Adjoints in a Commutative Involutive Algebra

Introduction

A non-degenerate reflexive pairing $B$ on a commutative involutive algebra $(A,\sigma)$ turns each $R$-linear operator $F : A\to A$ into a second operator $F^{\dagger}$, the adjoint, defined by moving $F$ from one side of the pairing to the other, $B(Fx,y) = B(x,F^{\dagger}y)$; the assignment $F\mapsto F^{\dagger}$ is an involution of the endomorphism algebra $\operatorname{End}_R(A)$, and it is the operator-level companion of the involution $\sigma$ of the elements. This article develops the adjoint for the commutative involutive algebras of Commutative Algebras with an Involution: it proves the existence and uniqueness of $F^{\dagger}$, establishes that ${}^{\dagger}$ is an anti-automorphism of order two, describes the self-adjoint part and the unitary operators, and computes the adjoint of a multiplication, $L_a^{\dagger} = L_{\sigma(a)}$. The last computation identifies the abstract involution of the multiplication operators of Involutions of the Multiplication Operators with the adjoint involution restricted to $\operatorname{Mult}(A)$, and it shows that every multiplication operator is normal, $L_aL_a^{\dagger} = L_a^{\dagger}L_a$, because of the commutativity of the algebra.

The article assumes Commutative Algebras with an Involution for $\sigma$, Involutions of the Multiplication Operators for $L_a$ and the induced involution, The Operators on an Algebra and The Adjoint of an Endomorphism for the adjoint of an endomorphism and the pairing, Modules over a Ring for the modules, and Involutive Linear Algebras for the involution of the endomorphism algebra. The forms, the Hilbert structure and the operator spectrum belong to Part II; the signed variants are The Signed Adjoint Sandwich, The Signed Adjoint of the Reflection and The Signed Adjoint of the Left Multiplication, later in this group. Throughout, $R$ is a commutative ring with identity in which $2$ is invertible, $(A,\sigma)$ is a commutative involutive $R$-algebra, $B$ is a non-degenerate reflexive $\sigma$-sesquilinear pairing on $A$ with $B(ax,y) = B(x,\sigma(a)y)$, and ${}^{\dagger}$ is the adjoint involution of $\operatorname{End}_R(A)$; no norm, distance, positivity or operator spectrum occurs.

The Pairing and the Adjoint

The Adjacent Structure

Definition. A $\sigma$-sesquilinear pairing on $A$ is an $R$-bilinear map $B : A\times A\to R$ with $B(ax,y) = B(x,\sigma(a)y)$ for all $a, x, y$; it is reflexive when $B(x,y) = 0 \iff B(y,x) = 0$, and non-degenerate when $B(x,y) = 0$ for all $y$ forces $x = 0$ and dually. The pairing is the algebra-level analogue of the pairing of The Adjoint of an Endomorphism, and it is not a form in the sense of Part II: no symmetry and no sign is assumed, only the reflexivity needed for the adjoint.

Theorem (existence and uniqueness). For every $F \in \operatorname{End}_R(A)$ there is a unique $F^{\dagger}\in\operatorname{End}_R(A)$ with

$$ B(Fx,y) = B(x,F^{\dagger}y) \qquad \text{for all } x, y \in A . $$

Proof. The map $y\mapsto B(F\,\cdot\,,y)$ is $R$-linear, so it is an element of the dual of $A$; non-degeneracy identifies the dual with $A$ through $z\mapsto B(\,\cdot\,,z)$, so there is a unique $z$ with $B(Fx,y) = B(x,z)$ for all $x$; put $F^{\dagger}y = z$. The assignment is additive and $R$-linear in $F$ and in $y$ because $B$ is bilinear. $\square$

The Adjoint Involution

Theorem. The map ${}^{\dagger} : \operatorname{End}_R(A)\to\operatorname{End}_R(A)$ is additive, $R$-linear, of order two, and anti-multiplicative:

$$ (F+G)^{\dagger} = F^{\dagger}+G^{\dagger}, \qquad (\lambda F)^{\dagger} = \lambda F^{\dagger}, \qquad (FG)^{\dagger} = G^{\dagger}F^{\dagger}, \qquad (F^\dagger)^\dagger = F . $$

Hence ${}^{\dagger}$ is an involution of the associative algebra $\operatorname{End}_R(A)$, in the sense of Involutive Linear Algebras.

Proof. Additivity and $R$-linearity are the bilinearity of $B$; the order two is the reflexivity: $B(Fx,y) = B(x,F^{\dagger}y) = B((F^\dagger)^\dagger x,y)$ gives $(F^\dagger)^\dagger = F$ by non-degeneracy; the anti-multiplicativity is $B(FGx,y) = B(Gx,F^{\dagger}y) = B(x,G^{\dagger}F^{\dagger}y)$. $\square$

The Adjoint Involution and the Self-Adjoint Part

Self-Adjoint and Skew-Adjoint Operators

Definition. An operator $F$ is self-adjoint when $F^{\dagger} = F$, skew-adjoint when $F^{\dagger} = -F$, and unitary when $F^{\dagger}F = FF^{\dagger} = 1$; the set of the self-adjoint operators is $E^+$ and the set of the skew-adjoint operators is $E^-$, with $E = \operatorname{End}_R(A)$.

