The Signed Adjoint Sandwich on a Banach Algebra

Introduction

The signed sandwich $\Sigma^\alpha_{a,b}(x) = a\,\alpha(x)\,b$ is the two-sided multiplication twisted by the grade involution $\alpha$, and its adjoint with respect to the form of the category is the signed sandwich of the conjugate parameters: $(\Sigma^\alpha_{a,b})^\dagger = \Sigma^\alpha_{\delta(a),\delta(b)}$, where $\delta = \sigma\alpha$ is the composite of the involution $\sigma$ with the grade involution. The formula is the adjoint of the general two-sided operator, and the unsigned, reflection and one-sided cases are its specialisations; from it one reads the whole dictionary — the inverse, the composition, the unitarity and the self-adjointness — and one meets the first genuinely signed phenomenon: the unitarity of the signed sandwich is governed not by the unitarity of the parameters but by the centrality of the defect $\delta(a)\alpha(a)$. This article computes the adjoint of the signed sandwich, derives the composition and unitarity conditions, and identifies the specialisations.

The article assumes the signed sandwich on a Banach algebra, its composition table, its inverse and its group of units from The Signed Sandwich on a Banach Algebra, which owns the operator; the grade involution $\alpha$ and the automorphism it induces from The Grade Involution on a Banach Algebra; the form of the category, the adjoint and the self-adjoint and skew operators from The Involution on the Operator Algebra; the adjoint of the left and right multiplications from The Adjoint of the Left Multiplication on an Involutive Banach Algebra; the involution $\sigma$ and the self-adjoint and unitary elements from Adjoints in a Banach Algebra; and the composite $\delta = \sigma\alpha$ from The Adjoint of the Left Multiplication on an Involutive Banach Algebra and The Signed Left Multiplication on an Involutive Banach Algebra. The graded module case is The Graded Adjoint Action on a Graded Module.

Throughout, $A$ is a unital Banach algebra over $\mathbb{C}$ with the grade involution $\alpha$, an automorphism of order two, and the involution $\sigma$, a continuous anti-automorphism of order two, with $\sigma\alpha = \alpha\sigma$, so that $\delta = \sigma\alpha = \alpha\sigma$ is a continuous anti-automorphism of order two; the form of the category is $\{x,y\} = \tau(x\sigma(y))$ for a continuous $\sigma$-invariant trace $\tau$, assumed also $\alpha$-invariant, $\tau\circ\alpha = \tau$, and $\alpha$ is assumed self-adjoint for the form, $\{\alpha x,y\} = \{x,\alpha y\}$, equivalently $\alpha^\dagger = \alpha$; the signed sandwich is

$$ \Sigma^\alpha_{a,b} = L_aR_b\alpha , \qquad \Sigma^\alpha_{a,b}(x) = a\,\alpha(x)\,b , $$

with $L_a(x) = ax$ and $R_b(x) = xb$; the conjugate parameters of $(a,b)$ are $(\delta(a),\delta(b))$.

The Adjoint of the Signed Sandwich

Theorem (the explicit form). For all $a,b$ the signed sandwich is adjointable, with

$$ \bigl(\Sigma^\alpha_{a,b}\bigr)^\dagger = \Sigma^\alpha_{\delta(a),\delta(b)} , $$

the signed sandwich of the conjugate parameters $(\delta(a),\delta(b))$; the order of the parameters is restored rather than reversed.

Proof. The adjoint is anti-multiplicative, and $\alpha$, $L$, $R$ have adjoints $\alpha^\dagger = \alpha$, $(L_a)^\dagger = L_{\sigma(a)}$, $(R_b)^\dagger = R_{\sigma(b)}$, so

$$ \bigl(L_aR_b\alpha\bigr)^\dagger = \alpha^\dagger R_b{}^\dagger L_a{}^\dagger = \alpha R_{\sigma(b)}L_{\sigma(a)} = \alpha L_{\sigma(a)}R_{\sigma(b)} , $$

using that the left and right multiplications commute. Moving $\alpha$ to the right through $L_{\sigma(a)}$ and $R_{\sigma(b)}$ gives $\alpha L_{\sigma(a)}R_{\sigma(b)} = L_{\alpha(\sigma(a))}R_{\alpha(\sigma(b))}\alpha = L_{\delta(a)}R_{\delta(b)}\alpha$, which is $\Sigma^\alpha_{\delta(a),\delta(b)}$; the adjoint exists and is this operator by the nondegeneracy of the form. $\square$

Corollary (the specialisations). The adjoint of the unsigned sandwich $\Sigma_{a,b} = L_aR_b$ is $\Sigma_{\sigma(a),\sigma(b)}$, recovered by $\alpha = \mathrm{id}$; the adjoint of the reflection $\rho_u = \Sigma^\alpha_{u,u^{-1}}$ is $\rho_{\delta(u)}$; and the adjoint of the signed left multiplication $\ell_a = \Sigma^\alpha_{a,1}$ is $\ell_{\delta(a)}$. The three computations agree with the direct ones on the one-sided and reflection operators.

