The Signed Left Multiplication on a Topological Ring

Introduction

The signed left multiplication is the one-sided operator obtained from the left multiplication by inserting the grade involution, $x \mapsto a\,\alpha(x)$; it is the signed sandwich with the right parameter equal to $1$, and it is the operator by which a graded ring acts on itself when the action is twisted by the grading. This article treats it on a topological ring: it fixes the operator, proves its continuity and its composition law, computes the group generated by the signed left multiplications with unit parameter, identifies the elements it fixes, relates it to the unsigned left multiplication — the two families differ exactly by the grade involution and coincide exactly when the grade involution is inner — and records the degenerate cases in which the signed one-sided operators carry no new information. The signed one-sided action is the model of a graded module action, and its two-sided counterparts are the signed sandwich and the reflections of the preceding articles of this group.

The article assumes the signed sandwich, its composition law, the signed sandwich group and the signed conjugations from The Signed Sandwich on a Topological Ring; the grade involution, its sign rule and its fixed and skew parts from The Grade Involution and Graded Rings; the abstract signed one-sided operators from The Signed Left Multiplication on a Ring; the one-sided multiplications, their composition, their failure of commutation and the sandwich from The Left and Right Multiplication Operators on a Topological Ring; the continuous additive operators, the homeomorphisms and the natural pairing from Operators on a Topological Ring; and the corresponding one-sided signed operators on a group from The Signed Left Multiplication on a Topological Group. The adjoint of the signed left multiplication is The Signed Adjoint of the Left Multiplication on a Topological Ring, later in this category, and the graded module action is The Graded Action on a Module over a Topological Ring; no involution in the sense of an anti-automorphism occurs here, and no form, measure or norm is used.

Throughout, $R$ is a topological ring, unital and Hausdorff, with continuous grade involution $\alpha$, group of units $R^\times$ and centre $Z(R)$; the unsigned left and right multiplications are $L_a(x) = ax$ and $R_a(x) = xa$, and the signed left multiplication carried by $a$ is

$$ \ell_a : R \longrightarrow R, \qquad \ell_a(x) = a\,\alpha(x) . $$

The signed right multiplication is $\varrho_b(x) = \alpha(x)\,b$, the mirror image.

Definition and Decomposition

Definition. The signed left multiplication with parameter $a \in R$ is the operator $\ell_a(x) = a\alpha(x)$; the signed right multiplication is $\varrho_b(x) = \alpha(x)b$. The unsigned left multiplication is $L_a$ and the unsigned right multiplication is $R_a$.

Proposition (decomposition and continuity). The signed left multiplication factors as

$$ \ell_a = L_a\circ\alpha = \Sigma^\alpha_{a,1} , $$

it is additive and continuous, and it is the signed sandwich with right parameter $1$; symmetrically $\varrho_b = \alpha\circ R_b = \Sigma^\alpha_{1,b}$. The signed left multiplications with a unit parameter are the invertible ones, $\ell_u^{-1} = \ell_{\alpha(u)^{-1}}$, and the signed left multiplications are the elements of the coset $L(R)\circ\alpha$ of the unsigned left multiplications.

Proof. $\ell_a(x) = a\alpha(x) = L_a(\alpha(x))$, and $\Sigma^\alpha_{a,1}(x) = a\alpha(x)1 = a\alpha(x)$; additivity and continuity follow from those of $L_a$ and $\alpha$. The inverse: $\ell_u\ell_{\alpha(u)^{-1}}(x) = u\alpha(\alpha(u)^{-1}\alpha(x)) = u\alpha(\alpha(u)^{-1})x = uu^{-1}x = x$ by the composition law below and $\alpha^2 = \mathrm{id}$, and likewise on the other side; conversely $\ell_a$ is invertible exactly when the sandwich $\Sigma^\alpha_{a,1}$ is, that is exactly when $a$ is a unit, by The Signed Sandwich on a Topological Ring.

