The Involution on the Measure Algebra

Introduction

The group algebra $L^1(G)$ is an ideal in the larger convolution algebra $M(G)$ of bounded Radon measures, and $M(G)$ carries an involution of its own: the reflection of a measure by the group inverse followed by conjugation. This involution is isometric and anti-multiplicative on the whole measure algebra, and it restricts on the absolutely continuous measures to the modular-corrected involution of the group algebra; it is therefore the universal source of the involution that the group algebra carries. The adjoint of convolution — the transpose of the operator of convolution by a measure with respect to the Haar pairing — is convolution by the involution, and the unimodular case is where the same identity holds for the right regular representation as well, the two-sided picture being unitary only there. This article fixes the measure algebra, its involution and its restriction, and states the adjoint of convolution; the operator-theoretic form of the adjoint is The Adjoint of a Convolution Operator, in the - * Operator Theory group.

The article assumes the group, its Haar measure and the modular function from Locally Compact Groups and Haar Measure; the algebra $L^1(G)$, its involution $f^*(x) = \overline{f(x^{-1})}\Delta(x)^{-1}$ and its properties from The Convolution Algebra $L^1(G)$; the involution, the element classes and the completions from The Group Algebra as an Involutive Algebra; the Hermitian forms and the forms of the positive functionals from Hermitian Forms and the Group Algebra, immediately preceding; the Radon measures, their total variation, the Riesz representation theorem and the Lebesgue decomposition from Measure Theory and Integration; the group von Neumann algebra and the left regular representation from The Group Algebra as an Algebra of Operators; and the *-representations and the bounded operators on $L^2$ from Operator Algebras. The grade involution $\alpha$ of the - Operator Theory group is a different order-two structure and is not used; the adjoint of a convolution operator on $L^p$ and its explicit form are The Adjoint of a Convolution Operator, in the - * Operator Theory group; the Plancherel measure and the unitary representations are the earlier articles of this group.

Throughout, $G$ is a locally compact Hausdorff group with left Haar measure $dx$ and modular function $\Delta$, and

$$ M(G) = \bigl\{\mu : \mu \text{ a bounded complex Radon measure on } G\bigr\}, \qquad \|\mu\| = |\mu|(G) , $$

with convolution

$$ (\mu*\nu)(E) = \int_G\int_G \mathbf{1}_E(xy)\,d\mu(x)\,d\nu(y) , $$

and involution

$$ \mu^*(E) = \overline{\mu(E^{-1})} \qquad (E\subseteq G \text{ Borel}). $$

The Haar pairing is $\langle\mu,f\rangle = \int_G f\,d\mu$ for $\mu\in M(G)$ and $f$ a bounded continuous function.

The Measure Algebra

Theorem (the algebra and its ideal). With the convolution and the total variation norm, $M(G)$ is a unital Banach algebra with identity $\delta_e$ and $\|\mu*\nu\|\leq\|\mu\|\,\|\nu\|$; the map $f\mapsto f\,dx$ is an isometric algebra isomorphism of $L^1(G)$ onto a closed two-sided ideal of $M(G)$, and the Lebesgue decomposition

$$ M(G) = L^1(G)\oplus M_s(G) $$

expresses every measure uniquely as an absolutely continuous part and a part singular with respect to $dx$.

Proof. The convolution inequalities follow from Fubini and the subadditivity of the total variation; associativity is the associativity of the product in $G$; the identity is $\delta_e$. The embedding is isometric by the definition of $L^1(G)$ and is multiplicative, and the ideal property is the translation invariance of $dx$. The Lebesgue decomposition is the Radon–Nikodym theorem for the pair $\mu$ and $dx$. $\square$

Example (the discrete and compact cases). For $G$ discrete with counting measure, $M(G) = \ell^1(G)$ is unital and the point masses $\delta_x$ are unitary elements; for $G$ compact, $M(G) = L^1(G)\oplus M_s(G)$ with $M_s(G)$ the discrete measures, and $\delta_e$ is a unit not in $L^1(G)$.

The Involution

Theorem (the involution axioms). The map $\mu\mapsto\mu^*$ is an involution on $M(G)$,

$$ (\mu^*)^* = \mu, \qquad (\lambda\mu + \nu)^* = \bar\lambda\,\mu^* + \nu^*, \qquad (\mu*\nu)^* = \nu^*\!*\mu^*, \qquad \|\mu^*\| = \|\mu\| , $$

so that $M(G)$ is a Banach *-algebra with isometric involution; the involution is an anti-automorphism of the full measure algebra, with no hypothesis on the group.

