The Left and Right Multiplication Operators on a Complex Vector Space

Introduction

Let $V$ be a complex vector space and $E = \operatorname{End}_{\mathbb C}(V)$ its algebra of $\mathbb{C}$-linear endomorphisms. The two one-sided multiplications are the operators on $E$ $$ L_A(X) = AX, \qquad R_B(X) = XB \qquad (A, B \in E), $$ the left and the right multiplication of the algebra, and they are the basic operators of the operator layer of the category. They commute, $L_A R_B = R_B L_A$, and they carry the complex structure: on $E$ the scalar $i$ may be multiplied on the left or on the right, and because $i$ is central the two multiplications agree, so the complex structure of $E$ is a single operator $J_E = L_i = R_i$ that commutes with every one-sided multiplication. When a Hermitian form $h$ is chosen on $V$ the endomorphism algebra $E$ inherits the Hermitian trace form $\langle X, Y\rangle = \operatorname{tr}(X^{\dagger}Y)$ and the conjugate-linear involution $A \mapsto A^{\dagger}$; the two one-sided multiplications by a unitary endomorphism are isometries of the trace form, and the operators $U \mapsto L_U$ and $U \mapsto R_U$ are the two commuting representations of the unitary group by which the group acts on the operator algebra.

The article has three sections: the one-sided multiplications and the algebra they generate; the trace form of the operator algebra and the unitary one-sided operators; and the complex structure of the operator algebra and the complex-linear operators. The algebra of endomorphisms, the double centraliser theorem and the trace are Algebras of Endomorphisms; the Hermitian form, the unitary group and the involution are The Unitary and Symplectic Groups and Hermitian Geometry and the Unitary Group; the complex structure of a linear space is The Involution on a Complex Vector Space, later in this category. The adjoints of the one-sided multiplications for the trace form are The Adjoint of the Left Multiplication on a Complex Vector Space, in the * Operator Theory group. The signed variants are The Signed Sandwich on a Complex Vector Space and The Signed Left Multiplication on a Complex Vector Space, and the module-level variant is The Graded Action on a Module over a Complex Vector Space, the other articles of this group.

Throughout, $V$ is a finite-dimensional complex vector space of dimension $n$ with a positive-definite Hermitian form $h$, $E = \operatorname{End}_{\mathbb C}(V)$ is its endomorphism algebra of complex dimension $n^2$, $A^\dagger$ is the adjoint of $A$ for $h$, $J$ is the complex structure of $V$ (multiplication by $i$), $L_A, R_B$ are the one-sided multiplications on $E$, and $\langle X, Y\rangle = \operatorname{tr}(X^{\dagger}Y)$ is the Hermitian trace form of $E$.

The One-Sided Multiplications and the Algebra They Generate

Definition. For $A \in E$ the left multiplication and for $B \in E$ the right multiplication are the operators on $E$ $$ L_A(X) = AX, \qquad R_B(X) = XB . $$ The left and right multiplication maps are $L : E \to \operatorname{End}(E)$, $A \mapsto L_A$ and $R : E \to \operatorname{End}(E)$, $B \mapsto R_B$.

Proposition (composition). For all $A, B \in E$, $$ L_A L_B = L_{AB}, \qquad R_A R_B = R_{AB}, \qquad L_A R_B = R_B L_A , \qquad L_A + R_B = R_B + L_A . $$ In particular $L$ is a representation of the algebra $E$ on itself, $R$ is a representation of the opposite algebra $E^{\mathrm{op}}$, and the left and right images commute; the composite action $A \otimes B \mapsto L_A R_B$ is a representation of $E \otimes_{\mathbb C} E^{\mathrm{op}}$.

Proof. $L_AL_B(X) = A(BX) = (AB)X = L_{AB}(X)$ by the associativity of composition; $R_AR_B(X) = (XB)A = X(BA) = R_{BA}(X)$, which is the representation of the opposite product; and $L_AR_B(X) = A(XB) = (AX)B = R_BL_A(X)$ by associativity. The last assertion is the associativity of the action.

Proposition (the images are the two copies). The maps $L$ and $R$ are injective, $\ker L = \ker R = 0$, and $L(E)$ and $R(E)$ are subalgebras of $\operatorname{End}(E)$ isomorphic to $E$ and $E^{\mathrm{op}}$.

Proof. $L_A = 0$ forces $AX = 0$ for all $X$; taking $X = \mathrm{id}$ gives $A = 0$. The image $L(E)$ is closed under composition and contains $L_{\mathrm{id}} = \mathrm{id}$, hence is a subalgebra isomorphic to $E$ by injectivity; the same holds for $R$.

Theorem (the double centraliser). The commutant of $L(E)$ in $\operatorname{End}(E)$ is $R(E)$, and the commutant of $R(E)$ is $L(E)$; moreover $$ \operatorname{End}(E) = L(E)\,R(E) \cong E \otimes_{\mathbb C} E^{\mathrm{op}} , $$ so every $\mathbb{C}$-linear operator on $E$ is a sum of two-sided multiplications.

