Fourier algebra

Informal

Let ${\displaystyle G}$ be a locally compact abelian group, and ${\displaystyle {\hat {G}}}$ the dual group of ${\displaystyle G}$. Then ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ is the space of all functions on ${\displaystyle {\hat {G}}}$ which are integrable with respect to the Haar measure on ${\displaystyle {\hat {G}}}$, and it has a Banach algebra structure where the product of two functions is convolution. We define ${\displaystyle A(G)}$ to be the set of Fourier transforms of functions in ${\displaystyle L_{1}({\hat {\mathit {G}}})}$, and it is a closed sub-algebra of ${\displaystyle CB(G)}$, the space of bounded continuous complex-valued functions on ${\displaystyle G}$ with pointwise multiplication. We call ${\displaystyle A(G)}$ the Fourier algebra of ${\displaystyle G}$.

Similarly, we write ${\displaystyle M({\hat {\mathit {G}}})}$ for the measure algebra on ${\displaystyle {\hat {G}}}$, meaning the convolution algebra of all finite regular Borel measures on ${\displaystyle {\hat {G}}}$. We define ${\displaystyle B(G)}$ to be the set of Fourier-Stieltjes transforms of measures in ${\displaystyle M({\hat {\mathit {G}}})}$. It is a closed sub-algebra of ${\displaystyle CB(G)}$, the space of bounded continuous complex-valued functions on ${\displaystyle G}$ with pointwise multiplication. We call ${\displaystyle B(G)}$ the Fourier-Stieltjes algebra of ${\displaystyle G}$. Equivalently, ${\displaystyle B(G)}$ can be defined as the linear span of the set ${\displaystyle P(G)}$ of continuous positive-definite functions on ${\displaystyle G}$.^[1]

Since ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ is naturally included in ${\displaystyle M({\hat {\mathit {G}}})}$, and since the Fourier-Stieltjes transform of an ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ function is just the Fourier transform of that function, we have that ${\displaystyle A(G)\subset B(G)}$. In fact, ${\displaystyle A(G)}$ is a closed ideal in ${\displaystyle B(G)}$.

Formal

While non-abelian groups don't have dual groups, we may still define the Fourier algebras for any locally compact group in terms of unitary representations.^[2] Let ${\displaystyle G}$ be a locally compact group. For any unitary representation ${\displaystyle \pi }$ of ${\displaystyle G}$ on a Hilbert space ${\displaystyle H_{\pi }}$, and any ${\displaystyle \xi ,\eta \in H_{\pi }}$, we denote by ${\displaystyle \pi _{\xi ,\eta }}$ the complex-valued function on ${\displaystyle G}$ defined by ${\displaystyle \pi _{\xi ,\eta }(g)=\langle \pi (g)\xi ,\eta \rangle }$. The Fourier-Stieltjes algebra ${\displaystyle B(G)}$ is then defined as the algebra of all complex functions on ${\displaystyle G}$ that arise as matrix coefficients ${\displaystyle \pi _{\xi ,\eta }}$ for some unitary representation ${\displaystyle \pi }$ of ${\displaystyle G}$ and some ${\displaystyle \xi ,\eta \in H_{\pi }}$. ${\displaystyle B(G)}$ forms an algebra with pointwise addition and multiplication, as for any unitary representations ${\displaystyle \pi }$ and ${\displaystyle \rho }$ of ${\displaystyle G}$, ${\displaystyle \pi _{\xi _{1},\eta _{1}}+\rho _{\xi _{2},\eta _{2}}=(\pi \oplus \rho )_{\xi _{1}\oplus \xi _{2},\eta _{1}\oplus \eta _{2}}}$ and ${\displaystyle \pi _{\xi _{1},\eta _{1}}\rho _{\xi _{2},\eta _{2}}=(\pi \otimes \rho )_{\xi _{1}\otimes \xi _{2},\eta _{1}\otimes \eta _{2}}}$. We define the norm in ${\displaystyle B(G)}$ to be given by ${\displaystyle \|u\|={\text{inf}}\{\|\xi \|\|\eta \|:u(g)=\langle \pi (g)\xi ,\eta \rangle {\text{ for some representation }}\pi \}}$. This norm makes ${\displaystyle B(G)}$ a Banach algebra. We define the Fourier algebra ${\displaystyle A(G)}$ to be the closed subalgebra spanned by the matrix coefficients of the left regular representation of ${\displaystyle G}$ on ${\displaystyle L^{2}(G)}$.

