The Signed Adjoint of the Reflection on a Bimodule over an Algebra

Introduction

A reflection of a graded bimodule is the signed sandwich $r_u(x)=u\,\alpha(x)\,u^{-1}$ by a unit $u$, an operator of order two when $u\alpha(u)$ is central. This article computes the adjoint of a reflection with respect to an $\alpha$-invariant pairing and asks when a reflection is self-adjoint. The adjoint of $r_u$ is the reflection $r_{\beta(u)}$ by the parameter transformed by the composite $\beta=\alpha\sigma$, so the reflections are permuted by the adjoint; a reflection is self-adjoint exactly when its parameter is fixed by $\beta$, and this criterion fails in the degenerate case where distinct parameters give the same reflection.

The article is the fifth of the * Operator Theory group of this category. It assumes the reflections of Reflections as Signed Two-Sided Operators on a Bimodule over an Algebra, the signed sandwich and its adjoint from The Signed Sandwich on a Bimodule over an Algebra and The Signed Adjoint of the Sandwich on a Bimodule over an Algebra, and the self-adjoint and unitary elements of Module Operators with an Involution. It is the operator-level counterpart, for the graded bimodule, of The Adjoint of the Sandwich on a Bimodule over an Algebra, and the one-sided partner is The Signed Adjoint of the Left Multiplication on a Module over an Algebra. The article stays inside Part I: no distance, norm, angle, form with a norm, topology or limit; a reflection means an operator of order two, with no geometric reading. Throughout, $R$ is a commutative ring with $1 \neq 0$ in which $2$ is invertible, $(A,\sigma)$ is an involutive $R$-algebra, $\alpha$ is a grade involution of $A$ commuting with $\sigma$, $\beta=\alpha\sigma$, ${}_A M_A$ is a graded $(A,A)$-bimodule with $\alpha$-invariant balanced $\sigma$-sesquilinear pairing, and for a unit $u \in A^{\times}$ the reflection is $r_u=S^{\alpha}_{u,u^{-1}}$, $r_u(x)=u\,\alpha(x)\,u^{-1}$.

The Reflection and Its Adjoint

The reflection recalled

Definition. For a unit $u \in A^{\times}$ the reflection with parameter $u$ is the signed sandwich

$$ r_u=S^{\alpha}_{u,u^{-1}}, \qquad r_u(x)=u\,\alpha(x)\,u^{-1} . $$

Proposition. The square of a reflection is the inner conjugation by $u\alpha(u)$,

$$ r_u^{2}=S_{u\alpha(u),\,(u\alpha(u))^{-1}}=\operatorname{conj}_{u\alpha(u)}, $$

so $r_u$ is an operator of order two, a genuine reflection, exactly when $u\alpha(u)$ is central in $A$.

Proof. This is Reflections as Signed Two-Sided Operators on a Bimodule over an Algebra. $\square$

The two conditions in play are therefore distinct from the outset: $r_u^{2}=\mathrm{id}$ asks $u\alpha(u)$ central, while self-adjointness below asks $\beta(u)=u$ up to centrality. Neither implies the other.

The adjoint of a reflection

Theorem. The adjoint of a reflection is the reflection by the parameter transformed by $\beta=\alpha\sigma$:

$$ (r_u)^{*}=r_{\beta(u)}, \qquad \beta(u)=\alpha(\sigma(u)) . $$

Proof. The adjoint of a signed sandwich is $\bigl(S^{\alpha}_{a,b}\bigr)^{*}=S^{\alpha}_{\beta(a),\beta(b)}$ by The Signed Adjoint of the Sandwich on a Bimodule over an Algebra; for $(a,b)=(u,u^{-1})$ this is $S^{\alpha}_{\beta(u),\beta(u^{-1})}$, and $\beta(u^{-1})=\beta(u)^{-1}$, so the adjoint is $r_{\beta(u)}$. $\square$

Corollary. The adjoint permutes the set of reflections: $\{r_u\}^{*}=\{r_u\}$, and the permutation is the map $u \mapsto \beta(u)$ on the parameters. Since $\beta$ is an involution, the adjoint acts on the reflections as an involution on the parameters, and $(r_u)^{**}=r_{\beta^{2}(u)}=r_u$.

