The Signed Adjoint of the Left Multiplication on an Involutive Banach Algebra

Introduction

The signed left multiplication by an element $r$ is the operator $\ell_r(x) = r\,\alpha(x)$, the left multiplication twisted by the grade involution; it is the one-sided operator of the signed theory, and its adjoint with respect to the form of the category is the signed left multiplication by the conjugate element, $(\ell_r)^\dagger = \ell_{\delta(r)}$, $\delta = \sigma\alpha$. From this formula the dictionary of the signed block follows: the signed left multiplication is self-adjoint exactly when $\delta(r) = r$ modulo the kernel, it is unitary exactly when the defect $\delta(r)\alpha(r)$ is the unit, and the two-sided signed sandwich is the composite of the signed left multiplication with a right multiplication. This article computes the adjoint of the signed left multiplication, derives its dictionary, and assembles the one-sided and two-sided operators.

The article assumes the signed left multiplication, its composition and its action on the regular module from The Signed Left Multiplication on an Involutive Banach Algebra; the grade involution $\alpha$, the composite $\delta = \sigma\alpha$ and the reflection from The Grade Involution on a Banach Algebra and The Signed Left Multiplication on an Involutive 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 (unsigned) left multiplication from The Adjoint of the Left Multiplication on an Involutive Banach Algebra; the adjoint of the signed sandwich from The Signed Adjoint Sandwich on a Banach Algebra; and the involution $\sigma$, the unitary elements and the Cayley transform from Adjoints in a 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 $\alpha$ and the involution $\sigma$, with $\sigma\alpha = \alpha\sigma$ and $\delta = \sigma\alpha = \alpha\sigma$; the form of the category is $\{x,y\} = \tau(x\sigma(y))$ with $\tau$ a continuous $\sigma$-invariant and $\alpha$-invariant trace, and $\alpha^\dagger = \alpha$; the signed left multiplication and the signed right multiplication are

$$ \ell_r(x) = r\,\alpha(x) , \qquad \varrho_s(x) = \alpha(x)\,s , $$

and the defect of $r$ is $\delta(r)\alpha(r)$; the signed sandwich is $\Sigma^\alpha_{r,s} = L_rR_s\alpha$.

The Adjoint of the Signed Left Multiplication

Theorem (the explicit form). For every $r$, the signed left multiplication is adjointable with

$$ (\ell_r)^\dagger = \ell_{\delta(r)} , \qquad (\ell_r)^\dagger(x) = \delta(r)\,\alpha(x) , $$

and the signed right multiplication satisfies $(\varrho_s)^\dagger = \varrho_{\delta(s)}$; hence the adjoint of a signed one-sided multiplication is the signed one-sided multiplication by the conjugate parameter.

Proof. $\ell_r = L_r\alpha = \Sigma^\alpha_{r,1}$, so its adjoint is $\Sigma^\alpha_{\delta(r),\delta(1)} = \Sigma^\alpha_{\delta(r),1} = \ell_{\delta(r)}$ by The Signed Adjoint Sandwich on a Banach Algebra, using $\delta(1) = 1$; the right-handed computation is the mirror, $\varrho_s = R_s\alpha = \Sigma^\alpha_{1,\delta^{-1}(s)}$ with the conjugate parameter $\delta(s)$. $\square$

Corollary (the regular graded module). The algebra $A$ is a graded left module over itself through $\ell$, and the map $r \mapsto \ell_r$ is a homomorphism of the graded algebra into the graded operator algebra: $\ell_{rs} = \ell_r\ell_{\delta^{-1}(s)}$, and the involution and the adjoint agree on the image, $(\ell_r)^\dagger = \ell_{\delta(r)}$. The signed sandwich is the one-sided operator followed by a right multiplication,

$$ \Sigma^\alpha_{r,s} = \ell_r\,R_s , $$

so the adjoint of the two-sided operator is read from the one-sided and the right multiplication.

Proof. $\ell_r\ell_{t}(x) = \ell_r(t\alpha(x)) = r\alpha(t\alpha(x)) = r\alpha(t)x = \Sigma^\alpha_{r\alpha(t),1}(x)$, giving the composition law $\ell_r\ell_t = \ell_{r\alpha(t)}$, that is $\ell_{rs} = \ell_r\ell_{\delta^{-1}(s)}$ after the substitution $s = \alpha(t)$; the sandwich identity is associativity, and the adjoint of the composite is the composite of the adjoints by anti-multiplicativity. $\square$

The Dictionary

Theorem (self-adjoint, unitary, involutive). For every $r$,

$$ \ell_r \text{ self-adjoint} \iff \delta(r) = r \ (\text{modulo the kernel}) , \qquad \ell_r \text{ unitary} \iff \delta(r)\,\alpha(r) = 1 , $$

$$ \ell_r \text{ an involution} \iff \ell_r^2 = \mathrm{id} , \ \ell_r \text{ self-adjoint} . $$

The self-adjoint signed left multiplications are those with $r$ fixed by $\delta$, the unitary ones are those with $\delta(r)\alpha(r) = 1$, and the involutions are the self-adjoint ones with $\ell_r^2 = \mathrm{id}$, that is $r\alpha(r) = 1$.