Proof. A map $T$ commuting with every $L_A$ is determined by $\Phi = T(\mathrm{id})$, since $T(X) = T(L_X\,\mathrm{id}) = L_X T(\mathrm{id}) = X\Phi = R_\Phi(X)$; this gives the commutant of $L(E)$ as $R(E)$, and symmetrically. For the spanning statement, both spaces have complex dimension $n^4 = \dim E \otimes E$ — indeed $\dim\operatorname{End}(E) = (n^2)^2 = n^4 = \dim E \cdot \dim E$ — and the map $A \otimes B \mapsto L_A R_B$ is injective because it sends the basis $E_{ij} \otimes E_{kl}$ to the operator $X \mapsto E_{ij}XE_{kl}$, whose matrix units are independent; an injective linear map between spaces of the same finite dimension is an isomorphism. This is the double centraliser theorem of Algebras of Endomorphisms.

Corollary (the centre and the scalars). The centre of $E$ is $\mathbb{C}\cdot\mathrm{id}$, and for a scalar $Z \in \mathbb{C}$ the two one-sided multiplications coincide, $L_{Z\,\mathrm{id}} = R_{Z\,\mathrm{id}}$; for a general $A \in E$ the two differ unless $A$ is central.

Proof. The centre statement is Algebras of Endomorphisms; $L_Z(X) = ZX = XZ = R_Z(X)$ for $Z$ central, and $L_A \neq R_A$ for noncentral $A$ because $L_A(\mathrm{id}) = A$ while $R_A(\mathrm{id}) = A$ — the difference is visible on a non-central element $X$ with $AX \neq XA$.

The Hermitian Trace Form and the Unitary Operators

Definition. The Hermitian trace form of $E$ is $$ \langle X, Y\rangle = \operatorname{tr}(X^{\dagger}Y) \qquad (X, Y \in E), $$ where $A^{\dagger}$ is the adjoint of $A$ for $h$, $h(Au,v) = h(u,A^{\dagger}v)$.

Proposition (the trace form is a Hermitian form). The trace form is Hermitian and positive definite, $\langle X, Y\rangle = \overline{\langle Y, X\rangle}$ and $\langle X, X\rangle = \operatorname{tr}(X^{\dagger}X) > 0$ for $X \neq 0$, and therefore makes $E$ a Hermitian space of dimension $n^2$.

Proof. $\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$ and $\overline{\langle Y,X\rangle} = \overline{\operatorname{tr}(Y^\dagger X)} = \operatorname{tr}((Y^\dagger X)^\dagger) = \operatorname{tr}(X^\dagger Y)$, using $(Y^\dagger X)^\dagger = X^\dagger Y$; the diagonal is the sum of the diagonal entries of $X^\dagger X$, which in an orthonormal basis of $V$ is $\sum_{i,j}|X_{ij}|^2 > 0$ for $X\neq0$. This is the Hilbert–Schmidt form of Hilbert Algebras.

Proposition (the unitary one-sided multiplications are isometries). Let $U \in E$ be unitary for $h$, $U^{\dagger}U = UU^{\dagger} = \mathrm{id}$. Then $L_U$ and $R_U$ are isometries of the trace form, $$ \langle L_U X, L_U Y\rangle = \langle X, Y\rangle, \qquad \langle R_U X, R_U Y\rangle = \langle X, Y\rangle , $$ and the maps $U \mapsto L_U$ and $U \mapsto R_U$ are commuting unitary representations of the unitary group $U(V,h)$ on the Hermitian space $E$.

Proof. $\langle L_UX,L_UY\rangle = \operatorname{tr}((UX)^\dagger UY) = \operatorname{tr}(X^\dagger U^\dagger U Y) = \operatorname{tr}(X^\dagger Y)$, and likewise on the right with $U^\dagger U$ placed on the other side; the representation property is the composition law, and the commutativity of the two images is the proposition above. The unitary group and the involution are The Unitary and Symplectic Groups and Hermitian Geometry and the Unitary Group.

Remark (the form of the category). The chosen form is the Hermitian form $h$ on $V$; it induces two forms on the operator algebra, the Hermitian trace form $\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$ used here and the bilinear trace pairing $\operatorname{tr}(XY)$ obtained by dropping the involution. The two agree in their real part on the self-adjoint part and differ in general; it is the Hermitian trace form that the one-sided operators are measured against in this category, and it is the form with respect to which the adjoints are computed.

The Complex Structure and the Complex-Linear Operators

Definition. The complex structure of the vector space $E$ is the operator $J_E(X) = iX$; the complex structure of $V$ is $J(v) = iv$.