Composition and the Group

Proposition (composition law). For all $a, b \in R$,

$$ \ell_a\circ\ell_b = L_{a\,\alpha(b)} = \ell_{a\alpha(b)} , $$

so the signed left multiplications compose among themselves and the family is closed under composition. The signed right multiplications compose by $\varrho_b\varrho_a = R_{\alpha(b)a}$, and a signed left and a signed right multiplication do not commute in general.

Proof. $\ell_a\ell_b(x) = a\alpha(b\alpha(x)) = a\alpha(b)\alpha(\alpha(x)) = a\alpha(b)x = L_{a\alpha(b)}(x)$; the right statement is the mirror image. For the failure of commutation, $\ell_a\varrho_b(x) = a\alpha(\alpha(x)b) = ax\alpha(b)$ and $\varrho_b\ell_a(x) = \alpha(a\alpha(x))b = \alpha(a)xb$, which differ in general.

Theorem (the signed left multiplication group). The signed left multiplications with unit parameter form a group $\ell(R^\times)$ under composition, isomorphic to $R^\times$ itself when the grade involution is not inner, and the map $R^\times \to \ell(R^\times)$, $u \mapsto \ell_u$, is a homomorphism with kernel $\{u : c_u = \alpha\}$. The grade involution normalises the group, $\alpha\ell_u\alpha = \ell_{\alpha(u)}$, so the group generated by $\ell(R^\times)$ and $\alpha$ is a semidirect product of $\ell(R^\times)$ with the two-element group $\{\mathrm{id}, \alpha\}$.

Proof. The composition law gives $\ell_u\ell_v = \ell_{u\alpha(v)}$, which is the group law of the image; for a unit $u$ the operator $\ell_u$ is invertible with $\ell_u^{-1} = \ell_{\alpha(u)^{-1}}$, so the image is a subgroup of the automorphism group. The map $u \mapsto \ell_u$ is injective exactly when $\ell_u = \mathrm{id}$ has no solution but $u = 1$, which is $u\alpha(x) = x$ for all $x$, that is $\alpha = c_u$; so the kernel is $\{u : c_u = \alpha\}$, trivial when $\alpha$ is not inner. Finally $\alpha\ell_u\alpha(x) = \alpha(u\alpha(\alpha(x))) = \alpha(ux) = \alpha(u)x = \ell_{\alpha(u)}(x)$, so $\ell(R^\times)$ is normalised by $\alpha$ and the generated group is a semidirect product.

Corollary (the signed left multiplication is the sandwich with a fixed side). The signed left multiplications are exactly the signed sandwiches whose right parameter is $1$, and the signed sandwich group surjects onto the signed left multiplication group by $(a, b) \mapsto a$; the signed left multiplication group is therefore a quotient of the signed sandwich group, with kernel the signed right multiplications. A signed left and a signed right multiplication compose to the unsigned sandwiches

$$ \ell_a\circ\varrho_b = \Sigma_{a,\,\alpha(b)} , \qquad \varrho_b\circ\ell_a = \Sigma_{\alpha(a),\,b} , $$

so the two-sided signed sandwich is not the product of a signed left and a signed right multiplication.

Proof. The identification $\ell_a = \Sigma^\alpha_{a,1}$ is the decomposition; the projection $(a,b)\mapsto a$ is a homomorphism of the sandwich group onto the left group by the composition law, and its kernel consists of the sandwiches with left parameter $1$, which are the signed right multiplications. For the products, $\ell_a\varrho_b(x) = a\alpha(\alpha(x)b) = ax\alpha(b)$ and $\varrho_b\ell_a(x) = \alpha(a\alpha(x))b = \alpha(a)xb$, which are the unsigned sandwiches displayed.

The Fixed Elements

Proposition (the fixed elements of a signed left multiplication). For $a \in R$ the fixed set of $\ell_a$ is

$$ R^{\ell_a} = \{ x : a\,\alpha(x) = x \} = \{ x : \alpha(x) = a^{-1}x \} $$

when $a$ is a unit, and it is the set of solutions of the linear equation $a\alpha(x) = x$ in general; it is closed, being the equalizer of the continuous maps $\ell_a$ and $\mathrm{id}$, and it contains $0$.

