The Signed Adjoint Sandwich on the Algebra of Random Variables

Introduction

The signed sandwich is the two-sided operator twisted by the grade involution, $$ S^\alpha_{a,b}(x)=a\,\alpha(x)\,b , $$ and this article computes its adjoint with respect to the form of the category, $$ \langle x,y\rangle=\varphi(xy^*)=\mathbb E\bigl[x\bar y\bigr]. $$ The result is $$ \bigl(S^\alpha_{a,b}\bigr)^*=S^\alpha_{\delta(a),\delta(b)},\qquad \delta=\sigma\alpha=\alpha\sigma , $$ the signed sandwich of the twisted images of the two parameters, where $\delta$ is the composite of the conjugation and the grade involution. The formula is the same as in the noncommutative signed theory of The Signed Adjoint Sandwich on a Ring, in Part I, but the reason is different: there the twisted involution $\delta$ performs the reversal of the product, while here the product is commutative and the invariance of the expectation form alone yields $L_c^*=L_{c^*}$, after which the twist $\delta$ on the parameters completes the computation. The article gives the closed form of the adjoint, the behaviour under products, the self-adjointness, the isometry, the unitarity and the involutions of the signed sandwiches, and the comparison with the unsigned sandwich and with the ring case.

The conventions are those fixed in The Signed Sandwich on the Algebra of Random Variables, earlier in this category: $\mathcal{A}=L^\infty(\Omega,\mathcal F,\mathbb P)$ with the pointwise product, the conjugation $\sigma(a)=\bar a$, the state $\varphi(a)=\mathbb E[a]$, the form of the category $\langle x,y\rangle=\varphi(xy^*)$, the grade involution $\alpha$ (an involutive, unitary, self-adjoint automorphism commuting with the conjugation), the grading $\mathcal{A}=\mathcal{A}_{\bar0}\oplus\mathcal{A}_{\bar1}$, and the twisted involution $\delta=\sigma\alpha=\alpha\sigma$. The signed sandwich collapses on the commutative algebra to $S^\alpha_c$ with $c=ab$ and $S^\alpha_c=L_c\alpha$, which is The Signed Sandwich on the Algebra of Random Variables, earlier in this category; the signed left multiplication is $T^\alpha_a=S^\alpha_{a,1}=L_a\alpha$, The Signed Left Multiplication on the Algebra of Random Variables, earlier in this category, and its adjoint is The Signed Adjoint of the Left Multiplication on the Algebra of Random Variables, the later article of this category; the reflections are Reflections as Signed Two-Sided Operators on the Algebra of Random Variables, earlier in this category, and their adjoints are The Signed Adjoint of the Reflection on the Algebra of Random Variables, later in this category. The adjoint of the unsigned left multiplication is The Adjoint of the Left Multiplication on the Algebra of Random Variables, earlier in this category, and the operator-algebra involution is The Involution on the Operator Algebra of a Process, earlier in this category. No physics is invoked.

Throughout, $\mathcal{A}$ is the algebra of random variables, $\mathcal{H}=L^2(\Omega,\mathbb P)$ with the form $\langle x,y\rangle=\varphi(xy^*)$ is the Hilbert space of the category, $\alpha$ is the grade involution and $\delta=\sigma\alpha$. The signed sandwich is $S^\alpha_{a,b}(x)=a\,\alpha(x)\,b$, the unsigned sandwich is $S_{a,b}(x)=axb$, and the twisted involution satisfies $\delta^2=\mathrm{id}$, $\delta\sigma=\sigma\delta=\alpha$ and $\delta\alpha=\alpha\delta=\sigma$.

The Signed Adjoint Sandwich

Definition and closed form

Definition. The adjoint of the signed sandwich is the operator $(S^\alpha_{a,b})^*$ with $$ \langle S^\alpha_{a,b}x,y\rangle=\langle x,(S^\alpha_{a,b})^*y\rangle\qquad\text{for all }x,y\in\mathcal{H}. $$

Theorem (the closed form). For all $a,b\in\mathcal{A}$, $$ \bigl(S^\alpha_{a,b}\bigr)^*=S^\alpha_{\delta(a),\delta(b)}=S^\alpha_{\alpha(a^*),\alpha(b^*)}, $$ with $\delta=\sigma\alpha$; on the commutative algebra, where $S^\alpha_{a,b}=S^\alpha_{ab}$, this is $\bigl(S^\alpha_c\bigr)^*=S^\alpha_{\delta(c)}$.

