The Signed Sandwich on an Ordered Algebra

Introduction

The unsigned sandwich of an algebra is the two-parameter operator

$$ \Theta_{a,b} : x\mapsto axb , $$

the composition of a left multiplication and a right multiplication, which for $b = a^{-1}$ is an inner automorphism and for $a = b$ is a square. The signed sandwich twists the argument by the grade involution $\alpha$ of a graded algebra:

$$ \Theta^{\alpha}_{a,b} : x\mapsto a\,\alpha(x)\,b . $$

The two are related by the grading: if $\Gamma$ is the operator that is $+1$ on the even part and $-1$ on the odd part, then $\Theta^{\alpha}_{a,b} = \Theta_{a,b}\circ\Gamma$, and when the grade involution is inner, $\alpha(x) = uxu^{-1}$, the signed sandwich is an unsigned sandwich with shifted parameters, $\Theta^{\alpha}_{a,b} = \Theta_{au,\,u^{-1}b}$. The signed sandwich therefore adds no new operator in those cases; what it adds is the twist of the parity, and its use is the study of the elements on which the twisted conjugation is an involution.

An element $a$ with $\alpha(a) = a^{-1}$ makes the signed conjugation

$$ x\mapsto a\,\alpha(x)\,a^{-1} $$

an operator of order two, that is, a reflection. This is the phenomenon the article isolates; the correspondence between the reflections of the algebra and the elements that realise them is the subject of Reflections as Signed Two-Sided Operators on an Ordered Algebra, which follows. The article also states when the signed sandwich is positive, which is exactly when the grade involution preserves the cone and the parameters are positive.

The ordered algebra, the left and right multiplications and their positivity are The Left and Right Multiplication Operators on an Ordered Algebra; the grading, the grade involution and the sign rule are Superalgebras and Graded Structures of Part I; the operator order is Positive Operators on an Ordered Space; and the involution of the corpus, which is an anti-automorphism and not the grade involution, is Ordered Involutive Algebras later in this category. The signed Hermitian sandwich $\Theta^{\alpha}_x(y) = \alpha(x)\,y\,x^{\dagger}$ of Two-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint of Part II carries the grade involution on the parameter; the present article carries it on the argument, and the two conventions are different objects with the same name, which is stated in a remark and reconciled in the companion file. The adjoint of the signed sandwich is The Signed Adjoint Sandwich on an Ordered Algebra at the end of this category.

The Grade Involution and the Grading

Definition. A grade involution of an algebra $A$ is an algebra automorphism $\alpha$ with $\alpha^2 = \mathrm{id}$. It is inner when $\alpha(x) = uxu^{-1}$ for a unit $u$, and positive when it preserves the cone, $\alpha(A_+)\subseteq A_+$.

Proposition (the grading). A grade involution decomposes $A$ as

$$ A = A_{\bar 0}\oplus A_{\bar 1}, \qquad A_{\bar 0} = \ker(\alpha - \mathrm{id}), \quad A_{\bar 1} = \ker(\alpha + \mathrm{id}) , $$

the even and odd parts, when $2$ is invertible in the scalars; $A_{\bar 0}$ is a subalgebra, $A_{\bar 0}A_{\bar 1}\cup A_{\bar 1}A_{\bar 0}\subseteq A_{\bar 1}$, and $A_{\bar 1}A_{\bar 1}\subseteq A_{\bar 0}$. A homogeneous element $x$ has a parity $\lvert x\rvert\in\{0,1\}$ with $\alpha(x) = (-1)^{\lvert x\rvert}x$, and the parity is additive, $\lvert xy\rvert = \lvert x\rvert+\lvert y\rvert$ modulo two.

Proof. The decomposition is the standard one for an involution with the two eigenvalues $\pm1$; the containment of the products is the multiplicativity of $\alpha$ applied to the eigenvalue equations, since $\alpha(xy) = \alpha(x)\alpha(y)$ multiplies the signs; the parity statement is the definition.

