Multipliers of a Banach Algebra

Introduction

The multiplications of a Banach algebra $A$ act on $A$ from the left and from the right, and an operator that acts like a multiplication without coming from an element of $A$ is a multiplier. In its cleanest form a multiplier is a double centraliser: a pair $(L,R)$ of bounded operators with $L(ab) = L(a)b$, $R(ab) = aR(b)$ and $aL(b) = R(a)b$. On a unital algebra there are no others, and the multipliers are exactly the multiplications; on a non-unital algebra they are genuinely more, and they form the multiplier algebra $M(A)$, a unital Banach algebra in which $A$ sits as an essential two-sided ideal. The multiplier algebra carries a natural locally convex topology, the strict topology, in which $A$ is dense and $M(A)$ complete, so that $M(A)$ is the completion of $A$ in the strict topology and the unitisation when $A$ has a bounded approximate identity. This article develops that construction: the double centralisers and their algebra, the multiplier algebra and its completeness, the strict topology and the density of $A$, and the relation to the double centraliser theorem of the one-sided multiplications.

The article assumes the Banach algebra, its norm, the unit group and the Banach-algebra structure from Topological Algebras and Banach Algebras; the bounded operators, the one-sided multiplications, the multiplier definitions, the separating space and the automatic continuity from Operators on a Banach Algebra, the first article of this group; the composition and centraliser theorem of the one-sided multiplications and the multiplication algebra from Left and Right Multiplication in a Banach Algebra; the closed ideals, the quotients and the unitisation from Ideals and Quotients of Algebras; and the bounded operators of a Banach space, the closed ideals and the compact operators from The Operator Algebra of a Banach Space. The involution on the elements and the multiplier algebra of a $\mathrm{C}^*$-algebra are the - * Theory and - * Operator Theory groups of this category and are not used; the strict topology of a Hilbert module and the von Neumann algebra are Operator Algebras. No form, no measure and no Fourier theory occurs.

Throughout, $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$; $A$ is a Banach algebra over $\mathbb{K}$ with submultiplicative norm $\lVert\cdot\rVert$, not assumed unital; $B(A)$ is the Banach algebra of bounded linear operators. A left multiplier is a bounded $L$ with $L(ab) = L(a)b$, a right multiplier a bounded $R$ with $R(ab) = aR(b)$, and a double centraliser a pair $(L,R)$ of a left and a right multiplier with $aL(b) = R(a)b$ for all $a,b$. The unitisation of $A$ is written $A^\sharp = A \oplus \mathbb{K}$.

Double Centralisers

Definition. A double centraliser on $A$ is a pair $(L,R)$ of bounded linear operators on $A$ with

$$ L(ab) = L(a)\,b , \qquad R(ab) = a\,R(b) , \qquad a\,L(b) = R(a)\,b \qquad \text{for all } a,b \in A . $$

The set of double centralisers is written $M(A)$, the multiplier algebra of $A$.

Proposition (the algebra structure). With the operations

$$ (L_1,R_1) + (L_2,R_2) = (L_1 + L_2,\,R_1 + R_2) , \qquad (L_1,R_1)(L_2,R_2) = (L_1L_2,\,R_2R_1) , \qquad \lambda(L,R) = (\lambda L,\,\lambda R) , $$

the set $M(A)$ is a unital associative algebra with identity $(\mathrm{id},\mathrm{id})$, and the two components determine each other on an algebra with $A^2 = A$.

Proof. The sum of two double centralisers is a double centraliser, and the product $(L_1L_2,R_2R_1)$ is one: $L_1L_2(ab) = L_1(L_2(a)b) = L_1L_2(a)b$, $R_2R_1(ab) = aR_2R_1(b)$, and $aL_1L_2(b) = aL_1(L_2(b))$ while $R_2R_1(a)b = R_2(R_1(a))b$; the two agree by the centralising relations applied twice. Associativity is the associativity of composition, and $(\mathrm{id},\mathrm{id})$ is the identity. The components determine each other when $A^2 = A$ because $aL(b) = R(a)b$ for all $a,b$ forces $L$ to determine $R$ on the ideal $A^2$ and $R$ to determine $L$. $\square$

Theorem ($M(A)$ is a Banach algebra). Give $M(A)$ the norm

$$ \lVert(L,R)\rVert = \max\{\lVert L\rVert,\lVert R\rVert\} , $$

where $\lVert L\rVert,\lVert R\rVert$ are the operator norms. Then $M(A)$ is a unital Banach algebra under the product above, the maps $(L,R) \mapsto L$ and $(L,R) \mapsto R$ are contractions, and $\lVert(L,R)\rVert = \lVert L\rVert = \lVert R\rVert$ on the double centralisers of a faithful algebra.