Proof. The adjoint of $r_u$ is $r_{\beta(u)}$, and $\beta^{2}=\mathrm{id}$ because $\alpha$ and $\sigma$ commute, so the two applications return $r_u$; the bijection $u\mapsto\beta(u)$ of $A^{\times}$ has inverse itself. $\square$

The reflection adjoint formula is the module-level form of the statement that conjugating by $u$ and taking adjoints uses the involution on the parameter; it is the same identity as $(L_a)^{*}=L_{\sigma(a)}$ for the one-sided case, with the grading inserted.

Self-Adjointness of a Reflection

The criterion

Definition. A reflection $r_u$ is self-adjoint when $(r_u)^{*}=r_u$. Its parameter $u$ is $\beta$-symmetric when $\beta(u)=u$; more generally $u$ is $\beta$-symmetric modulo the centre when $\beta(u)u^{-1} \in Z(A)$.

Theorem. If the map $u \mapsto r_u$ is injective on $A^{\times}$, then a reflection is self-adjoint exactly when its parameter is $\beta$-symmetric:

$$ (r_u)^{*}=r_u \iff \beta(u)=u \iff \alpha\sigma(u)=u . $$

Proof. By the adjoint formula, $(r_u)^{*}=r_u$ is $r_{\beta(u)}=r_u$, which by injectivity is $\beta(u)=u$. $\square$

Corollary. The $\beta$-symmetric units form a subgroup $A^{\times}_{\beta}=\{u \in A^{\times} : \beta(u)=u\}$ of the unit group, because $\beta$ is an involution of the algebra, and the reflection map restricts to a map $A^{\times}_{\beta}\to\{r_u : (r_u)^{*}=r_u\}$ onto the self-adjoint reflections when the parameter map is injective. The restriction is multiplicative only when $u\alpha(v)$ is central for all $u,v \in A^{\times}_{\beta}$, as it is in a commutative algebra; in general the product of two reflections is a conjugation and not a reflection.

Proof. $A^{\times}_{\beta}$ is a subgroup because $\beta$ is an algebra automorphism with $\beta^{2}=\mathrm{id}$; the image lies in the self-adjoint reflections by the theorem; the product $r_ur_v=S_{u\alpha(v),\,\alpha(v)^{-1}u^{-1}}$ is a reflection exactly when $u\alpha(v)$ is central, by the criterion for a diagonal sandwich to be a reflection. $\square$

The corollary stops short of a group isomorphism, because $r_ur_v$ is a reflection only under the centrality condition; the self-adjoint reflections therefore do not automatically form a group.

Self-adjointness and the order of the reflection

Proposition. The two conditions

$$ \text{order two: } r_u^{2}=\mathrm{id} \iff u\alpha(u) \in Z(A), \qquad \text{self-adjoint: } (r_u)^{*}=r_u \iff \beta(u)=u, $$

are independent: either can hold without the other.

Proof. Order two is $u\alpha(u)\in Z(A)$ and self-adjointness is $\alpha\sigma(u)=u$; these involve $u\alpha(u)$ and $\alpha\sigma(u)$ respectively, and neither condition forces the other, as the examples below show. $\square$

Example. For $\sigma=\mathrm{id}$ and $\alpha\neq\mathrm{id}$, self-adjointness is $\alpha(u)=u$ while order two is $u\alpha(u)\in Z(A)$; an element of even degree that is central is self-adjoint of order larger than two under $r$, and an odd element of order two in the grading satisfies order two without being self-adjoint.

In particular a self-adjoint reflection need not be a genuine reflection, and a genuine reflection need not be self-adjoint; the two properties are the "graded" analogue of the fact that a symmetric operator need not be an involution.