Proposition (the grading operator). The grading operator $\Gamma$ defined by $\Gamma x = (-1)^{\lvert x\rvert}x$ on the homogeneous elements is linear, satisfies $\Gamma^2 = \mathrm{id}$, and is the diagonalisable operator whose eigenspaces are the even and odd parts; it is an involution of the vector space, not of the algebra, and it equals $\alpha$ as a map of the underlying vector space.

Proof. Linearity on the homogeneous pieces and the extension by linearity give the first two claims; the identification with $\alpha$ is the parity formula.

The Signed Sandwich

Definition and Linearity

Definition. For $a,b\in A$ the signed sandwich is the operator

$$ \Theta^{\alpha}_{a,b} : A\to A, \qquad \Theta^{\alpha}_{a,b}(x) = a\,\alpha(x)\,b . $$

When $\alpha = \mathrm{id}$ it is the unsigned sandwich $\Theta_{a,b}(x) = axb$.

Proposition. $\Theta^{\alpha}_{a,b}$ is linear in $x$ and in each parameter; it is $\Theta_{a,b}\circ\Gamma = \Gamma_{l}\circ\Theta_{a,b}$ for the grading operator $\Gamma$ on the argument, so it factors through the unsigned sandwich; and

$$ \Theta^{\alpha}_{a,b} = 0 \iff a = 0 \ \text{or}\ b = 0 $$

in a unital algebra.

Proof. Linearity in the parameters is the bilinearity of the multiplication and the linearity of $\alpha$. The factorisation is $\alpha = \Gamma$ on the underlying vector space. The vanishing statement uses $\Theta^\alpha_{a,b}(1) = ab$ in a unital algebra, so $a = 0$ or $b = 0$.

Composition and the Parameter Law

Theorem (the composition law). The signed sandwiches compose by the twisted parameter rule

$$ \Theta^{\alpha}_{a,b}\,\Theta^{\alpha}_{c,d} = \Theta^{\alpha}_{a\,\alpha(c),\,\alpha(d)\,b} , $$

which for $\alpha = \mathrm{id}$ is the unsigned law $\Theta_{a,b}\Theta_{c,d} = \Theta_{ac,\,db}$.

Proof. Compute $\Theta^{\alpha}_{a,b}\bigl(\Theta^{\alpha}_{c,d}(x)\bigr) = a\,\alpha\bigl(c\,\alpha(x)\,d\bigr)b = a\,\alpha(c)\,\alpha^2(x)\,\alpha(d)\,b = a\alpha(c)\,x\,\alpha(d)\,b$, using that $\alpha$ is an algebra homomorphism and $\alpha^2 = \mathrm{id}$; the result is $\Theta^{\alpha}_{a\alpha(c),\,\alpha(d)b}(x)$.

Corollary (the parameter algebra). With the composition as the product the signed sandwiches form a semigroup isomorphic to $A\times A$ with the twisted multiplication $(a,b)(c,d) = (a\alpha(c),\alpha(d)b)$; when $\alpha$ is inner, $\alpha(x) = uxu^{-1}$, the same law is the ordinary multiplication after the shift $(a,b)\mapsto(au,u^{-1}b)$, and the signed sandwiches are the unsigned sandwiches of the shifted pairs.

Proof. The semigroup statement is the composition law. For the inner case, $a\alpha(c) = au c u^{-1}$ and $\alpha(d)b = u d u^{-1}b$, so $\Theta^\alpha_{a,b} = \Theta_{au,\,u^{-1}b}$ and the twisted law becomes the ordinary one.

Positivity

Proposition (positivity of the signed sandwich). If the grade involution is positive and $a\geq0$, $b\geq0$, then $\Theta^{\alpha}_{a,b}\geq0$. If the grade involution is not positive the signed sandwich of positive parameters need not be positive, and the obstruction is exactly the sign of $\alpha$ on the odd part of the cone; in the graded case with the parity $\varepsilon$, the signed sandwich acts as

$$ \Theta^{\alpha}_{a,b}(x) = \varepsilon_x\,\Theta_{a,b}(x) \quad (x \ \text{homogeneous}) , $$

so on the odd part it is the negative of the unsigned sandwich.

