The Left and Right Multiplication Operators on an Ordered Algebra

Introduction

An ordered algebra is an algebra whose underlying vector space is ordered by a cone that the multiplication respects: the product of two positive elements is positive. This is the structure on which every one-sided operator of the category is written. The left multiplication $L_a$ and the right multiplication $R_a$,

$$ L_a x = ax, \qquad R_a x = xa , $$

are the elementary operators of the structure: they are linear in $x$, they are positive as soon as $a$ is positive, they commute with each other by associativity, and together they produce the multiplication algebra of $A$. The assignment $a\mapsto L_a$ is the left regular representation; it is an algebra homomorphism, it is injective when $A$ is unital, and — this is the point of the ordered setting — it is an order isomorphism onto its image, because a left multiplication by $a$ is positive exactly when $a$ is positive, the identity being available to evaluate it.

The article states these facts and organises the multiplication algebra. It is the base of the operator theory of the category: the sandwich articles add a second parameter and a grade involution to the two-sided multiplication, the reflection articles read the reflections of the algebra as two-sided operators, the signed left multiplication article twists this one-sided action by the grade involution, and the graded-action article carries the action to a module. The order and the order unit are Ordered Vector Spaces and the Order Unit; the positivity and the operator order are Positive Operators on an Ordered Space and The Cone of Positive Operators; the function algebra $C(X)$ and the matrix algebra are the instances of Jordan Algebras and the Positive Cone and of the algebra articles of Part II; the sandwich operator of an unordered algebra is The Sandwich Operator on an Algebra, and the graded structures and the grade involution are Superalgebras and Graded Structures, both of Part I. The involution on the algebra is Ordered Involutive Algebras later in this category, and the adjoints of the one-sided multiplications are the * Operator Theory articles at the end of the category.

Ordered Algebras

The Structure

Definition. An ordered algebra over $\mathbb{R}$ is an associative algebra $A$ together with a positive cone $A_+$ that is a cone of the underlying ordered vector space and is closed under multiplication:

$$ A_+\cdot A_+\subseteq A_+ . $$

It is unital when it has an identity $1$, and an ordered algebra with order unit when $1$ is an order unit of the underlying ordered vector space. The order is $a\leq b$ when $b-a\in A_+$, and the algebra is Archimedean when the underlying order is.

Proposition (the order is compatible with the multiplication). In an ordered algebra the multiplication is positive bilinear: for $a,b\geq0$ and $x\leq y$,

$$ ax\leq ay, \qquad xa\leq ya, \qquad ab\geq0 , $$

and more generally $a\leq b$ implies $ax\leq bx$ and $xa\leq xb$ for $x\geq0$.

Proof. The cone is closed under products, so $a(y-x)\in A_+$ and $(y-x)a\in A_+$ when $a\geq0$ and $y-x\geq0$; this is the two inequalities, and $ab\in A_+$ for positive $a,b$ is the defining closure.

Example. The algebra $C_{\mathbb{R}}(X)$ of continuous real functions on a compact space, with the cone of nonnegative functions, is a commutative ordered algebra with order unit $1$. The matrix algebra $M_n(\mathbb{R})$ with the Loewner cone of positive semidefinite matrices is an ordered algebra, because a product of two positive semidefinite matrices is not symmetric in general but the symmetrised product cone is the one that is closed; the correct ordered algebra in the matrix case is therefore the Jordan algebra of Jordan Algebras and the Positive Cone, and this is the reason the category separates the associative and the Jordan orders. The polynomials with a real positivity cone are not an ordered algebra in an obvious way, which is one of the reasons the theory is carried by the function and the matrix instances.

The One-Sided Multiplications

Definition. For $a\in A$ the left multiplication and the right multiplication are the operators

$$ L_a : A\to A, \quad L_a x = ax , \qquad R_a : A\to A, \quad R_a x = xa . $$

Proposition (linearity and positivity). $L_a$ and $R_a$ are linear in $x$. In a unital ordered algebra, for every $a$,

$$ L_a\geq0 \iff a\geq0 \iff R_a\geq0 , $$

so the maps $a\mapsto L_a$ and $a\mapsto R_a$ are order isomorphisms from $A$ onto their images, and each is an isomorphism of ordered vector spaces when $A$ is Archimedean.

Proof. Linearity is the distributivity and the compatibility of the scalars. If $a\geq0$ then $ax\geq0$ and $xa\geq0$ for $x\geq0$ by the positive bilinearity; conversely $L_a\geq0$ evaluated at $1\geq0$ gives $a = L_a1\geq0$, and $R_a\geq0$ gives $a = R_a1\geq0$. The order isomorphism statement is the two implications, and the Archimedean statement is the remark that no further relation is imposed on the cone.