Abelian case

Let ${\displaystyle B({\mathit {G}})}$ be a Fourier–Stieltjes algebra and ${\displaystyle A({\mathit {G}})}$ be a Fourier algebra such that the locally compact group ${\displaystyle {\mathit {G}}}$ is abelian. Let ${\displaystyle M({\widehat {\mathit {G}}})}$ be the measure algebra of finite measures on ${\displaystyle {\widehat {G}}}$ and let ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$ be the convolution algebra of integrable functions on ${\displaystyle {\widehat {G}}}$, where ${\displaystyle {\widehat {\mathit {G}}}}$ is the character group of the Abelian group ${\displaystyle {\mathit {G}}}$.

The Fourier–Stieltjes transform of a finite measure ${\displaystyle \mu }$ on ${\displaystyle {\widehat {\mathit {G}}}}$ is the function ${\displaystyle {\widehat {\mu }}}$ on ${\displaystyle {\mathit {G}}}$ defined by

${\displaystyle {\widehat {\mu }}(x)=\int _{\widehat {G}}{\overline {X(x)}}\,d\mu (X),\quad x\in G}$

The space ${\displaystyle B({\mathit {G}})}$ of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra ${\displaystyle M({\widehat {\mathit {G}}})}$. Restricted to ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$, viewed as a subspace of ${\displaystyle M({\widehat {\mathit {G}}})}$, the Fourier–Stieltjes transform is the Fourier transform on ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$ and its image is, by definition, the Fourier algebra ${\displaystyle A({\mathit {G}})}$. The generalized Bochner theorem states that a measurable function on ${\displaystyle {\mathit {G}}}$ is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on ${\displaystyle {\widehat {G}}}$ if and only if it is positive definite. Thus, ${\displaystyle B({\mathit {G}})}$ can be defined as the linear span of the set of continuous positive-definite functions on ${\displaystyle {\mathit {G}}}$. This definition is still valid when ${\displaystyle {\mathit {G}}}$ is not Abelian.

Helson–Kahane–Katznelson–Rudin theorem

Let ${\displaystyle A(G)}$ be the Fourier algebra of a compact group ${\displaystyle G}$. Building upon the work of Wiener, Lévy, Gelfand, and Beurling, in 1959 Helson, Kahane, Katznelson, and Rudin proved that, when ${\displaystyle G}$ is compact and abelian, a function f defined on a closed convex subset of the plane operates in ${\displaystyle A(G)}$ if and only if ${\displaystyle f}$ is real analytic.^[3] In 1969 Dunkl proved the result holds when ${\displaystyle G}$ is compact and contains an infinite abelian subgroup.

Relationship to group operator algebras

If ${\displaystyle G}$ is a locally compact group, we may define the associated group algebras ${\displaystyle C^{*}(G)}$ to be the enveloping C*-algebra of ${\displaystyle L^{1}(G)}$ and the von Neumann algebra ${\displaystyle VN(G)}$ associated to the left regular representation ${\displaystyle \lambda }$ of ${\displaystyle G}$ on ${\displaystyle L^{2}(G)}$. Then ${\displaystyle B(G)}$ is the Banach space dual of ${\displaystyle C^{*}(G)}$ and ${\displaystyle VN(G)}$ is the dual of ${\displaystyle A(G)}$. The pairing between ${\displaystyle A(G)}$ and ${\displaystyle VN(G)}$ is defined by, for ${\displaystyle T\in VN(G)}$ and ${\displaystyle u\in A(G)}$ defined ${\displaystyle u(x)=\langle \lambda (x)\xi ,\eta \rangle }$, by ${\displaystyle \langle T,u\rangle =\langle T\xi ,\eta \rangle }$.

In the case where ${\displaystyle G}$ is abelian, we have ${\displaystyle A(G)\cong L^{1}({\hat {G}})}$ and ${\displaystyle VN(G)\cong L^{\infty }({\hat {G}})}$, so this reduces to the fact that ${\displaystyle L^{\infty }({\hat {G}})}$ is the dual of ${\displaystyle L^{1}({\hat {G}})}$. Likewise, we have ${\displaystyle C^{*}(G)\cong C_{0}({\hat {G}})}$ (the algebra of continuous functions on ${\displaystyle {\hat {G}}}$ vanishing at infinity) and ${\displaystyle B(G)\cong M({\hat {G}})}$, so that ${\displaystyle B(G)}$ is the dual of ${\displaystyle C^{*}(G)}$ reduces to the fact that ${\displaystyle M({\hat {G}})}$ is the dual of ${\displaystyle C_{0}({\hat {G}})}$.