Proof. The equation $a\alpha(x) = x$ is the definition of the fixed set; it is closed as the equalizer of two continuous maps; for a unit $a$ the equation is equivalent to $\alpha(x) = a^{-1}x$. The set contains $0$ because $\ell_a$ is additive. It is a subring only in special cases: $\ell_a$ is a homomorphism only when $a$ is central and fixed by $\alpha$, and then the fixed set is the fixed set of the composite automorphism.

Corollary (the fixed subring of the grade involution). The signed left multiplication $\ell_1 = \alpha$ has fixed set the fixed subring $R^\alpha$; the unsigned left multiplication $L_1 = \mathrm{id}$ fixes everything; and the fixed set of $\ell_a$ contains $R^\alpha$ exactly when $a$ acts trivially on $R^\alpha$, that is when $a x = x$ for all $x \in R^\alpha$.

Proof. $\ell_1 = \alpha$ has fixed set $\{x : \alpha(x) = x\} = R^\alpha$; the identity fixes all of $R$; an element $x \in R^\alpha$ satisfies $a\alpha(x) = ax$, which equals $x$ for all such $x$ exactly when $a$ fixes $R^\alpha$ pointwise.

The Relation to the Unsigned Left Multiplication

Proposition (the two families differ by the grade involution). The signed left multiplications are the composite of the unsigned left multiplications with the grade involution,

$$ \ell_a = L_a\circ\alpha , \qquad \varrho_b = \alpha\circ R_b , $$

so the signed family is the coset $L(R)\alpha$ and the unsigned family is the subgroup $L(R)$ of the automorphism group of the additive group. The signed and unsigned left multiplications of a unit coincide only when $\alpha$ stabilises the corresponding operator, $\ell_u = L_u$ exactly when $L_u(\alpha(x) - x) = 0$ for all $x$, that is when $\alpha = \mathrm{id}$ on the relevant part.

Proof. The first identity is the decomposition; the second distinguishes the families: an unsigned left multiplication is $L_u$, a signed one is $L_u\alpha$, and they are equal exactly when $u\alpha(x) = ux$ for all $x$, that is $\alpha(x) = x$ for all $x$ in the range of $L_u^{-1}$, which is all of $R$ when $u$ is a unit, so the equality forces $\alpha = \mathrm{id}$.

Theorem (the degenerate case: the inner grade involution). When the grade involution is inner, $\alpha = c_z$ with $z \in R^\times$, the signed left multiplication is an unsigned two-sided sandwich,

$$ \ell_a(x) = a z x z^{-1} = \Sigma_{az,\,z^{-1}}(x) , $$

so the signed left multiplications are the unsigned sandwiches with second parameter fixed at $z^{-1}$, and the signed family is contained in the unsigned sandwich family rather than in the unsigned left family. When the ring is commutative, $\ell_a(x) = a\alpha(x) = \alpha(a x) = \alpha L_a(x)$, so the signed and unsigned left multiplications differ by the fixed automorphism $\alpha$ and the signed left multiplication group is abelian.

Proof. For $\alpha = c_z$, $\ell_a(x) = azxz^{-1}$, which is the unsigned sandwich $\Sigma_{az,z^{-1}}(x)$; the signed family is therefore contained in the unsigned sandwich family with second parameter $z^{-1}$. In a commutative ring $\ell_a = \alpha L_a$ because $\alpha$ is a ring automorphism and $a$ is central; the group is abelian because the ring is.

Remark (the boundary to the graded action and the adjoint). The signed left multiplication $\ell_a = a\alpha(\,\cdot\,)$ is the operator of a graded module action twisted by the grading involution, and the full compatibility of such an action with the grading is The Graded Action on a Module over a Topological Ring, later in this group. The adjoint of the signed left multiplication with respect to the form of the category, and its explicit expression in terms of the signed sandwich, are The Signed Adjoint of the Left Multiplication on a Topological Ring, later in this category; this article fixes the operator and its one-sided calculus, which the adjoint article uses.