Failure in the Degenerate Case

Non-injective parameters

Proposition. For the regular bimodule $M={}_A A_A$ the parameters of a reflection are determined modulo the centre:

$$ r_u=r_v \iff v^{-1}u \in Z(A) . $$

Proof. $r_u=r_v$ is $u\alpha(x)u^{-1}=v\alpha(x)v^{-1}$ for all $x$, that is $w\alpha(x)=\alpha(x)w$ for all $x$ with $w=v^{-1}u$. Since $\alpha$ is onto, this is $wx=xw$ for all $x$, that is $w\in Z(A)$. $\square$

Theorem (the criterion in general). Without injectivity of $u \mapsto r_u$ the self-adjointness criterion is the congruence

$$ (r_u)^{*}=r_u \iff \beta(u)u^{-1} \in Z(A), $$

where $Z(A)$ is replaced by the centraliser of $M$ in $A$ when $M$ is not faithful.

Proof. $(r_u)^{*}=r_{\beta(u)}$ and $r_{\beta(u)}=r_u$ is $\beta(u)u^{-1}$ in the centraliser, by the same computation as in the proposition with $v=u$, $w=u^{-1}\beta(u)$. $\square$

So a reflection can be self-adjoint although its parameter is not $\beta$-symmetric, when the two parameters differ by a central element; the criterion $\beta(u)=u$ is the special case of an injective parameter map.

The commutative case

Proposition. If $A$ is commutative then every reflection equals the grade involution,

$$ r_u=\alpha \quad (u \in A^{\times}), $$

the parameter map $u \mapsto r_u$ is constant, and every reflection is self-adjoint, $(r_u)^{*}=\alpha^{*}=\alpha=r_u$.

Proof. In a commutative algebra $u$ commutes with $\alpha(x)$, so $u\alpha(x)u^{-1}=\alpha(x)$; the adjoint of $\alpha$ is $\alpha$ by the $\alpha$-invariance of the pairing. $\square$

The commutative case is the extreme degeneracy: the reflections collapse to a single self-adjoint operator, and the criterion $\beta(u)=u$ is vacuous. This is why the self-adjointness statement must be formulated modulo the centraliser and not as a condition on the parameter alone.

The degenerate pairing

Proposition. If the pairing is not $\alpha$-invariant then $\alpha^{*}\neq\alpha$ and $(r_u)^{*}=\alpha^{*}S_{\sigma(u),\sigma(u)^{-1}}$, which need not be a reflection; in this case the adjoint of a reflection need not be a reflection at all.

Proof. The adjoint of a signed sandwich is $\alpha^{*}S_{\sigma(a),\sigma(b)}$ without invariance; for a reflection this is $\alpha^{*}r'$ with $r'$ the reflection by $\sigma(u)$ in the unsigned sense, and $\alpha^{*}r'$ is a reflection only when $\alpha^{*}$ is a signed sandwich, which fails when $\alpha$ is not self-adjoint. $\square$

Unitarity and the Self-Adjoint Reflections

Unitary reflections

Proposition. A reflection is unitary exactly when its parameter is a unitary element of $A$:

$$ r_u^{*}r_u=r_ur_u^{*}=\mathrm{id} \iff \sigma(u)u=u\sigma(u)=1 . $$

Proof. This is the unitarity criterion of The Signed Adjoint of the Sandwich on a Bimodule over an Algebra with $(a,b)=(u,u^{-1})$. $\square$

Self-adjoint unitary reflections

Proposition. A reflection is simultaneously self-adjoint and unitary exactly when its parameter is unitary and $\beta$-symmetric:

$$ (r_u)^{*}=r_u \text{ and } r_u^{*}r_u=r_ur_u^{*}=\mathrm{id} \iff \sigma(u)u=u\sigma(u)=1 \text{ and } \beta(u)=u . $$

A simultaneously self-adjoint and unitary reflection is automatically of order two, and its parameter is a $\beta$-symmetric unitary element; the unitaries of $A$ that are $\beta$-symmetric form a subgroup of the unitary group of $A$ on which the reflection map is defined, and the unitary group generated by the simultaneously self-adjoint and unitary reflections consists of their products, an even number at a time.