Proof. The norm is a norm because it is the max of two norms, and it is submultiplicative: $\lVert L_1L_2\rVert \leq \lVert L_1\rVert\lVert L_2\rVert$ and $\lVert R_2R_1\rVert \leq \lVert R_2\rVert\lVert R_1\rVert$, so $\lVert(L_1,R_1)(L_2,R_2)\rVert \leq \lVert(L_1,R_1)\rVert\lVert(L_2,R_2)\rVert$. Completeness: a Cauchy sequence of double centralisers has Cauchy components in the complete spaces $B(A)$, with limits $L,R$; the defining relations pass to the limit by continuity of the products, so the limit is a double centraliser, and the norm converges. The identity has norm one. The equality of the norms on a faithful algebra is the standard consequence of $aL(b) = R(a)b$. $\square$

Proposition (the canonical embedding). The map

$$ \iota : A \longrightarrow M(A) , \qquad \iota(a) = (L_a,R_a) , $$

is an isometric algebra homomorphism when $A$ is unital and a bounded algebra homomorphism in general, with image $\iota(A)$ a two-sided ideal of $M(A)$; it is injective on a faithful algebra and, in particular, on a unital one. The identity of $M(A)$ lies in $\iota(A)$ exactly when $A$ is unital, and then $\iota$ is onto.

Proof. $(L_a,R_a)$ is a double centraliser by Operators on a Banach Algebra, and $\iota(ab) = (L_{ab},R_{ab}) = (L_aL_b,R_bR_a) = \iota(a)\iota(b)$ by the composition laws, so $\iota$ is a homomorphism. On a unital algebra $\lVert L_a\rVert = \lVert R_a\rVert = \lVert a\rVert$, giving the isometry, and $\iota$ is onto by the unital identification of the multipliers. That $\iota(A)$ is an ideal: for a double centraliser $(L,R)$ and $a \in A$, $(L,R)(L_a,R_a) = (LL_a, R_aR)$, and $LL_a = L_{L(a)}$, $R_aR = R_{R(a)}$ by the multiplier identities, so the product lies in $\iota(A)$, and likewise on the left. $\square$

Corollary (the unital case). If $A$ is unital then $M(A) = \iota(A) \cong A$, the multiplier algebra is the algebra itself, and the construction adds nothing. The multipliers are genuinely new only for a non-unital algebra.

Proof. On a unital algebra every left multiplier is $L_{L(1)}$ and every right multiplier is $R_{R(1)}$, and the centralising relation forces $L(1) = R(1)$; hence every double centraliser is $\iota(c)$ with $c = L(1)$, and $\iota$ is a bijection. $\square$

The Multiplier Algebra

Theorem (the multiplier algebra is unital with essential ideal $A$). Let $A$ be a Banach algebra with $A^2 = A$ (in particular a non-unital algebra with a bounded approximate identity). Then $\iota(A)$ is a two-sided ideal of $M(A)$ with $\iota(A)^2 = \iota(A)$, the algebra $M(A)$ is unital, and $\iota(A)$ is essential: a double centraliser $(L,R)$ with $(L,R)\iota(A) = 0$ is zero, and the same on the other side.

Proof. $\iota(A)$ is an ideal by the proposition, and it is essential because $(L,R)\iota(a) = 0$ for all $a$ gives $(LL_a, R_aR) = 0$, so $LL_a = 0$ for all $a$ and $L = 0$ on $A^2 = A$, that is $L = 0$; then $R = 0$ by the centralising relation on a faithful algebra. The unit is $(\mathrm{id},\mathrm{id})$. $\square$

Theorem (the multiplier algebra as the double centraliser algebra). Let $A$ be a unital Banach algebra and let $\lambda(A)$ and $\rho(A)$ be the two regular representations in $B(A)$. Then the double centraliser algebra of the pair $(\lambda(A),\rho(A))$ is $\lambda(A)' = \rho(A)$ and $\rho(A)' = \lambda(A)$, and $M(A) = \iota(A)$. In the non-unital case, if the algebra is faithful and has a bounded approximate identity, then

$$ M(A) \cong \{\,T \in B(A) : T\lambda(A) \subseteq \lambda(A),\ \lambda(A)T \subseteq \lambda(A),\ T\rho(A) \subseteq \rho(A),\ \rho(A)T \subseteq \rho(A)\,\} , $$

the operators normalising both regular representations, and the isomorphism pairs $T$ with the double centraliser of the two inclusions.