Proof. The total variation of $\mu^*$ is $|\mu^*|(E) = |\mu|(E^{-1})$, so $|\mu^*|(G) = |\mu|(G)$ and the involution is isometric; the order-two and linearity statements are immediate. For the anti-automorphism, test against $f\in C_{00}(G)$ using the pairing identity $\langle\lambda^*,f\rangle = \overline{\langle\lambda,\tilde f\rangle}$ with $\tilde f(x) = \overline{f(x^{-1})}$. Then

$$ \bigl\langle(\mu*\nu)^*,f\bigr\rangle = \overline{\bigl\langle\mu*\nu,\tilde f\bigr\rangle} = \overline{\int\!\!\int \overline{f(y^{-1}x^{-1})}\,d\mu(x)\,d\nu(y)} = \int\!\!\int f(y^{-1}x^{-1})\,d\overline{\mu(x)}\,d\overline{\nu(y)} , $$

and the substitution $u = y^{-1}$, $v = x^{-1}$ turns the last expression into $\int\!\!\int f(uv)\,d\nu^*(u)\,d\mu^*(v) = \langle\nu^*\!*\mu^*,f\rangle$. Since $C_{00}(G)$ separates measures, the identity follows. $\square$

Corollary (the isometry does not need unimodularity). The involution of $M(G)$ preserves the norm for every locally compact $G$; the role of the modular function is not here but in the compatibility of the involution with the Haar pairing and with the $L^2$ structure, below.

Proof. The computation $|\mu^*|(G) = |\mu|(G)$ above uses only the fact that inversion is a homeomorphism, which holds without any unimodularity. $\square$

The Involution on the Group Algebra

Theorem (the restriction reproduces the group-algebra involution). Under the identification $f\leftrightarrow f\,dx$ the involution of $M(G)$ restricts to the involution of $L^1(G)$,

$$ (f\,dx)^* = f^*\,dx, \qquad f^*(x) = \overline{f(x^{-1})}\,\Delta(x)^{-1} , $$

so the involution of the group algebra is exactly the restriction of the measure-algebra involution, and the modular factor is generated by the group inverse together with the change of variable $x\mapsto x^{-1}$.

Proof. By definition $(f\,dx)^*(E) = \overline{\int_{E^{-1}}f(x)\,dx}$. Since $E^{-1}$ is the image of $E$ under inversion, the change of variable $x = y^{-1}$ in the integral has Jacobian $\Delta(y)^{-1}$, giving $\overline{\int_{E^{-1}}f(x)dx} = \int_E\overline{f(y^{-1})}\Delta(y)^{-1}dy$, which is the density $f^*$ of $(f\,dx)^*$. $\square$

Corollary (why $L^1$ has the modular factor). The four involution axioms of The Group Algebra as an Involutive Algebra are the restrictions of the axioms of the theorem above; in particular the modular factor is not an extra convention of the group algebra but the Jacobian of the inversion, and it disappears exactly when $\Delta\equiv1$, that is when $G$ is unimodular.

Proof. Restrict the axioms $(\mu*\nu)^* = \nu^*\!*\mu^*$, $(\mu^*)^* = \mu$ and $\|\mu^*\| = \|\mu\|$ to absolutely continuous measures and read the identity $(f*g)\,dx = (f\,dx)*(g\,dx)$. The factor $\Delta(x)^{-1}$ is the Jacobian computed in the theorem. $\square$

The Adjoint of Convolution

Theorem (the adjoint of convolution). Let $\mu\in M(G)$ and let $\lambda(\mu)$ be the operator of left convolution, $\lambda(\mu)\xi = \mu*\xi$, on $L^2(G)$. Then

$$ \lambda(\mu)^* = \lambda(\mu^*) , $$

the adjoint with respect to the Haar pairing $\langle\xi,\eta\rangle = \int_G\xi(x)\overline{\eta(x)}\,dx$; consequently $\lambda : M(G)\to B(L^2(G))$ is a *-representation for the measure-algebra involution, and its restriction to $L^1(G)$ is the left regular representation with $\lambda(f)^* = \lambda(f^*)$. No hypothesis on $G$ is needed.