Proof. The operator is the composite $S^\alpha_{a,b}=L_a\,\alpha\,R_b$, and the adjoint of a product is the product of the adjoints in reverse order, $(L_a\alpha R_b)^*=R_b^*\alpha^*L_a^*$. By The Adjoint of the Left Multiplication on the Algebra of Random Variables, earlier in this category, $L_a^*=L_{a^*}$ and $R_b^*=R_{b^*}=L_{b^*}$, and $\alpha^*=\alpha$ because the grade involution is self-adjoint; hence $(S^\alpha_{a,b})^*=L_{b^*}\alpha L_{a^*}$. Using $\alpha L_{a^*}=L_{\alpha(a^*)}\alpha$ and the commutativity, this is $L_{b^*\alpha(a^*)}\alpha$. A direct computation of $S^\alpha_{\delta(a),\delta(b)}=L_{\delta(a)}\alpha L_{\delta(b)}=L_{\delta(a)}L_{\alpha\delta(b)}\alpha$ gives the coefficient $\delta(a)\,\alpha\delta(b)$. Now $\delta(a)=(\alpha(a))^*=\alpha(a^*)$ because $\alpha$ commutes with the conjugation, and $\alpha\delta(b)=\alpha\sigma\alpha(b)=\sigma(b)=b^*$ because $\alpha^2=\mathrm{id}$ and $\alpha\sigma=\sigma\alpha$; the coefficient is therefore $\alpha(a^*)\,b^*=b^*\,\alpha(a^*)$ by the commutativity, which matches the coefficient $b^*\alpha(a^*)$ of $L_{b^*}\alpha L_{a^*}$.

Corollary (the two readings). The adjoint of the sandwich with the parameters $(a,b)$ is the sandwich with the parameters $(\alpha(a^*),\alpha(b^*))$; when $\alpha=\mathrm{id}$ this is the unsigned sandwich adjoint $S_{a,b}^*=S_{a^*,b^*}$, and the twist is the whole effect of the grading.

Proof. The formula with $\alpha=\mathrm{id}$ gives $\delta=\sigma$ and $S_{\delta(a),\delta(b)}=S_{a^*,b^*}$, the adjoint of the unsigned sandwich.

Products and the involution

Theorem (products). The signed sandwiches compose by $$ S^\alpha_{a,b}\,S^\alpha_{c,d}=S^\alpha_{a\alpha(c),\alpha(d)b}, $$ and the adjoint reverses the product, $(S^\alpha_{a,b}S^\alpha_{c,d})^*=(S^\alpha_{c,d})^*(S^\alpha_{a,b})^*$; the map $(a,b)\mapsto S^\alpha_{a,b}$ is a representation of the semidirect product of the algebra with its grade involution.

Proof. Composing, $(S^\alpha_{a,b}S^\alpha_{c,d})(x)=a\alpha(c\alpha(x)d)b=a\alpha(c)\,\alpha^2(x)\,\alpha(d)b=S^\alpha_{a\alpha(c),\alpha(d)b}(x)$; the anti-multiplicativity of the adjoint is the general property, and the representation statement is the multiplicativity just computed.

Theorem (square and involution). On the commutative algebra the sandwich is $S^\alpha_c$ with $c=ab$, its square is $$ \bigl(S^\alpha_c\bigr)^2=S^\alpha_{c\,\alpha(c)}, $$ and it is an involution exactly when $c\,\alpha(c)=1$, that is when $c$ is an $\alpha$-cocycle; the involutive signed sandwiches form a group under multiplication.

Proof. The square is $(L_c\alpha)^2=L_c\alpha L_c\alpha=L_cL_{\alpha(c)}\alpha^2=L_{c\alpha(c)}=S^\alpha_{c\alpha(c)}$; the involution is the square equal to the identity, which is $c\alpha(c)=1$; the $\alpha$-cocycles are closed under multiplication and inversion because $\alpha$ is an involution.

The Comparison with the Ring

Theorem (the same formula, a different reason). The adjoint of the signed sandwich on the algebra of random variables is the sandwich of the twisted images, exactly as on a ring; in the ring the formula arises from the reversal of the product by the transposition and the twist $\delta$, while here the product is commutative, the transposition is trivial, and the formula arises from the invariance of the expectation form, $L_a^*=L_{a^*}$, together with the twist $\delta$ on the parameters.

Proof. The ring statement is The Signed Adjoint Sandwich on a Ring, in Part I, where the adjoint of $S^\alpha_{a,b}$ is $S^\alpha_{\delta(a),\delta(b)}$ with $\delta=\sigma\alpha$; the commutative computation is the theorem above, and the comparison isolates the invariance of the form as the substitute for the transposition.

Corollary (the arithmetic contrast). Over the algebra of arithmetic functions under the coefficient form the adjoint of a signed sandwich is the transposed sandwich and is not a signed sandwich in general; here it is a signed sandwich for every pair of parameters, which is the commutativity and the traciality of the state.

Proof. The arithmetic statement is that of The Signed Sandwich on the Algebra of Arithmetic Functions, written, and The Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions, written; the difference is the invariance of the form.

Worked Examples

Example (the signed left multiplication). For $b=1$ the sandwich is $S^\alpha_{a,1}=T^\alpha_a=L_a\alpha$, and the formula gives $(T^\alpha_a)^*=S^\alpha_{\delta(a),1}=T^\alpha_{\delta(a)}$; this is the adjoint computed independently in The Signed Adjoint of the Left Multiplication on the Algebra of Random Variables, the later article of this category, and the agreement is the consistency of the two routes.