Proof. The centraliser identities in the unital case are Left and Right Multiplication in a Banach Algebra, and the identification with $\iota(A)$ is the corollary above. In the non-unital case a double centraliser $(L,R)$ defines, through the inclusions $L\lambda(a) = \lambda(L(a))$ and $\rho(a)R = \rho(R(a))$, an operator normalising both families, and conversely such an operator gives a double centraliser by its actions; the correspondence is bijective on a faithful algebra with an approximate identity. $\square$

Remark (the boundary). The identification of the multiplier algebra with the operators normalising the regular representations is the operator form of the double centraliser theorem; the abstract form is in Left and Right Multiplication in a Banach Algebra. The multiplier algebra of a $\mathrm{C}^*$-algebra is again a $\mathrm{C}^*$-algebra, and the multiplier algebra of a Hilbert $\mathrm{C}^*$-module is the source of the strongly continuous strict topology, both owned by the involutive groups of this category and by Operator Algebras.

The Strict Topology

Definition. The strict topology on $M(A)$ is the locally convex topology generated by the seminorms

$$ (L,R) \longmapsto \lVert L(a)\rVert , \qquad (L,R) \longmapsto \lVert R(a)\rVert , \qquad a \in A . $$

The strict topology on $A$ is the topology generated by the seminorms $x \mapsto \lVert ax\rVert$ and $x \mapsto \lVert xa\rVert$ for $a \in A$, pulled back along $\iota$.

Proposition (the strict topology is well defined and the multiplications are strictly continuous). The family of seminorms above separates the points of $M(A)$ on a faithful algebra with $A^2 = A$, so the strict topology is Hausdorff; for each fixed $a$ the maps $(L,R) \mapsto L(a)$ and $(L,R) \mapsto R(a)$ are strictly continuous, and the product of $M(A)$ is strictly continuous in each variable separately.

Proof. If all seminorms vanish then $L(a) = 0$ for every $a$, so $L = 0$, and $R = 0$ by faithfulness; hence the topology is Hausdorff. Continuity of evaluation is the definition of the seminorms. For the product in the first variable: $((L_1,R_1)(L,R))(a) = L_1L(a) = L_1(L(a))$, which depends strictly continuously on $(L,R)$ through $L(a)$ and then on $L_1$ through its value at $L(a)$; the second variable is the mirror. $\square$

Theorem ($A$ is strictly dense in $M(A)$). Let $A$ be a Banach algebra with a bounded approximate identity $(u_\lambda)$. Then $\iota(A)$ is dense in $M(A)$ in the strict topology, and $M(A)$ is complete in the strict topology; consequently $M(A)$ is the strict completion of $A$ and the identity of $M(A)$ is the strict limit of the approximate identity.

Proof. For a double centraliser $(L,R)$ and $a \in A$ one has $L(u_\lambda a) = L(u_\lambda)a \to L(a)$ and $L(u_\lambda)$ is a bounded net in $A$, so $\iota(L(u_\lambda)) \to (L,R)$ in the seminorms generated by $L$, using the boundedness of $L$; the right component is handled by $R(u_\lambda)$. Hence $\iota(A)$ is strictly dense. Completeness is the standard statement that a strict Cauchy net has strictly convergent left and right components by completeness of $B(A)$ and the boundedness of the approximate identity; the limit double centraliser is $(L,R)$, and the identity is the limit of $\iota(u_\lambda)$. $\square$

Corollary (the strict topology and the norm topology). The strict topology is coarser than the norm topology on $M(A)$, and it agrees with the norm topology on a unital algebra; on the unit ball of $M(A)$ the strict topology is the topology of pointwise convergence on the image of $\iota$, and the closed unit ball is strictly bounded but need not be strictly compact.

Proof. The seminorms are dominated by the norm, since $\lVert L(a)\rVert \leq \lVert L\rVert\lVert a\rVert$, so strict is coarser; on a unital algebra $a = 1$ gives $\lVert L\rVert \leq \lVert L(1)\rVert$ and the two topologies agree. The ball statement is the definition of the seminorms. $\square$

Remark (bounded approximate identities). The existence of a bounded approximate identity is a hypothesis, not automatic for a Banach algebra; it holds for $c_0$, for $K(H)$, for $C_0(X)$ and for the group algebra $\ell^1(G)$ for an amenable $G$. The strict density theorem is stated under this hypothesis, and the multiplier algebra is defined in general.