Example (the commutative case). When $A$ is commutative, $L_a = R_a$ for every $a$, and the one-sided multiplications are the single multiplication operators of the algebra. For $A = C(X)$ the operator $L_f = R_f$ is the multiplication by $f$, positive exactly when $f\geq0$, which is the operator of Multiplication Operators on an $L^1$ Function of Part II read on the continuous functions.

The Regular Representation and the Commutation

Theorem (the left and right regular representations). The map

$$ L : A\to L(A), \qquad L(a) = L_a , $$

is an injective algebra homomorphism in a unital algebra,

$$ L_aL_b = L_{ab}, \qquad L_{a+b} = L_a+L_b, \qquad L_{\lambda a} = \lambda L_a, \qquad L_1 = I , $$

the map $R : a\mapsto R_a$ is an injective anti-homomorphism,

$$ R_aR_b = R_{ba} , $$

and the two images commute:

$$ L_aR_b = R_bL_a \qquad \text{for all } a,b\in A , $$

because both sides send $x$ to $axb$.

Proof. For the composition, $L_aL_bx = a(bx) = (ab)x = L_{ab}x$; the additive and scalar claims are the bilinearity of the multiplication; $L_1 = I$ is the identity axiom. For the anti-homomorphism, $R_aR_bx = (xb)a = x(ba) = R_{ba}x$. The commutation is $a(xb) = (ax)b$, which is associativity.

Corollary (the multiplication algebra). The algebra generated by the $L_a$ and the $R_a$ is the image of the homomorphism

$$ A\otimes A^{\mathrm{op}} \to L(A), \qquad a\otimes b\mapsto L_aR_b , $$

it contains the identity when $A$ is unital, and it is the multiplication algebra of $A$. When $A$ is commutative it is the image of $A\otimes A$, and when $A$ is a division algebra it is a tensor product of two copies of the structure.

Proof. The stated map is well defined and is a homomorphism by the two representation theorems and the commutation; its image is generated by the two families by definition; the remaining statements follow from the identification of the opposite algebra in the commutative case.

The Multipliers

Definition. A double centralizer, or multiplier, of $A$ is a pair $(S,T)$ of linear maps with

$$ x\,S(y) = T(x)\,y \qquad \text{for all } x,y\in A . $$

Proposition. For every $a\in A$ the pair $(L_a,R_a)$ is a double centralizer, and the map $a\mapsto(L_a,R_a)$ is injective in a unital algebra; the double centralizers form an algebra under the componentwise operations, containing the image of $A$ as a two-sided ideal, the multiplier algebra $M(A)$.

Proof. The centralizer equation for $(L_a,R_a)$ is $x(ay) = (xa)y$, which is associativity; injectivity is $L_a1 = a$. The componentwise operations make the centralizers an algebra, and for $a\in A$ and a centralizer $(S,T)$ the products $L_a(S,T) = (L_aS, \cdot)$ and $(S,T)L_a$ are again centralizers, so the image of $A$ is an ideal.

Proposition (positivity of the multipliers). In a unital ordered algebra a double centralizer $(S,T)$ with $S\geq0$ and $T\geq0$ is a positive pair, and the multiplier algebra is ordered by the cone of the positive pairs; when $A$ is an ordered algebra with order unit and the multipliers are bounded for the order-unit norm, $M(A)$ is an ordered algebra with order unit $(L_1,R_1) = (I,I)$.

Proof. The positivity of the pair is the positivity of its two components, and the operations of the multiplier algebra preserve it by the positive bilinearity of the multiplication; the order unit statement is that $I\geq0$ and that every centralizer is between multiples of $(I,I)$ when it is bounded, which is the order-unit estimate of Positive Operators on an Ordered Space.

The Order and the Norm

The Regular Representation is an Order Isomorphism

Theorem. In a unital ordered algebra the left regular representation is an isomorphism of ordered algebras from $A$ onto $L(A)$ with the pointwise order: it preserves the order, the products and the identity in both directions. The same holds for the right regular representation onto $R(A)$ with the anti-multiplication.

Proof. The order statement is the proposition on positivity: $a\geq0$ iff $L_a\geq0$. The product and identity statements are the representation theorem.

The Operator Norm of a Multiplication

Definition. On a unital ordered algebra define

$$ \lVert a\rVert_L = \lVert L_a\rVert , $$

the operator norm of the left multiplication for the order-unit norm, when the latter is finite.