Proof. Positivity of $\alpha$ gives $\alpha(x)\geq0$ for $x\geq0$, and the product of two positive elements with a positive middle is positive by the positive bilinearity of the ordered algebra. For the parity formula, $\alpha(x) = \varepsilon_x x$, which is the second display; on the odd part $\varepsilon_x = -1$, and the unsigned sandwich of positive parameters is positive while the signed one is negative.

The Reflections the Sandwich Realises

Definition. An element $a\in A$ is a reflection element when it satisfies

$$ \alpha(a) = a^{-1} , $$

that is, when the grade involution inverts it; it is normalised when in addition $a^2 = 1$, and skew when $a^2 = -1$.

Theorem (the signed conjugation is an involution exactly on the reflection elements). The signed sandwich with $b = a^{-1}$, the signed conjugation

$$ \tau_a : x\mapsto a\,\alpha(x)\,a^{-1} , $$

is an involution, $\tau_a^2 = \mathrm{id}$, if and only if $\alpha(a) = a^{-1}$. When $\alpha = \mathrm{id}$ the same operator is the ordinary inner conjugation $\tau_a(x) = axa^{-1}$, an automorphism of the algebra for every invertible $a$, and it is an involution if and only if $a^2$ is central.

Proof. By the composition law, $\tau_a^2 = \Theta^{\alpha}_{a\alpha(a),\,a^{-1}\alpha(a)^{-1}}$, which is the identity exactly when $a\alpha(a) = 1$ and its inverse, that is, $\alpha(a) = a^{-1}$. In the unsigned case $\tau_a = \Theta_{a,a^{-1}}$ is the conjugation by $a$, and $\tau_a^2 = \Theta_{a^2,\,a^{-2}}$ is the identity exactly when $a^2$ is central.

Corollary (the odd reflections). In a graded algebra whose odd part generates, an odd element $a$ with $a^2 = -1$ is a skew reflection element: $\alpha(a) = -a = a^{-1}$, so the signed conjugation $\tau_a(x) = a\alpha(x)a^{-1}$ is an involution, and it acts as the ordinary conjugation on the even part and as its negative on the odd part, $\tau_a = \Theta_{a,a^{-1}}\circ\Gamma$. A normalised even element $a$ with $a^2 = 1$ is a reflection element with $\alpha(a) = a = a^{-1}$, and it acts by the ordinary conjugation.

Proof. For odd $a$ with $a^2 = -1$ the inverse is $a^{-1} = -a = \alpha(a)$, so the involution condition holds, and the parity formula of the positivity proposition gives the two cases. For even $a$ with $a^2 = 1$ the inverse is $a = \alpha(a)$ and the signed conjugation is the ordinary one.

Remark (the boundary of the correspondence). The theorem gives a reflection operator for every reflection element, but the converse is delicate: an involutive signed sandwich $\Theta^{\alpha}_{a,b}$ with $b\neq a^{-1}$ need not come from a reflection element, and in characteristic two every element is even and the involution condition collapses; the correspondence is therefore a statement about the elements and not about all involutive signed sandwiches, and it is the subject of the next article.

Worked Cases

The Biquaternion Algebra

Let $A = \mathbb{B} = M_2(\mathbb{C})$ with the grade involution $\alpha$ the conjugation by $\operatorname{diag}(1,-1)$, the parity grading of the biquaternions. The even part is the span of $\operatorname{diag}(1,-1)$ and the identity, the odd part is the span of the off-diagonal matrix units, and an odd $a$ with $a^2 = -1$, such as an off-diagonal matrix unit with a suitable sign, realises the signed conjugation as an involution of order two. The signed sandwich $\Theta^{\alpha}_{a,b}$ with general parameters is an unsigned sandwich after the inner shift, since $\alpha$ is inner in $M_2(\mathbb{C})$; this is the model in which the signed and the unsigned sandwiches are the same operator family under a shift of the parameters, and it shows that the interest of the signed sandwich lies in the fixed parameters, not in the operator family.

