The Adjoint of the Left Multiplication on an Algebra

Introduction

The left multiplication $L_a(x)=ax$ is the simplest operator an algebra carries, and its adjoint with respect to the pairing of the category is explicit: it is the right multiplication by the same element, $L_a^{*}=R_a$. The pairing moves the factor $a$ across the product, and the adjoint records the move as a change of side; the left and the right multiplications are therefore mutual adjoints, and the regular representation is its own dual read through the pairing. Against an involution $\sigma$ the answer changes side: the twisted pairing sends $L_a$ to the left multiplication by $\sigma(a)$, $L_a^{*_\sigma}=L_{\sigma(a)}$, so that the twisted adjoint of a left multiplication is again a left multiplication, and the element involution is the operator involution of the regular representation. The self-adjoint and the unitary left multiplications are read off at once — the central elements and the symmetric elements for the first, the central involutions and the unitary elements for the second — and the inner derivation $L_a-R_a$ is skew-adjoint.

This article computes the adjoint of the left and the right multiplications for the two pairings of the category, records the compatibility with the involution, identifies the self-adjoint and the unitary left multiplications, and relates the inner derivations to the skew-adjoint operators. It assumes Left and Right Multiplication for $L_a$ and $R_a$ and the anti-homomorphism $a\mapsto R_a$ onto $A^{\mathrm{op}}$, The Sandwich Operator on an Algebra for the two-sided operators $T_{a,b}=L_aR_b$, The Commutator Operator for the inner derivations $D_a=L_a-R_a$, Frobenius Algebras for the trace and the pairing, Involutive Linear Algebras for the involution, and Involutions of the Operator Algebra and The Adjoint in an Involutive Algebra for the adjoint operation and the twisted pairing. The signed version, in which the grade involution is inserted between the two factors, is The Signed Adjoint of the Left Multiplication on an Algebra and The Signed Adjoint Sandwich on an Algebra; the graded version is The Graded Adjoint Action on a Module over an Algebra; and the analytic left multiplication, the norm and the positivity are Part II. This article stays inside Part I: no distance, norm, form with a norm, topology or limit.

Throughout, $k$ is a field of characteristic not two, $A$ is a finite-dimensional unital associative $k$-algebra, $\tau$ is a trace whose pairing $\langle x,y\rangle=\tau(xy)$ is nondegenerate, $\sigma$ is an involution of $A$ with $\tau(\sigma(x))=\tau(x)$, the twisted pairing is $\{x,y\}=\tau(x\sigma(y))$, the operator algebra is $E=\operatorname{End}_k(A)$, and the adjoints for the two pairings are $T^{*}$ and $T^{*_\sigma}$.

The Adjoints of the One-Sided Multiplications

Theorem. For all $a,b \in A$,

$$ \langle L_ax,y\rangle=\langle x,R_ay\rangle, \qquad \langle R_bx,y\rangle=\langle x,L_by\rangle, $$

so $L_a^{*}=R_a$ and $R_b^{*}=L_b$: the adjoint of the left multiplication is the right multiplication by the same element, and conversely. The adjoint operation therefore exchanges the two sides of the regular representation.

Proof. By the compatibility of the pairing with the product, $\langle L_ax,y\rangle=\langle ax,y\rangle=\langle x,ya\rangle=\langle x,R_ay\rangle$; the second identity is the same computation with the sides exchanged, $\langle R_bx,y\rangle=\langle xb,y\rangle=\langle x,by\rangle=\langle x,L_by\rangle$, using the compatibility in its other form.

Corollary. The subalgebra of $E$ generated by the left and the right multiplications is stable under the adjoint, and the adjoint exchanges its two generating families; a product $L_{a_1}R_{b_1}\cdots L_{a_n}R_{b_n}$ has adjoint $L_{b_n}R_{a_n}\cdots L_{b_1}R_{a_1}$, by anti-multiplicativity.

Proof. Anti-multiplicativity and the two identities give $(L_aR_b)^{*}=R_b^{*}L_a^{*}=L_bR_a$, and the general product follows by induction.

Example (the matrix algebra). For $A=M_n(k)$ with the trace pairing, $L_X^{*}=R_X$, so the adjoint of the left multiplication by $X$ is the right multiplication by $X$; the self-adjoint left multiplications are those with $X$ scalar.

Compatibility with the Involution

Theorem. Let $\sigma$ be an involution of $A$ with $\tau(\sigma(x))=\tau(x)$, and let $c_{\sigma}(T)=\sigma T\sigma$ be the conjugation of the operators. Then

$$ R_a=\sigma\,L_{\sigma(a)}\,\sigma=c_{\sigma}\bigl(L_{\sigma(a)}\bigr), \qquad\text{and}\qquad L_a^{*_\sigma}=L_{\sigma(a)}, \qquad R_b^{*_\sigma}=R_{\sigma(b)} $$

for the twisted pairing $\{x,y\}=\tau(x\sigma(y))$.