Proof. Combine the unitarity criterion with the self-adjointness criterion. If $r_u^{*}=r_u$ and $r_u^{*}r_u=\mathrm{id}$ then $r_u^{2}=\mathrm{id}$, so the operator is of order two. The $\beta$-symmetric unitaries are closed under multiplication because $\beta$ is an algebra automorphism and the unitary elements of $A$ form a group. $\square$

The self-adjoint unitary reflections are the "graded orthogonal" operators of the bimodule: they preserve the pairing and are fixed by the involution, and their parameters are the unitary elements fixed by $\beta$.

Examples

(a) The trivial grading. For $\alpha=\mathrm{id}$ the reflection is the inner conjugation $r_u(x)=uxu^{-1}$, $\beta=\sigma$, $(r_u)^{*}=r_{\sigma(u)}$, and $r_u$ is self-adjoint exactly when $\sigma(u)=u$: the conjugation by a symmetric unit.

(b) The trivial involution. For $\sigma=\mathrm{id}$ and a nontrivial grading, $(r_u)^{*}=r_{\alpha(u)}$, so a reflection is self-adjoint exactly when its parameter is even, $\alpha(u)=u$, and the self-adjoint reflections are the conjugations by even units.

(c) The commutative algebra. For $A$ commutative every reflection is $\alpha$; the parameter map is constant and the self-adjointness is automatic.

(d) The Clifford algebra. For a Clifford algebra with its canonical involution $\sigma$ and grade involution $\alpha$, the reflection $r_v$ by a unit versor $v$ has adjoint $r_{\alpha\sigma(v)}$, and it is self-adjoint exactly when $\alpha\sigma(v)=v$; the standard examples of self-adjoint reflections are the conjugations by vectors, which have $v\alpha(v)=\pm1$ central and are of order two.

(e) A matrix algebra. For $A=M_n(k)$ with the transpose involution $\sigma$ and a diagonal grading $\alpha$, $(r_u)^{*}=r_{\beta(u)}$ with $\beta(X)=\alpha(X^{\mathsf{T}})$; the self-adjoint reflections are those with $\alpha(u)=u^{\mathsf{T}}$, the graded-symmetric units.

Summary

For a graded bimodule with an $\alpha$-invariant balanced $\sigma$-sesquilinear pairing, the adjoint of the reflection $r_u(x)=u\alpha(x)u^{-1}$ is the reflection $r_{\beta(u)}$ by the composite $\beta=\alpha\sigma$, so the adjoint permutes the reflections and acts on the parameters as the involution $\beta$. When the parameter map $u \mapsto r_u$ is injective, a reflection is self-adjoint exactly when $\beta(u)=u$; in general the criterion is $\beta(u)u^{-1}\in Z(A)$ — that is, $\beta(u)$ and $u$ differ by a central element — and in the commutative case all reflections collapse to the grade involution and all are self-adjoint. Self-adjointness and being an operator of order two are independent conditions, the first involving $\beta(u)=u$ and the second $u\alpha(u)$ central. A reflection is unitary exactly when its parameter is unitary, and it is simultaneously self-adjoint and unitary exactly when the parameter is unitary and $\beta$-symmetric; these are the graded isometries among the reflections. If the pairing is not $\alpha$-invariant the adjoint of a reflection need not be a reflection, the failure being governed by the adjoint of the grade involution.

Summary of Notation

Symbol	Meaning
$R$, $(A,\sigma)$	base ring, involutive $R$-algebra
$\alpha$	grade involution, commuting with $\sigma$
$\beta=\alpha\sigma$	the composite involution
$r_u=S^{\alpha}_{u,u^{-1}}$	the reflection, $r_u(x)=u\alpha(x)u^{-1}$
$r_u^{2}=S_{u\alpha(u),(u\alpha(u))^{-1}}$	the square is a conjugation
$(r_u)^{*}=r_{\beta(u)}$	the adjoint of the reflection
$\beta(u)=u$	the $\beta$-symmetry of the parameter
$\beta(u)u^{-1}\in Z(A)$	the self-adjointness criterion in general
$\sigma(u)u=u\sigma(u)=1$	unitarity of the parameter
$A^{\times}_{\beta}$	the $\beta$-symmetric units