Examples

Example (the polynomial ring with $f(x)\mapsto f(-x)$). On $k[x]$ with the grade involution $\alpha(f)(x) = f(-x)$, the signed left multiplication by $a = x$ is $\ell_x(f) = x f(-x)$; the ring is commutative, so $\ell_x = \alpha L_x$ and the signed left multiplication is the sign change followed by the unsigned multiplication; the fixed set of $\ell_x$ is $\{f : x f(-x) = f\}$, a closed subset.

Example (the Clifford algebra). In $Cl(V,q)$ with the grade involution, the signed left multiplication by a vector $v$ is $\ell_v(x) = v\alpha(x)$; for $x = v$ it gives $v\alpha(v) = -q(v)$, so $\ell_v$ inverts the vector $v$ up to the scalar $-q(v)$, which is the statement that the vector acts on itself by the signed left multiplication.

Example (the matrix ring). On $M_n(k)$ with $\alpha(A) = A^{\top}$, the signed left multiplication by $A$ is $\ell_A(X) = AX^{\top}$; it is invertible for invertible $A$ with inverse $\ell_{(A^{\top})^{-1}}$, and the signed left multiplication group is isomorphic to $GL_n(k)$.

Example (the inner grade involution). On $M_n(k)\times M_n(k)$ with $\alpha(A,B) = (B,A)$ and the two factors conjugate, $\alpha = c_z$, and $\ell_a = \Sigma_{az,z^{-1}}$ is a two-sided sandwich; the signed left multiplication is genuinely two-sided in the degenerate case, which shows that the one-sided signed calculus is only one-sided when the grade involution is not inner.

Summary

The signed left multiplication of a topological ring with a continuous grade involution $\alpha$ is the operator $\ell_a(x) = a\alpha(x)$, equal to $L_a\circ\alpha$ and to the signed sandwich $\Sigma^\alpha_{a,1}$; it is additive and continuous, it composes by $\ell_a\ell_b = \ell_{a\alpha(b)}$, and the signed left multiplications with unit parameter form a group $\ell(R^\times)$ that is a homomorphic image of $R^\times$ with kernel the units inducing the grade involution, $\{u : c_u = \alpha\}$. The fixed set of $\ell_a$ is the closed set $\{x : a\alpha(x) = x\}$; it is the fixed subring of the grade involution for $a = 1$, and it contains $R^\alpha$ exactly when $a$ fixes $R^\alpha$ pointwise. The signed right multiplications are the mirror image, $\varrho_b = \alpha\circ R_b$, the signed left group is a quotient of the signed sandwich group with kernel the signed right multiplications, and the two-sided sandwich is the product of a signed left and a signed right multiplication in either order.

The signed and unsigned left multiplications differ by the grade involution, $\ell_a = L_a\alpha$, and they coincide only when $\alpha$ is trivial; when the grade involution is inner, $\alpha = c_z$, the signed left multiplication becomes the two-sided sandwich $\Sigma_{az,z^{-1}}$, so the one-sided signed calculus is one-sided exactly when the grade involution is not inner, and in the commutative case $\ell_a = \alpha L_a$ and the signed left group is abelian. The graded module action and the adjoint of the signed left multiplication are the neighbouring articles.

Summary of Notation

Symbol	Meaning
$\alpha$	The continuous grade involution
$L_a(x) = ax$, $R_a(x) = xa$	The unsigned one-sided multiplications
$\ell_a(x) = a\alpha(x)$	The signed left multiplication
$\varrho_b(x) = \alpha(x)b$	The signed right multiplication
$\ell_a = L_a\alpha = \Sigma^\alpha_{a,1}$	Decomposition of the signed left multiplication
$\ell_a\ell_b = \ell_{a\alpha(b)}$	The composition law
$\ell_u^{-1} = \ell_{\alpha(u)^{-1}}$	The inverse for a unit
$\ell(R^\times)$	The signed left multiplication group
$\{u : c_u = \alpha\}$	The kernel of $u \mapsto \ell_u$
$R^{\ell_a} = \{x : a\alpha(x) = x\}$	The closed fixed set
$\alpha = c_z$	The degenerate inner grade involution