Example (the grade involution). For $a=b=1$ the sandwich is $\alpha=S^\alpha_1$, and $\delta(1)=1$, so $\alpha^*=\alpha$; the involution is self-adjoint, unitary and an involution, and its adjoint is itself.

Example (the swapped two-atom algebra). For $\Omega=\{\omega_-,\omega_+\}$ with the uniform probability, $\alpha(x_-,x_+)=(x_+,x_-)$ and the parameters $a=(a_-,a_+)$, $b=(b_-,b_+)$, the sandwich is $S^\alpha_{a,b}(x_-,x_+)=(a_-b_+x_+,a_+b_-x_-)$; the adjoint has the parameters $\delta(a)=(\bar a_+,\bar a_-)$ and $\delta(b)=(\bar b_+,\bar b_-)$, that is $S^\alpha_{\delta(a),\delta(b)}(x_-,x_+)=(\delta(a)_-\delta(b)_+x_+,\delta(a)_+\delta(b)_-x_-)=(\bar a_+\bar b_-x_+,\bar a_-\bar b_+x_-)$; the direct adjoint of the diagonal action $(x_-,x_+)\mapsto(a_-b_+x_+,a_+b_-x_-)$ is the diagonal action by the conjugates $(\overline{a_+b_-},\overline{a_-b_+})$ $=(\bar a_+\bar b_-,\bar a_-\bar b_+)$, in agreement.

Example (the unitary sandwiches). On the commutative algebra the sandwich $S^\alpha_c$ is unitary exactly when $|c|=1$ and self-adjoint exactly when $\alpha(c)=c^*$; the two conditions together give the unitary self-adjoint signed sandwiches, the moduli-one $\alpha$-Hermitian elements, among which the reflection $\alpha=S^\alpha_1$ is the trivial one and the unimodular cocycles $c\alpha(c)=1$ are the involutions.

Failure of the Degenerate Cases

The signed adjoint sandwich degenerates in four configurations. First, the closed form holds for the form of the category and the grade involution; a different grade involution or a different form changes the twisted involution $\delta$ and the formula, and the adjoint is not a signed sandwich at all for a general form. Second, on the commutative algebra the two parameters of the sandwich collapse to the product $c=ab$, and the adjoint depends on $c$ alone; the two-sided information of the ring case is lost, so the adjoint cannot distinguish different factorisations of the same $c$. Third, the involution condition $c\alpha(c)=1$ is not the unitarity condition $|c|=1$; the involutive sandwiches and the unitary sandwiches are different families, intersecting in the moduli-one cocycles, and confusing the two is the standard error. Fourth, the self-adjointness $c=\delta(c)$ is the $\alpha$-Hermitian condition $\alpha(c)=c^*$, which is not the reality $c=c^*$ unless $\alpha=\mathrm{id}$; the grading shifts the self-adjointness of the signed operators off the real elements.

Summary

The adjoint of the signed sandwich $S^\alpha_{a,b}(x)=a\alpha(x)b$ on the algebra of random variables with respect to the form $\langle x,y\rangle=\varphi(xy^*)$ is the signed sandwich of the twisted images of the parameters, $$ \bigl(S^\alpha_{a,b}\bigr)^*=S^\alpha_{\delta(a),\delta(b)},\qquad \delta=\sigma\alpha=\alpha\sigma , $$ the same formula as on a noncommutative ring but derived here from the invariance of the expectation form, $L_a^*=L_{a^*}$, together with the twist $\delta$. The sandwiches compose by $S^\alpha_{a,b}S^\alpha_{c,d}=S^\alpha_{a\alpha(c),\alpha(d)b}$, the adjoint reverses the product, the square on the commutative algebra is $(S^\alpha_c)^2=S^\alpha_{c\alpha(c)}$, and the sandwich is an involution exactly for the $\alpha$-cocycles $c\alpha(c)=1$, self-adjoint exactly for the $\alpha$-Hermitian elements $\alpha(c)=c^*$, and unitary exactly for the moduli-one elements $|c|=1$. The signed left multiplication and its adjoint are The Signed Adjoint of the Left Multiplication on the Algebra of Random Variables, the later article of this category, and the reflections and their adjoints are The Signed Adjoint of the Reflection on the Algebra of Random Variables, the later article of this category.

Summary of Notation

Symbol	Meaning
$S^\alpha_{a,b}(x)=a\alpha(x)b$	the signed sandwich
$\langle x,y\rangle=\varphi(xy^*)$	the form of the category
$\delta=\sigma\alpha=\alpha\sigma$	the twisted involution
$(S^\alpha_{a,b})^*=S^\alpha_{\delta(a),\delta(b)}$	the adjoint
$S^\alpha_{a,b}S^\alpha_{c,d}=S^\alpha_{a\alpha(c),\alpha(d)b}$	the product
$(S^\alpha_c)^2=S^\alpha_{c\alpha(c)}$	the square
$c\alpha(c)=1$	the involutions
$\alpha(c)=c^*$	the self-adjoint sandwiches
$\|c\|=1$	the unitary sandwiches