Examples

Example ($c_0$ and $\ell^\infty$). Let $A = c_0$ with the sup norm; the approximate identity is the sequence of truncations, which is bounded. Every multiplier of $c_0$ is multiplication by a bounded sequence, so $M(c_0) = \ell^\infty$, the strict topology is the topology of pointwise convergence on the coordinates, and $c_0$ is strictly dense in $\ell^\infty$. The unit of $M(c_0)$ is the constant sequence $1$.

Example ($K(H)$ and $B(H)$). Let $A = K(H)$ be the compact operators on an infinite-dimensional Hilbert space. Its multiplier algebra is $M(K(H)) = B(H)$, the bounded operators, with the strict topology the strong-$*$ topology of pointwise convergence; $K(H)$ is strictly dense in $B(H)$, and the unit of $B(H)$ is the strict limit of the finite-rank projections. The double centraliser of $T \in B(H)$ is the pair of left and right multiplication by $T$.

Example ($C_0(X)$ and $C_b(X)$). Let $A = C_0(X)$ for a locally compact Hausdorff space $X$, with the sup norm. The multipliers are the multiplication operators by bounded continuous functions, $M(C_0(X)) = C_b(X)$, and the strict topology is that of uniform convergence on compact subsets of $X$; the unitisation of $C_0(X)$ is contained in $C_b(X)$ and is strictly dense.

Example (the unitisation). For any Banach algebra $A$, the unitisation $A^\sharp = A \oplus \mathbb{K}$ is a unital Banach algebra containing $A$ as an ideal of codimension one, and $M(A^\sharp) = A^\sharp$; the multiplier algebra of $A$ contains the unitisation when the latter contains $A$ as an essential ideal, and for $A$ with a bounded approximate identity the relation $M(A) = M(A^\sharp)$ holds.

Summary

A double centraliser on a Banach algebra $A$ is a pair $(L,R)$ of bounded operators with $L(ab) = L(a)b$, $R(ab) = aR(b)$ and $aL(b) = R(a)b$; the double centralisers form the multiplier algebra $M(A)$ under the product $(L_1,R_1)(L_2,R_2) = (L_1L_2,R_2R_1)$, a unital Banach algebra for the max norm, in which $A$ embeds by $a \mapsto (L_a,R_a)$ as a two-sided ideal, essential when $A^2 = A$ and faithful. On a unital algebra the embedding is an isomorphism $M(A) \cong A$, and the multipliers are exactly the multiplications; the construction is new only in the non-unital case. The multiplier algebra is the algebra of bounded operators normalising both regular representations, which is the operator form of the double centraliser theorem of Left and Right Multiplication in a Banach Algebra. The strict topology, generated by the seminorms $(L,R) \mapsto \lVert L(a)\rVert$ and $(L,R) \mapsto \lVert R(a)\rVert$, is Hausdorff and coarser than the norm topology and agrees with it in the unital case; when $A$ has a bounded approximate identity, $A$ is strictly dense in $M(A)$, the multiplier algebra is strictly complete, and it is the strict completion of $A$, with the unit as the strict limit of the approximate identity. The multiplier algebra of a $\mathrm{C}^*$-algebra and the strict topology of a Hilbert module are the involutive refinements, owned by the later groups of this category and by Operator Algebras.

Summary of Notation

Symbol	Meaning
$A$, $\lVert\cdot\rVert$	Banach algebra, not assumed unital; its norm
$L$, $R$	Left multiplier $L(ab)=L(a)b$; right multiplier $R(ab)=aR(b)$
$(L,R)$, $aL(b)=R(a)b$	A double centraliser
$M(A)$	The multiplier algebra of double centralisers
$(L_1,R_1)(L_2,R_2) = (L_1L_2,R_2R_1)$	Product in $M(A)$
$\lVert(L,R)\rVert = \max\{\lVert L\rVert,\lVert R\rVert\}$	The norm, submultiplicative and complete
$\iota : a \mapsto (L_a,R_a)$	The embedding of $A$ as an essential ideal
$(\mathrm{id},\mathrm{id})$	The unit of $M(A)$
$\lVert L(a)\rVert$, $\lVert R(a)\rVert$	The seminorms of the strict topology
$x \mapsto \lVert ax\rVert$, $\lVert xa\rVert$	The strict topology on $A$
$(u_\lambda)$	A bounded approximate identity, strictly convergent to the unit
$A^\sharp = A \oplus \mathbb{K}$	The unitisation