Proof. Set $\alpha = \mathrm{id}$ in the theorem for the unsigned case, giving $\delta = \sigma$; set $b = a^{-1}$ and use that $\Sigma^\alpha_{a,a^{-1}}$ is the reflection; set $b = 1$. $\square$

Composition, Inverse and Unitarity

Theorem (composition and inverse). The signed sandwiches form a monoid under composition with

$$ \Sigma^\alpha_{p,q}\circ\Sigma^\alpha_{r,s} = \Sigma^\alpha_{p\,\alpha(r),\,\alpha(s)\,q} , $$

as in the operator theory of the signed sandwich; the inverse of $\Sigma^\alpha_{a,b}$ for invertible $a,b$ is $\Sigma^\alpha_{\alpha(a)^{-1},\alpha(b)^{-1}}$, and the adjoint of the inverse is $(\Sigma^\alpha_{a,b})^{\dagger\,-1} = \Sigma^\alpha_{\delta(a)^{-1},\delta(b)^{-1}} = \bigl(\Sigma^\alpha_{a,b}{}^{-1}\bigr)^\dagger$ when $a,b$ and $\delta(a),\delta(b)$ are invertible.

Proof. The composition table is the computation $\Sigma^\alpha_{p,q}\Sigma^\alpha_{r,s} = L_pR_q\alpha L_rR_s\alpha = L_pR_qL_{\alpha(r)}R_{\alpha(s)}\alpha^2 = L_{p\alpha(r)}R_{\alpha(s)q}$, using $\alpha L_r = L_{\alpha(r)}\alpha$, $R_qL_{\alpha(r)} = L_{\alpha(r)}R_q$ and $\alpha^2 = \mathrm{id}$; the inverse is the two-sided application, and the adjoint of the inverse is the inverse of the adjoint because the adjoint is an anti-automorphism of order two. $\square$

Theorem (the unitarity defect). The signed sandwich $\Sigma^\alpha_{a,b}$ is unitary, $\bigl(\Sigma^\alpha_{a,b}\bigr)^\dagger\Sigma^\alpha_{a,b} = \Sigma^\alpha_{a,b}\bigl(\Sigma^\alpha_{a,b}\bigr)^\dagger = \mathrm{id}$, if and only if the defect $\delta(a)\alpha(a)$ is a central unit and

$$ \alpha(b)\,\delta(b) = \bigl(\delta(a)\,\alpha(a)\bigr)^{-1} . $$

Equivalently, $\Sigma^\alpha_{a,b}$ is unitary if and only if $\delta(a)\alpha(a)$ is central and $\alpha(b)\delta(b)$ is its inverse; in particular, when the parameters are unitary and $\delta = \sigma\alpha$ acts as inversion on them, $\delta(a)\alpha(a) = 1 = \alpha(b)\delta(b)$ and the sandwich is unitary.

Proof. By the composition table, $\bigl(\Sigma^\alpha_{a,b}\bigr)^\dagger\Sigma^\alpha_{a,b} = \Sigma^\alpha_{\delta(a),\delta(b)}\Sigma^\alpha_{a,b} = \Sigma^\alpha_{\delta(a)\alpha(a),\,\alpha(b)\delta(b)}$. The operator $\Sigma^\alpha_{u,v} = L_uR_v$ evaluated on $1$ is $uv$, so $\Sigma^\alpha_{u,v} = \mathrm{id}$ forces $uv = 1$ and hence $v = u^{-1}$ and $u$ central (from the identity $uxv = x$ for all $x$); thus the product is the identity exactly when $u = \delta(a)\alpha(a)$ is a central unit with inverse $v = \alpha(b)\delta(b)$. The second product gives the same condition, and the specialisation is $\delta(a)\alpha(a) = \sigma(\alpha(a))\alpha(a) = 1$ when $\alpha(a)$ is unitary. $\square$

Examples

Example ($\alpha = \mathrm{id}$: the unsigned sandwich). The signed sandwich reduces to $L_aR_b$, the adjoint to $\Sigma_{\sigma(a),\sigma(b)}$, and the unitarity condition to $a$ and $b$ unitary with $\sigma$ the inverse; the unsigned case is The Adjoint of the Left Multiplication on an Involutive Banach Algebra.

Example ($b = a^{-1}$: the reflection). The reflection $\rho_u = \Sigma^\alpha_{u,u^{-1}}$ has adjoint $\rho_{\delta(u)}$, so it is self-adjoint exactly when $\rho_{\delta(u)} = \rho_u$, that is when the defect $\delta(u)u^{-1}$ is central; a computation deferred to The Signed Adjoint of the Reflection on a Banach Algebra.