Proof. For $f\in L^1(G)$ the identity $\lambda(f)^* = \lambda(f^*)$ follows from the unitary representation $\lambda$ by Unitary Representations and the Adjoint: $\lambda(f)^* = \int\overline{f(x)}\lambda(x)^{-1}dx$, and the substitution $x\to x^{-1}$ with the inversion identity $dx = \Delta(x)^{-1}dx$ at $x = y^{-1}$ turns this into $\int\overline{f(y^{-1})}\Delta(y)^{-1}\lambda(y)dy = \lambda(f^*)$. The map $\lambda : M(G)\to B(L^2(G))$ is norm-continuous, and $L^1(G)$ is weak-* dense in $M(G) = C_0(G)^*$ with convolution and the involution weak-* continuous; passing to the limit gives the identity for every $\mu\in M(G)$. $\square$

Remark (where the unimodularity does appear). The measure-algebra involution itself needs no hypothesis; the identity $\lambda(\mu)^* = \lambda(\mu^*)$ needs none either, because the left regular representation is unitary for every $G$. The unimodular case is distinguished by the fact that the right regular representation is also unitary on $L^2(G)$ for the left Haar measure, so that right convolution is a *-representation as well, $\rho(\mu)^* = \rho(\mu^*)$; when $G$ is not unimodular, $\rho(x)$ is unitary only for the weighted inner product $\langle\xi,\eta\rangle_\Delta = \int\xi\overline{\eta}\,\Delta$, and the right-handed adjoint identity must be read there. The exact operator form of the adjoint, on $L^p$ and for the two-sided case, is The Adjoint of a Convolution Operator, in the - * Operator Theory group.

Remark (what the article does not do). The operator-theoretic adjoint of a convolution operator, its explicit form on $L^p$, the case of a two-sided measure and the involutive algebra structure it gives are The Adjoint of a Convolution Operator, in the - * Operator Theory group below, where the theory is developed on the operator level; the adjoint of a representation operator is Unitary Representations and the Adjoint, there. The grade involution of the signed block is not the measure involution: the first is an automorphism, the second an anti-automorphism, and the two are distinguished as in The Group Algebra as an Involutive Algebra.

Summary

The bounded Radon measures on $G$ form a unital Banach algebra $M(G)$ under convolution, with identity $\delta_e$ and $\|\mu*\nu\|\leq\|\mu\|\|\nu\|$, in which $L^1(G)$ sits as a closed two-sided ideal, $M(G) = L^1(G)\oplus M_s(G)$ by Lebesgue decomposition. The involution $\mu^*(E) = \overline{\mu(E^{-1})}$ is isometric and satisfies the anti-automorphism $(\mu*\nu)^* = \nu^*\!*\mu^*$ on the whole of $M(G)$, with no hypothesis on the group: the isometry uses only that inversion is a homeomorphism. Restricted to the absolutely continuous measures the involution gives $(f\,dx)^* = f^*\,dx$ with $f^*(x) = \overline{f(x^{-1})}\Delta(x)^{-1}$, so the modular factor of the group-algebra involution is the Jacobian of the inversion and vanishes exactly in the unimodular case. The operator of left convolution by $\mu$ on $L^2(G)$ has adjoint $\lambda(\mu)^* = \lambda(\mu^*)$ for every $G$, so $\lambda$ is a *-representation of the measure algebra and the left regular representation satisfies $\lambda(f)^* = \lambda(f^*)$; the unimodular case is the one in which the right regular representation is also unitary on $L^2(G)$ and right convolution is a *-representation with the same involution. The operator-theoretic adjoint is the neighbouring - * Operator Theory article.

Summary of Notation

Further Reading

• Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis I (Springer, second edition, 1979), for the measure algebra, its convolution, its involution and the Lebesgue decomposition.
• Walter Rudin, Fourier Analysis on Groups (Wiley, 1962), for the measure algebra of an abelian group and the involution on it.
• Hans Reiter and Jan D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups (Oxford University Press, second edition, 2000), for the modular function, the unimodular case and the modular twisting of the adjoint.
• Jacques Dixmier, $\mathrm{C}^*$-Algebras (North-Holland, 1977), for the *-representation of the measure algebra on $L^2(G)$ and the group von Neumann algebra.
• Sterling K. Berberian, Baer ${}^*$-Rings (Springer, 1972), for involutions on algebras of measures and the anti-automorphism axioms.