for its own signed structure. For an arbitrary trace-preserving grade involution $\alpha$ the adjoint of the signed inner sandwich is $\Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)}$, the sandwich by the image of $r$; the sandwich is unitary for every $\alpha$, because both parameters $u = \alpha(r)^{-1}\alpha(r) = 1$ and $w = rr^{-1} = 1$ are central, and it is self-adjoint exactly when $\alpha(r) = r$, the reflection being fixed by the grade involution. The degenerate cases are the trivial reflection $r = 1$, where the sandwich is the grade involution itself, and the reflection moved by $\alpha$, where the sandwich is unitary but not self-adjoint with defect $\alpha(r)r^{-1}$; in characteristic two the involution is unipotent, the fixed and negated splitting fails, and the two-sided reading is the only one available. The module-level variant of the group is The Graded Adjoint Action on a Module over a Topological Vector Space.

Summary of Notation

Symbol	Meaning
$r$, $r^{2} = 1$	reflection, a continuous involution
$\alpha_{r}(x) = rxr^{-1}$	the grade involution of $r$
$\Theta^{\alpha}_{a,b}(x) = a\alpha(x)b$	signed sandwich
$\Theta^{\alpha_{r}}_{r,r^{-1}} = \mathrm{id}$	a reflection is fixed by its own signed sandwich
$(\Theta^{\alpha}_{r,r^{-1}})^{\dagger} = \Theta^{\alpha}_{\alpha(r)^{-1},\alpha(r)}$	its adjoint for an arbitrary $\alpha$
$\alpha(r) = r$	the self-adjointness criterion
$u = 1$, $w = 1$	the unitarity, always
$\langle x,y\rangle = \tau(xy)$	trace form