Proof. $\ell_r$ is self-adjoint iff $\ell_{\delta(r)} = \ell_r$, which is the kernel condition on the parametrisation of the one-sided operators, and it is the identity $\delta(r) = r$ modulo the kernel; $\ell_r$ is unitary iff $\ell_{\delta(r)}\ell_r = \ell_{\delta(r)\alpha(r)} = \mathrm{id}$, that is $\delta(r)\alpha(r) = 1$; and $\ell_r^2 = \mathrm{id}$ is $r\alpha(r) = 1$, with self-adjointness from the previous line. $\square$

Theorem (assembly of the two-sided operator). Every signed sandwich is the composite of a signed left multiplication and a signed right multiplication,

$$ \Sigma^\alpha_{r,s} = \ell_r\,R_s = L_r\,\varrho_s , $$

and its adjoint is $\Sigma^\alpha_{\delta(r),\delta(s)} = \ell_{\delta(r)}R_{\delta(s)}$, so the two-sided adjoint is recovered from the one-sided adjoints and the commutation of the left and right multiplications.

Proof. The identities are associativity; the adjoint of the composite is the composite of the adjoints by anti-multiplicativity, $(L_r\varrho_s)^\dagger = \varrho_{\delta(s)}^\dagger L_{\delta(r)}^\dagger = R_{\delta(s)}L_{\delta(r)} = \Sigma^\alpha_{\delta(r),\delta(s)}$. $\square$

Examples

Example (the matrix algebra). For $A = M_n(\mathbb{C})$ with the transpose-involution and the trace form, the signed left multiplication by $X$ has adjoint the signed left multiplication by $\delta(X) = X^{\mathsf{T}}$ in the graded sense; the self-adjoint signed left multiplications are the symmetric graded multiplications, and the unitary ones satisfy $\delta(X)\alpha(X) = 1$.

Example (the commutative algebra). For a commutative $A$ the kernel of the one-sided parametrisation is trivial on the units, the self-adjoint signed left multiplication is exactly $\delta(r) = r$, and the unitarity condition reads $\delta(r)\alpha(r) = 1$; the Cayley transform of Adjoints in a Banach Algebra produces a family of unitary signed left multiplications from the self-adjoint ones.

Example (the group algebra). For the group algebra of a finite group with $\alpha(g) = g$, $\delta(g) = g^{-1}$ on the group elements, the signed left multiplication by a group element $g$ has adjoint the signed left multiplication by $g^{-1}$; it is unitary because $\delta(g)\alpha(g) = 1$, and the regular graded module is the group algebra acting on itself.

Summary

The signed left multiplication $\ell_r(x) = r\alpha(x)$ is adjointable with $(\ell_r)^\dagger = \ell_{\delta(r)}$, and the signed right multiplication satisfies $(\varrho_s)^\dagger = \varrho_{\delta(s)}$; the adjoint sends the parameter to its conjugate under $\delta = \sigma\alpha$. The algebra is a graded left module over itself through $\ell$, the composition is $\ell_{rs} = \ell_r\ell_{\delta^{-1}(s)}$, and the signed sandwich is the composite $\Sigma^\alpha_{r,s} = \ell_rR_s$ of the one-sided and the right multiplication, so the two-sided adjoint is recovered from the one-sided ones. The dictionary holds: $\ell_r$ is self-adjoint exactly when $\delta(r) = r$ modulo the kernel, unitary exactly when the defect $\delta(r)\alpha(r)$ is the unit, and an involution exactly when it is self-adjoint with $r\alpha(r) = 1$. The reflection is the special case $s = \delta(r)^{-1}$ of The Signed Adjoint of the Reflection on a Banach Algebra, and the module case is The Graded Adjoint Action on a Graded Module.

Summary of Notation

Symbol	Meaning
$\ell_r(x) = r\alpha(x)$	The signed left multiplication
$\varrho_s(x) = \alpha(x)s$	The signed right multiplication
$\delta = \sigma\alpha = \alpha\sigma$	The composite involution
$(\ell_r)^\dagger = \ell_{\delta(r)}$	The signed adjoint of the left multiplication
$\ell_{rs} = \ell_r\ell_{\delta^{-1}(s)}$	Composition of the signed left multiplications
$\delta(r) = r$ modulo kernel	Criterion for self-adjointness
$\delta(r)\alpha(r) = 1$	Criterion for unitarity
$\Sigma^\alpha_{r,s} = \ell_rR_s$	Assembly of the two-sided operator