Proof. For the first identity, $\sigma L_{\sigma(a)}\sigma(x)=\sigma(\sigma(a)\sigma(x))=\sigma^{2}(x)\sigma^{2}(a)=xa=R_ax$, using the anti-multiplicativity of $\sigma$ and $\sigma^{2}=\mathrm{id}$. For the second, use the dictionary $T^{*_\sigma}=\sigma T^{*}\sigma=c_{\sigma}(T^{*})$ of The Adjoint in an Involutive Algebra with $T=L_a$: $L_a^{*_\sigma}=c_{\sigma}(R_a)=c_{\sigma}(L_{\sigma(a)})=L_{\sigma(a)}$, the middle step by the first identity; the right-multiplication case is the same computation.

Corollary (the regular representation is a *-representation for the twisted pairing). With the twisted pairing the assignment $a\mapsto L_a$ is a *-homomorphism,

$$ L_{\sigma(a)}=L_a^{*_\sigma}, $$

from the involutive algebra $(A,\sigma)$ to the involutive algebra $(E,*_\sigma)$; the element involution and the operator adjoint therefore agree on the image of the regular representation, and this agreement is a theorem, not a definition.

Proof. The intertwining is the theorem; it is multiplicative, $L_{ab}=L_aL_b$, additive and unital, so it is a homomorphism of algebras carrying the involution $\sigma$ to the involution $*_\sigma$.

Corollary (the right multiplication). The right multiplication satisfies $R_{\sigma(b)}=R_b^{*_\sigma}$, and $R$ is an injective anti-homomorphism of $(A,\sigma)$ into $(E,*_\sigma)$; the two images are exchanged by the adjoint for the plain pairing and preserved by it for the twisted one.

Proof. The identity is the theorem; $R$ reverses products, $R_{ab}=R_bR_a$, and the compatibility with the involution is the same computation as for $L$.

Self-Adjointness

Theorem. The left multiplication $L_a$ is self-adjoint for the plain pairing exactly when $a$ is central, $a \in Z(A)$, and then $L_a=R_a$; it is self-adjoint for the twisted pairing exactly when $\sigma(a)=a$, that is when $a$ is a symmetric element.

Proof. $L_a^{*}=R_a$, so $L_a=L_a^{*}$ is $L_a=R_a$, which is $ax=xa$ for all $x$, that is the centrality of $a$. For the twisted pairing $L_a^{*_\sigma}=L_{\sigma(a)}$, so self-adjointness is $L_{\sigma(a)}=L_a$, which is injective in the parameter, hence $\sigma(a)=a$.

Corollary. The self-adjoint part of the left multiplication $L_a$ for the plain pairing is the symmetrised operator

$$ \tfrac12\bigl(L_a+L_a^{*}\bigr)=\tfrac12\bigl(L_a+R_a\bigr), $$

which is the two-sided multiplication by $a$ on the two sides; the skew part is $\tfrac12(L_a-R_a)$, half the inner derivation of The Commutator Operator.

Proof. The averaging is the decomposition of an operator into its self-adjoint and skew-adjoint parts from Involutions of the Operator Algebra, and $R_a=L_a^{*}$ makes the two terms explicit.

Theorem (the inner derivation is skew-adjoint). The inner derivation $D_a=L_a-R_a$ satisfies

$$ D_a^{*}=R_a-L_a=-D_a $$

for the plain pairing, so the inner derivations are skew-adjoint operators; for the twisted pairing $D_a^{*_\sigma}=D_{\sigma(a)}$, so $D_a$ is skew-adjoint for the twisted pairing exactly when $\sigma(a)=-a$, that is when $a$ is a skew element.

Proof. $D_a^{*}=L_a^{*}-R_a^{*}=R_a-L_a=-D_a$, and for the twisted pairing $D_a^{*_\sigma}=L_{\sigma(a)}-R_{\sigma(a)}=D_{\sigma(a)}$; the relation $D_{\sigma(a)}=-D_a$ is $\sigma(a)=-a$, by injectivity of the parameter in $L$ and $R$.