Proposition. The assignment $a\mapsto\lVert a\rVert_L$ is a submultiplicative algebra norm,

$$ \lVert ab\rVert_L\leq\lVert a\rVert_L\,\lVert b\rVert_L, \qquad \lVert 1\rVert_L = 1 , $$

and the left regular representation is isometric for it. In an ordered Banach algebra with a monotone norm the two norms agree on the positive cone.

Proof. Submultiplicativity is $\lVert L_{ab}\rVert = \lVert L_aL_b\rVert\leq\lVert L_a\rVert\lVert L_b\rVert$ and the identity is $\lVert L_1\rVert = \lVert I\rVert = 1$. Isometry is the definition. Monotonicity of the norm gives $\lVert a\rVert = \lVert L_a1\rVert\leq\lVert L_a\rVert$ and the reverse follows from the order-unit estimate, so the two agree.

Worked Cases

The Continuous Functions

For $A = C(X)$ with $X$ compact every multiplication is commutative, $L_f = R_f$ is the multiplication operator by $f$, and the left regular representation is the isometric isomorphism of $C(X)$ onto the algebra of multiplication operators on itself. The algebra is unital, so its multiplier algebra is itself; the double centralizer condition is automatic by commutativity, and a multiplier is positive exactly when its function is nonnegative.

The Triangular Matrices

For $A = T_n(\mathbb{R})$ the upper triangular matrices with the cone of entrywise nonnegative matrices, the left and right multiplications by a nonnegative matrix are positive, and they do not commute with each other except through the centre, which for $T_n$ is the span of the identity. The multiplier algebra is $T_n$ itself, since the algebra is unital. This is the smallest noncommutative example in which $L(A)$ and $R(A)$ are genuinely different, and it shows that the commutation $L_aR_b = R_bL_a$ is a statement about the two families and not about the elements.

The Matrix Algebra Revisited

For $A = M_n(\mathbb{R})$ with the cone of symmetric positive semidefinite matrices inside the self-adjoint part, the left and right multiplications preserve the cone of $A$ when $a$ is a positive semidefinite matrix, but the product of two positive semidefinite matrices need not be self-adjoint; the ordered algebra of the associative product is therefore the symmetrised structure, and the one-sided multiplications of the associative algebra are used together with the Jordan product of Jordan Algebras and the Positive Cone to recover the positive cone. This is the reason the later articles of the category distinguish the one-sided from the sandwich action.

Summary

An ordered algebra is an associative algebra whose cone is closed under multiplication, so that the multiplication is positive bilinear and the order is compatible with it. The left and right multiplications $L_a x = ax$ and $R_a x = xa$ are linear, they are positive exactly when $a$ is positive in a unital ordered algebra, so that $a\mapsto L_a$ and $a\mapsto R_a$ are order isomorphisms onto their images. The left multiplication is an algebra homomorphism $L_{ab} = L_aL_b$ with $L_1 = I$, the right multiplication is an anti-homomorphism $R_{ab} = R_bR_a$, and the two families commute, $L_aR_b = R_bL_a$, by associativity. They generate the multiplication algebra, the image of $A\otimes A^{\mathrm{op}}$, and the double centralizers extend it to the multiplier algebra. The operator norm $\lVert a\rVert_L = \lVert L_a\rVert$ is a submultiplicative algebra norm for which the left regular representation is isometric. The order and the order unit are Ordered Vector Spaces and the Order Unit; the positivity and the operator order are Positive Operators on an Ordered Space and The Cone of Positive Operators; the sandwich operator of an unordered algebra is The Sandwich Operator on an Algebra; the graded structures are Superalgebras and Graded Structures; and the involution version and the adjoints are Ordered Involutive Algebras and the * Operator Theory articles of this category.

Summary of Notation

Symbol	Meaning
$A_+\cdot A_+\subseteq A_+$	Defining closure of the cone of an ordered algebra
$L_a x = ax$, $R_a x = xa$	Left and right multiplication operators
$L_a\geq0\iff a\geq0$	Positivity of the one-sided multiplications in a unital algebra
$L_{ab} = L_aL_b$, $R_{ab} = R_bR_a$	Regular representations, homomorphism and anti-homomorphism
$L_aR_b = R_bL_a$	Commutation by associativity
$A\otimes A^{\mathrm{op}}$	Multiplication algebra of the one-sided multiplications
$(S,T)$, $xS(y) = T(x)y$	Double centralizer, multiplier
$M(A)$	Multiplier algebra
$\lVert a\rVert_L = \lVert L_a\rVert$	Operator norm of the left multiplication