The Group Algebra

Let $A = \mathbb{R}[G]$ for a group $G$ with a homomorphism $\lvert\cdot\rvert : G\to\mathbb{Z}/2$ to the parity group, extended to the grade involution $\alpha(g) = (-1)^{\lvert g\rvert}g$. An even involution $g$ with $\lvert g\rvert = 0$ and $g^2 = 1$ is a reflection element, since $\alpha(g) = g = g^{-1}$, and the signed conjugation is the ordinary conjugation by $g$. An odd element with $g^2 = -1$ is a skew reflection element, since then $g^{-1} = -g = \alpha(g)$; in the group algebra of the quaternion group such an element exists and the signed conjugation is an involution. An odd element of order two is never a reflection element in characteristic zero, because $\alpha(g) = -g$ while $g^{-1} = g$ and $2g\neq0$. This is the group-theoretic instance of the two classes of reflection elements.

The Self-Adjoint Matrices

Let $A = H_n(\mathbb{R})$ as the ordered algebra of Jordan Algebras and the Positive Cone, with the grade involution the identity and the order the Loewner one. The signed sandwich is then the ordinary sandwich $\Theta_{a,b}(x) = axb$ of the symmetrised product, the reflection elements are the involutions $a = a^{-1}$, and the signed conjugation is the ordinary conjugation by an involution. The positivity of the sandwich is not automatic in the associative matrix algebra — the product of positive matrices is not symmetrised-positive — which is why the ordered matrix algebra is the Jordan structure and not the associative one.

Summary

A grade involution is an algebra automorphism $\alpha$ of order two; it decomposes the algebra into the even and odd parts, it is inner when it is a conjugation and positive when it preserves the cone, and the grading operator $\Gamma$ realises it on the underlying vector space. The signed sandwich $\Theta^{\alpha}_{a,b}(x) = a\alpha(x)b$ is linear, it factors as the unsigned sandwich $\Theta_{a,b}(x) = axb$ composed with $\Gamma$, it is an unsigned sandwich with shifted parameters when $\alpha$ is inner, and it composes by the twisted law $\Theta^\alpha_{a,b}\Theta^\alpha_{c,d} = \Theta^\alpha_{a\alpha(c),\,\alpha(d)b}$. It is positive exactly when the grade involution preserves the cone; in the graded case it is the unsigned sandwich on the even part and its negative on the odd part, $\Theta^\alpha_{a,b}(x) = \varepsilon_x\Theta_{a,b}(x)$. The signed conjugation $x\mapsto a\alpha(x)a^{-1}$ is an involution exactly on the reflection elements $\alpha(a) = a^{-1}$; odd skew elements with $a^2 = -1$ and even normalised elements with $a^2 = 1$ are the two classes, and the correspondence between the reflections and the elements that realise them is the subject of the next article. The unsigned sandwich and the one-sided multiplications are The Left and Right Multiplication Operators on an Ordered Algebra; the grading and the grade involution are Superalgebras and Graded Structures; the positivity is Positive Operators on an Ordered Space; the involution of the corpus is Ordered Involutive Algebras; the Hermitian signed sandwich of Part II is Two-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint; and the adjoint of the signed sandwich is The Signed Adjoint Sandwich on an Ordered Algebra.

Summary of Notation

Symbol	Meaning
$\alpha$	Grade involution, an algebra automorphism of order two
$A_{\bar 0}, A_{\bar 1}$	Even and odd parts of the graded algebra
$\lvert x\rvert$, $\varepsilon_x = (-1)^{\lvert x\rvert}$	Parity and its sign
$\Gamma$	Grading operator, $\Gamma x = (-1)^{\lvert x\rvert}x$
$\Theta_{a,b}(x) = axb$	Unsigned sandwich
$\Theta^{\alpha}_{a,b}(x) = a\alpha(x)b$	Signed sandwich
$\Theta^\alpha_{a,b}\Theta^\alpha_{c,d} = \Theta^\alpha_{a\alpha(c),\,\alpha(d)b}$	Composition law
$\tau_a(x) = a\alpha(x)a^{-1}$	Signed conjugation
$\alpha(a) = a^{-1}$	Reflection element, making $\tau_a$ an involution