The Unitary Left Multiplications

Theorem. The left multiplication $L_a$ is unitary for the plain pairing exactly when $a$ is a central involution,

$$ L_a^{*}L_a=L_aL_a^{*}=\mathrm{id} \iff a^{2}=1 \text{ and } a \in Z(A), $$

and unitary for the twisted pairing exactly when $a$ is unitary in the algebra,

$$ L_a^{*_\sigma}L_a=L_aL_a^{*_\sigma}=\mathrm{id} \iff \sigma(a)a=1 . $$

Proof. For the plain pairing $L_a^{*}L_a=R_aL_a$ is the operator $x\mapsto axa$, which is the identity exactly when $axa=x$ for all $x$; the case $x=1$ gives $a^{2}=1$, and then $axa=x$ is $ax=xa^{-1}=xa$, the centrality of $a$. For the twisted pairing $L_a^{*_\sigma}L_a=L_{\sigma(a)}L_a=L_{\sigma(a)a}$, which is the identity exactly when $\sigma(a)a=1$ and $\sigma(a)a$ is central, the second condition being automatic for the unit $1$; the other order is the same equation.

Corollary. The unitary left multiplications with respect to the twisted pairing are the images of the unitary elements of the algebra, so the map $a\mapsto L_a$ carries $U(A,\sigma)$ isomorphically onto a subgroup of $U(E,*_\sigma)$; for the plain pairing the unitary left multiplications form the smaller group of the central involutions, and the two groups coincide exactly when every unitary element is central.

Proof. The criterion $\sigma(a)a=1$ is the unitarity of $a$ in $A$; the injectivity of $L$ and the group structure are Unitary Operators of an Involutive Algebra. A central involution is unitary, so the plain group is contained in the twisted one under the identification, and equality is the centrality of the unitary elements.

Examples

(a) The matrix algebra. $A=M_n(k)$ with the trace and the transpose: $L_X^{*}=R_X$, $L_X^{*_\sigma}=L_{X^{\mathsf{T}}}$, so $L_X$ is self-adjoint for the plain pairing exactly when $X$ is scalar and for the twisted pairing exactly when $X$ is symmetric; $L_X$ is unitary for the twisted pairing exactly when $X^{\mathsf{T}}X=1$, that is when $X$ is orthogonal over $k$.

(b) The group algebra. $A=k[G]$ with $\sigma(g)=g^{-1}$ and $\tau$ the coefficient of the identity: $L_g^{*}=R_g$ and $L_g^{*_\sigma}=L_{g^{-1}}$, so the twisted adjoint of the left multiplication by a group element is the left multiplication by its inverse, and every $L_g$ is unitary for the twisted pairing because every group element is unitary.

(c) The commutative algebra. For a commutative $A$ one has $L_a=R_a$ and every $L_a$ is self-adjoint for the plain pairing; the adjoint operation is the identity on the image of $A$ in $E$, which is the centrality criterion of the self-adjointness theorem read as a definition.

(d) The connection with the derivation. The inner derivation $D_a=L_a-R_a$ is the difference of a left and a right multiplication, and it is skew-adjoint for the plain pairing; the self-adjoint part of $L_a$ is the symmetrised operator $\tfrac12(L_a+R_a)$ and the skew part is $\tfrac12 D_a$, so the derivation is the skew shadow of the left multiplication.

Summary

For the pairing $\langle x,y\rangle=\tau(xy)$ of the category the adjoint of the left multiplication is the right multiplication, $L_a^{*}=R_a$, and conversely $R_b^{*}=L_b$: the adjoint operation exchanges the two sides of the regular representation, and a product $L_{a_1}R_{b_1}\cdots$ is sent to the product of the adjoints in the reverse order. Against an involution $\sigma$ the right multiplication is the conjugate of a left one, $R_a=\sigma L_{\sigma(a)}\sigma$, and the twisted pairing $\{x,y\}=\tau(x\sigma(y))$ sends $L_a$ to $L_{\sigma(a)}$ and $R_b$ to $R_{\sigma(b)}$, so that the twisted adjoint preserves each side; with the twisted pairing the regular representation is a *-representation, $L_{\sigma(a)}=L_a^{*_\sigma}$, and the agreement of the element involution with the operator adjoint is proved. The left multiplication is self-adjoint for the plain pairing exactly when $a$ is central and for the twisted pairing exactly when $a$ is symmetric; it is unitary for the plain pairing exactly when $a$ is a central involution and for the twisted pairing exactly when $\sigma(a)a=1$, that is when $a$ is a unitary element of the algebra. The inner derivation $D_a=L_a-R_a$ is skew-adjoint for the plain pairing, and its twisted adjoint is $D_{\sigma(a)}$.

Summary of Notation

Further Reading

• Nathan Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37 (1964), for the left and right multiplications, the trace form and the regular representation.
• I. N. Herstein, Rings with Involution (University of Chicago Press, 1976), for the involutive compatibility of the left and right multiplications and the unitary elements.
• Nicolas Bourbaki, Algebra I, Chapters 1–3 (Springer, 1998), for the opposite algebra, the regular representation and the adjoint under a symmetric pairing.
• Matej Brešar, Introduction to Noncommutative Algebra (Springer, 2014), for the inner derivations $L_a-R_a$ and their adjoint properties.