The Left and the Right Regular Representation

Introduction

A Hilbert algebra acts on itself in two ways: on the left by $L_x(y) = xy$ and on the right by $R_x(y) = yx$. The left regular representation $x\mapsto L_x$ is a $\ast$-representation, by the adjoint axiom; the right regular representation $x\mapsto R_x$ is an anti-$\ast$-representation, since $R_{xy} = R_yR_x$; and the two commute, $L_xR_y = R_yL_x$, which is nothing but the associativity of the product read in two orders. This triple of facts – representation, antirepresentation, commutation – is the whole algebraic content of the pair.

After completion, the two representations become two families of bounded operators on the Hilbert space $H$, and the von Neumann algebras they generate carry the structure: the algebra generated by the left multiplications has, as its commutant, the algebra generated by the right multiplications, and vice versa. This is the operator-algebraic form of the statement that a product is associative, and it is the mechanism by which a Hilbert algebra produces a von Neumann algebra and its commutant from a single object: the left representation gives the algebra, the right representation gives its commutant, and the modular conjugation is the map that exchanges them.

This article fixes the two representations, their commutation, the von Neumann algebras they generate and the commutant relation, and it identifies the modular conjugation as the exchange between the left and the right pictures.

The Hilbert algebra is Hilbert Algebras; the completion on which the representations become operators is The Completion of a Hilbert Algebra; the algebra and commutant produced are Von Neumann Algebras and the Hilbert Algebra Completeness; the conjugation that exchanges the two pictures is The Modular Operator and Tomita-Takesaki Theory; the construction from a state is The GNS Construction. Those are cited. The algebra is $A$, the completion is $H$, and $\mathcal{M}_L$, $\mathcal{M}_R$ are the generated von Neumann algebras.

The Two Representations

Definition. On $A$ the left regular representation and the right regular representation are

$$ L_x(y) = xy, \qquad R_x(y) = yx = L_x^{\mathrm{right}}(y). $$

Proposition (representation and antirepresentation). $L$ is a representation, $L_{xy} = L_xL_y$; $R$ is an antirepresentation, $R_{xy} = R_yR_x$; and both are $\ast$-compatible on the left in the sense

$$ L_{x^{\dagger}} = L_x^{*}, \qquad R_{x^{\dagger}} = R_x^{*}, $$

the adjoints being taken for the form of the algebra.

Proof. The representation and antirepresentation laws are associativity in its two readings. For the adjoints, $\langle L_xy,z\rangle = \langle xy,z\rangle = \langle y,x^{\dagger}z\rangle = \langle y,L_{x^{\dagger}}z\rangle$ by the adjoint axiom; and $\langle R_xy,z\rangle = \langle yx,z\rangle = \langle x^{\dagger}y,\ldots\rangle$, which is the adjoint axiom read from the other side, giving $R_{x^{\dagger}} = R_x^{*}$.

Proposition (commutation). The two representations commute:

$$ L_xR_y = R_yL_x \qquad \text{for all } x, y , $$

because both sides send $z$ to $xyz$.

Proof. $L_xR_y(z) = x(yz)$ and $R_yL_x(z) = (xz)y$, and these agree by associativity.

Remark (why the commutation is the crux). The commutation is the only structural relation between the two representations, and it is exactly associativity. Everything that follows – the commutant statement, the standard form, the exchange by the modular conjugation – is a consequence of this one identity.

The Generated Algebras

Definition. After completion the extended representations give bounded operators on $H$; the von Neumann algebras they generate are

$$ \mathcal{M}_L = \overline{\{\bar L_x : x\in A\}}^{\text{weak}}, \qquad \mathcal{M}_R = \overline{\{\bar R_x : x\in A\}}^{\text{weak}} . $$

Proposition (they are von Neumann algebras). $\mathcal{M}_L$ and $\mathcal{M}_R$ are unital von Neumann algebras of $B(H)$, each closed under the Hilbert adjoint, and $\mathcal{M}_L$ is the strong closure of the left representation.

Proof. The weak closure of a self-adjoint algebra of operators is a von Neumann algebra; self-adjointness is the identity $L_{x^{\dagger}} = L_x^{*}$ extended by continuity.

Theorem (the commutant relation). With $\xi$ the cyclic vector of the completion and $\mathcal{M}$ the von Neumann algebra generated by the left multiplications, the commutant is generated by the right multiplications:

$$ \mathcal{M}_L^{c} = \mathcal{M}_R , $$

and symmetrically $\mathcal{M}_R^{c} = \mathcal{M}_L$; consequently $\mathcal{M}_L = \mathcal{M}_L^{cc}$.

Proof. Every right multiplication commutes with every left multiplication, so $\mathcal{M}_R\subseteq\mathcal{M}_L^{c}$. For the reverse inclusion, an operator $T$ commuting with $\mathcal{M}_L$ is determined by its value at $\xi = \iota(1)$, and the commutation with the left multiplications gives $T(x\xi) = T(L_x\xi) = L_x(T\xi)$, so $T$ is right multiplication by $T\xi$; if $T\xi = \iota(y)$ then $T = R_y$, and in general $T\xi$ is a limit of elements of the algebra and $T$ lies in the strong closure $\mathcal{M}_R$. The symmetric statement is identical, and the bicommutant statement follows.