Proposition (the complex structure is the central one-sided multiplication). $J_E = L_i = R_i$, and it commutes with every one-sided multiplication, $$ J_E L_A = L_A J_E = L_{iA}, \qquad J_E R_B = R_B J_E = R_{iB} ; $$ the complex structure of $E$ is thus the one-sided multiplication by the central scalar $i$, and it is a complex-linear operator of $E$ on itself. On $V$ the complex structure is $J$, and $J$ is skew-adjoint and unitary for $h$, $J^\dagger = -J$, $J^2 = -\mathrm{id}$, so $J \in E$ is an element of the operator algebra; left multiplication by $i$ and the operator $J$ are related by $L_i = i\,\mathrm{id}_{\operatorname{End}(E)}$, while $J$ acts on $V$ and not on $E$.

Proof. $J_E(X) = iX = Xi = R_i(X)$ because $i$ is central, and $iA = Ai$ gives the commutation. For $J$, $h(Ju,v) = h(iu,v) = i\,h(u,v)$ and $h(u,J^\dagger v) = h(u,-iv) = ih(u,v)$, so $J^\dagger = -J$; $J^2 = -\mathrm{id}$ and unitarity follow. The distinction between the scalar $i$ multiplying $E$ and the operator $J$ acting on $V$ is the distinction between the complex structure of the operator algebra and the complex structure of the space it acts on.

Proposition (the complex-linear operators are those commuting with $J_E$). A real-linear operator $T$ on $E$ is complex-linear for the complex structure $J_E$ exactly when $T J_E = J_E T$; the one-sided multiplications $L_A, R_B$ and every two-sided multiplication are complex-linear, and the complex-linear operators on $E$ form the algebra $\operatorname{End}_{\mathbb C}(E)$.

Proof. The equivalence of complex-linearity and commuting with the complex structure is the definition of a complex-linear map on a complex vector space; the one-sided multiplications commute with $J_E$ because $L_AJ_E(X) = A(iX) = i\,AX = J_EL_A(X)$ and $R_BJ_E(X) = (iX)B = i\,XB = J_ER_B(X)$, using the centrality of $i$. The two-sided multiplications are sums of composites and hence complex-linear too.

Remark (holomorphic and antiholomorphic operators on $E$). The complex structure $J$ of $V$ extends to $E$ in two ways: the conjugation $E \to E$, $A \mapsto JA$ or $A \mapsto AJ$ (which is the left or right multiplication by the element $J \in E$), and the $\mathbb C$-linear structure $J_E = L_i$. The two are different: $L_J(A) = JA$ is a complex-linear operator on $E$ for the structure $J_E$, whereas the conjugation by $J$, $\mathrm{Ad}_J(A) = JAJ^{-1} = -JAJ$, is its own inverse and is the involution associated with $J$; the one-sided multiplications by the elements of $E$ and the inner automorphisms by the unitary elements are the two families of operators of the operator layer, the latter developed in The Signed Sandwich on a Complex Vector Space.

Summary

On the endomorphism algebra $E = \operatorname{End}_{\mathbb C}(V)$ of a complex vector space the left and right multiplications $L_A(X) = AX$ and $R_B(X) = XB$ satisfy $L_AL_B = L_{AB}$, $R_AR_B = R_{AB}$ and $L_AR_B = R_BL_A$, so $L$ is a representation of $E$ and $R$ of the opposite algebra $E^{\mathrm{op}}$ with commuting images, and the double centraliser theorem gives $\operatorname{End}(E) = L(E)R(E)\cong E\otimes E^{\mathrm{op}}$ with the commutant of $L(E)$ equal to $R(E)$ and conversely. The complex structure of $E$ is the central one-sided multiplication $J_E = L_i = R_i$, which commutes with every one-sided multiplication, whereas the complex structure of $V$ is the skew-adjoint unitary element $J \in E$; a real-linear operator on $E$ is complex-linear exactly when it commutes with $J_E$. When $h$ is a chosen Hermitian form on $V$, the operator algebra carries the Hermitian trace form $\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$, positive definite, and the one-sided multiplications $L_U, R_U$ by a unitary $U$ are commuting isometries of it, so the unitary group acts on $E$ through two commuting unitary representations. The adjoints of the one-sided multiplications for the trace form, the signed variants and the module-level graded action are the companion articles of this group.

Summary of Notation

Symbol	Meaning
$V$, $h$, $n$	the complex vector space, its Hermitian form, its dimension
$E = \operatorname{End}_{\mathbb C}(V)$	the endomorphism algebra, of complex dimension $n^2$
$A^\dagger$	the adjoint of $A$ for $h$
$L_A(X) = AX$, $R_B(X) = XB$	the one-sided multiplications
$J_E = L_i = R_i$	the complex structure of the operator algebra
$J$	the complex structure of $V$, an element of $E$
$\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$	the Hermitian trace form of $E$
$\operatorname{End}(E) = L(E)R(E)$	the double centraliser