Proposition. The self-adjoint operators form a Jordan algebra under $F\bullet G = \tfrac12(FG+GF)$ and the skew-adjoint operators form a Lie algebra under the commutator; $E = E^+\oplus E^-$ when $2$ is invertible, and the unitary operators form a group.

Proof. This is Involutive Linear Algebras applied to the involution ${}^{\dagger}$ of $E$: the fixed set is closed under the symmetrised product, the anti-fixed under the commutator, and the units fixed up to the inverse form the unitary group. $\square$

The Adjoint of a Multiplication

Theorem. For every $a \in A$ the adjoint of the multiplication $L_a$ is the multiplication by the image of $a$ under the involution: $$ L_a^{\dagger} = L_{\sigma(a)} . $$

Proof. $B(L_ax,y) = B(ax,y) = B(x,\sigma(a)y) = B(x,L_{\sigma(a)}y)$ for all $x,y$, so $L_{\sigma(a)}$ satisfies the defining relation of the adjoint, and the adjoint is unique. $\square$

Corollary (consistency with the induced involution). The restriction of the adjoint involution ${}^{\dagger}$ to the multiplication operators is the induced involution of Involutions of the Multiplication Operators: $L_a^{\dagger} = L_a^{*}$. Hence the fixed part of the multiplication algebra is the multiplication algebra of the fixed subalgebra, and the two constructions of the involution of $\operatorname{Mult}(A)$ coincide.

Corollary (normality). Every multiplication operator is normal with respect to ${}^{\dagger}$, $$ L_aL_a^{\dagger} = L_{a\sigma(a)} = L_a^{\dagger}L_a , $$ because the algebra is commutative; the multiplication operators therefore lie in the normal part of $\operatorname{End}_R(A)$, and $L_a$ is self-adjoint exactly when $\sigma(a) = a$ and skew-adjoint exactly when $\sigma(a) = -a$.

Proof. $L_aL_a^{\dagger} = L_aL_{\sigma(a)} = L_{a\sigma(a)}$ and $L_a^{\dagger}L_a = L_{\sigma(a)}L_a = L_{\sigma(a)a}$, and $a\sigma(a) = \sigma(a)a$. $\square$

Examples

Example (the coordinate algebra). Let $A = R^n$ with the standard pairing $B(x,y) = \sum_ix_iy_i$ and let $\sigma$ be an involution of $A$ permuting or negating the coordinates. Then the adjoint of an operator $F$, represented by a matrix, is $F^{\dagger} = \sigma F^{\mathsf{T}}\sigma^{-1}$, that is, the transpose composed with the coordinate involution on both sides. The multiplications $L_a$ for $a$ a diagonal matrix are diagonal, and $L_a^{\dagger} = L_{\sigma(a)}$ is the diagonal matrix with the permuted or negated diagonal, confirming the theorem.

Example (the polynomial ring). Let $A = R[x]$ with $\sigma(x) = -x$ and let $B$ be the $R$-bilinear pairing with $B(x^i,x^j) = \delta_{ij}$, reflexive and non-degenerate on the polynomials of bounded degree; then $L_x^{\dagger} = L_{-x} = -L_x$, so the multiplication by $x$ is skew-adjoint, and the multiplication by $x^2$ is self-adjoint. The fixed subalgebra of the multiplication operators is the multiplication by the even polynomials.

Summary

A non-degenerate reflexive $\sigma$-sesquilinear pairing on a commutative involutive algebra $(A,\sigma)$ defines the adjoint $F^{\dagger}$ of every operator by $B(Fx,y) = B(x,F^{\dagger}y)$; the adjoint exists, is unique, and the assignment ${}^{\dagger}$ is an involution of $\operatorname{End}_R(A)$, additive, $R$-linear, of order two and anti-multiplicative. The self-adjoint operators form a Jordan algebra under the symmetrised product and the skew-adjoint operators a Lie algebra under the commutator, with the decomposition into the two when $2$ is invertible; the unitary operators form a group. The adjoint of a multiplication is $L_a^{\dagger} = L_{\sigma(a)}$, so the adjoint involution restricted to the multiplication operators is the induced involution of Involutions of the Multiplication Operators and every multiplication operator is normal. The coordinate algebra and the polynomial ring are the worked examples. No norm, distance, positivity or operator spectrum occurs.

Summary of Notation

Symbol	Meaning
$(A,\sigma)$	Commutative involutive $R$-algebra
$B(ax,y) = B(x,\sigma(a)y)$	$\sigma$-sesquilinear pairing
$B(Fx,y) = B(x,F^{\dagger}y)$	Definition of the adjoint
$(FG)^{\dagger} = G^{\dagger}F^{\dagger}$, $(F^\dagger)^\dagger = F$	The adjoint involution
$E^+, E^-$	Self-adjoint and skew-adjoint operators
$F^{\dagger}F = FF^{\dagger} = 1$	Unitary operator
$L_a^{\dagger} = L_{\sigma(a)} = L_a^{*}$	Adjoint of a multiplication
$L_aL_a^{\dagger} = L_a^{\dagger}L_a$	Normality of the multiplications