Example ($b = 1$: the signed left multiplication). The signed left multiplication $\ell_a = \Sigma^\alpha_{a,1}$ has adjoint $\ell_{\delta(a)}$, with unitarity condition the centrality of $\delta(a)\alpha(a)$, deferred to The Signed Adjoint of the Left Multiplication on an Involutive Banach Algebra.

Example (the matrix algebra). For $A = M_n(\mathbb{C})$ with the transpose-involution and the trace form, the signed sandwich displays the centrality condition concretely: the defect is a scalar multiple of the identity exactly when the sandwich is unitary, and the graded sandwich is unitary precisely when its matrix is orthogonal or unitary in the graded sense.

The Central Defect and the Unitarity Condition

Definition. The unitarity defect of the pair $(a,b)$ is the pair of products

$$ u = \delta(a)\alpha(a) , \qquad v = \alpha(b)\delta(b) , $$

and the self-adjointness defect is the departure of the parameters from being fixed by $\delta$.

Proposition (the defects govern unitarity and self-adjointness). The sandwich $\Sigma^\alpha_{a,b}$ is unitary if and only if $u$ is central and $v = u^{-1}$; it is self-adjoint if and only if $\delta(a) = a$ and $\delta(b) = b$ modulo the central kernel of the parametrisation. If the parameters are fixed by $\delta$ and by $\alpha$ and are involutions, both conditions hold.

Proof. The unitarity condition is the theorem of the article. For self-adjointness, $(\Sigma^\alpha_{a,b})^\dagger = \Sigma^\alpha_{\delta(a),\delta(b)} = \Sigma^\alpha_{a,b}$ holds exactly when the two parameter pairs differ by the kernel of the parametrisation, which is the set of central parameters; and when $\alpha(a) = \delta(a) = a$ with $a^2 = 1$ the defect is $u = a^2 = 1$, central with inverse $1$. $\square$

Corollary (the centrality obstruction). The unitarity of the signed sandwich fails exactly when the defect $\delta(a)\alpha(a)$ is not central, and its self-adjointness fails exactly when $\delta$ moves a parameter by more than a central unit; both obstructions vanish for a commutative algebra and for the parameters fixed by $\delta$ with an $\alpha$-symmetric defect.

Summary

The signed sandwich $\Sigma^\alpha_{a,b}(x) = a\alpha(x)b$ is adjointable with respect to the form of the category, and its adjoint is the signed sandwich of the conjugate parameters, $(\Sigma^\alpha_{a,b})^\dagger = \Sigma^\alpha_{\delta(a),\delta(b)}$, $\delta = \sigma\alpha$, provided the grade involution is self-adjoint for the form and $\alpha\sigma = \sigma\alpha$. The composition is $\Sigma^\alpha_{p,q}\circ\Sigma^\alpha_{r,s} = \Sigma^\alpha_{p\alpha(r),\alpha(s)q}$, the inverse is $\Sigma^\alpha_{\alpha(a)^{-1},\alpha(b)^{-1}}$, and the adjoint of the inverse is the inverse of the adjoint. The sandwich is unitary exactly when the defect $\delta(a)\alpha(a)$ is a central unit with inverse $\alpha(b)\delta(b)$; the unsigned case, the reflection and the signed left multiplication are the specialisations $\alpha = \mathrm{id}$, $b = a^{-1}$ and $b = 1$. The reflection case is The Signed Adjoint of the Reflection on a Banach Algebra and the one-sided case The Signed Adjoint of the Left Multiplication on an Involutive Banach Algebra.

Summary of Notation

Symbol	Meaning
$\alpha$, $\sigma$, $\delta = \sigma\alpha = \alpha\sigma$	Grade involution, involution, their composite
$\Sigma^\alpha_{a,b}(x) = a\alpha(x)b$	The signed sandwich
$\bigl(\Sigma^\alpha_{a,b}\bigr)^\dagger = \Sigma^\alpha_{\delta(a),\delta(b)}$	The adjoint of the signed sandwich
$\Sigma^\alpha_{p,q}\circ\Sigma^\alpha_{r,s} = \Sigma^\alpha_{p\alpha(r),\alpha(s)q}$	Composition
$\Sigma^\alpha_{a,b}{}^{-1} = \Sigma^\alpha_{\alpha(a)^{-1},\alpha(b)^{-1}}$	Inverse
$\delta(a)\alpha(a)$ central, $\alpha(b)\delta(b) = (\delta(a)\alpha(a))^{-1}$	Unitarity condition
$\alpha = \mathrm{id}$, $b = a^{-1}$, $b = 1$	Unsigned, reflection, signed left multiplication