Corollary (a Hilbert algebra produces a pair). From a single Hilbert algebra one obtains a von Neumann algebra $\mathcal{M}_L$, its commutant $\mathcal{M}_R$ and a cyclic and separating vector $\xi$: the pair $(\mathcal{M}_L,\mathcal{M}_R)$ is the standard form of the algebra.

Proof. Cyclicity for $\mathcal{M}_L$ is the density of the orbit; separatingness follows from the same density for $\mathcal{M}_R$, since an element annihilated by both left and right multiplications is zero.

The Exchange by the Modular Conjugation

Proposition (the two representations are exchanged). Let $\jmath$ be the modular conjugation of the completion. Then

$$ \jmath\,\bar L_x\,\jmath = \bar R_{x^{\dagger}} , \qquad \jmath\,\bar R_x\,\jmath = \bar L_{x^{\dagger}} , $$

so the modular conjugation exchanges the left and the right pictures and with them the algebra and its commutant, $\jmath\mathcal{M}_L\jmath = \mathcal{M}_R$.

Proof. The modular conjugation carries the algebra to its commutant, by The Modular Operator and Tomita-Takesaki Theory; since it fixes the vector up to the modular operator and reverses products, it sends the left multiplications to the right ones twisted by the involution, which is the displayed identity.

Remark (left and right are one object). The left representation and the right representation are not two structures but one seen from two sides: they commute, they are exchanged by the modular conjugation, and each is the commutant of the other. This is the reason the standard form is described as a single object, and the reason the left and the right cannot be chosen independently.

Worked Cases

The Group Algebra

For a finite group $G$ and $A = \mathbb{C}[G]$, the left multiplications give the left regular representation of $G$ and the right multiplications the right regular representation; they commute, and on the completion $\mathbb{C}^{G}$ their generated algebras are each other's commutants.

Matrix Algebras

For $A = M_n(\mathbb{C})$ with the Hilbert–Schmidt form and $H = M_n(\mathbb{C})$ the completion, the left multiplications are $x\mapsto ax$ and the right ones $y\mapsto ya$. The von Neumann algebra generated by the left multiplications is the left-action copy of $M_n(\mathbb{C})$ on $H$, its commutant is the right-action copy $\{y\mapsto yb\}$, which is anti-isomorphic to $M_n(\mathbb{C})$, and the two have trivial intersection; the commutant relation $\mathcal{M}_L^{c} = \mathcal{M}_R$ is exact and neither algebra is the scalars.

The Abelian Case

For a commutative Hilbert algebra the left and the right representations coincide, $L_x = R_x$, the generated algebra is commutative and equal to its commutant, and the modular conjugation is the identity on the algebra; the exchange statement degenerates.

Summary

The left regular representation $L_x(y) = xy$ is a $\ast$-representation of a Hilbert algebra and the right regular representation $R_x(y) = yx$ is an anti-$\ast$-representation, with $L_{x^{\dagger}} = L_x^{*}$ and $R_{x^{\dagger}} = R_x^{*}$ for the adjoint axiom. The two commute, $L_xR_y = R_yL_x$, which is associativity, and this single identity drives everything: after completion the generated von Neumann algebras satisfy $\mathcal{M}_L^{c} = \mathcal{M}_R$ and $\mathcal{M}_R^{c} = \mathcal{M}_L$, so a Hilbert algebra produces a von Neumann algebra, its commutant and a cyclic and separating vector at once. The modular conjugation exchanges the two pictures, $\jmath\bar L_x\jmath = \bar R_{x^{\dagger}}$, and with them the algebra and its commutant, so the left and the right are one structure seen from two sides and cannot be chosen independently. The algebra is Hilbert Algebras, the completion on which the multiplications become operators is The Completion of a Hilbert Algebra, the resulting operator algebras are Von Neumann Algebras and the Hilbert Algebra Completeness, and the exchange is the modular conjugation of The Modular Operator and Tomita-Takesaki Theory.

Summary of Notation

Symbol	Meaning
$L_x(y) = xy$	Left regular representation, a $\ast$-representation
$R_x(y) = yx$	Right regular representation, an anti-$\ast$-representation
$L_{x^{\dagger}} = L_x^{}$, $R_{x^{\dagger}} = R_x^{}$	The adjoint axiom in operator form
$L_xR_y = R_yL_x$	Associativity, the crux
$\mathcal{M}_L$, $\mathcal{M}_R$	Generated von Neumann algebra and its commutant
$\mathcal{M}_L^{c} = \mathcal{M}_R$	The commutant relation
$\jmath\bar L_x\jmath = \bar R_{x^{\dagger}}$	The modular conjugation exchanges the two
$(\mathcal{M}_L,\mathcal{M}_R,\xi)$	The standard form produced by the algebra