Correction to: On Harmonic Hilbert Spaces on Compact Abelian Groups

Correction to: Journal of Fourier Analysis and Applications https://doi.org/10.1007/s00041-023-09992-4

We correct an error in Theorem 6 of [Das, S., Giannakis, D.: On harmonic Hilbert spaces on compact abelian groups. J. Fourier Anal. Appl. 29, 12 (2023)]. That theorem claimed that given a suitable (e.g., absolutely summable, symmetric, and subconvolutive) function $\lambda : {\hat{G}} \rightarrow {\mathbb {C}}$ on the dual group ${\hat{G}}$, there is an associated harmonic Hilbert space ${\mathcal {H}}_\lambda$ of complex-valued functions on G, which is a Banach $^*$-algebra with respect to pointwise function multiplication and complex conjugation, and the Gelfand spectrum $\sigma ({\mathcal {H}}_\lambda )$ is homeomorphic to G. However, the stated assumptions on $\lambda$ in that theorem are actually not sufficient to deduce that $G \cong \sigma ({\mathcal {H}}_\lambda )$. Here, we show that [DG23, Theorem 6] remains valid if and only if $\lambda$ satisfies the Gelfand–Raikov–Shilov condition. Aside from this modification on the assumptions of Theorem 6, the results of [DG23] remain unchanged.

1 Introduction

Let G be a compact abelian group with (discrete) Pontryagin dual group ${\hat{G}}$. Given a strictly positive function $\lambda \in L^1({\hat{G}})$, define the space

$$\begin{aligned} {\mathcal {H}}_\lambda = \left\{ f: G \rightarrow {\mathbb {C}}: \; \sum _{\gamma \in {\hat{G}}} \frac{|\langle \gamma , f\rangle _{L^2(G)}|^2}{\lambda (\gamma )} < \infty \right\} , \end{aligned}$$

where $\gamma : G \rightarrow S^1 \subset {\mathbb {C}}^\times$ are unitary characters, and $\langle f, g\rangle _{L^2(G)} = \int _G f^*(x) g(x)\,d\mu (x)$ is the $L^2$ inner product associated with the Haar probability measure $\mu$ of G. Recently, [2, Theorem 4] showed that if $\lambda$ satisfies the symmetry and subconvolutivity conditions

$$\begin{aligned} \lambda (\gamma ) = \lambda (-\gamma ), \quad (\lambda * \lambda ) \le {\tilde{C}}\lambda (\gamma ), \quad \forall \gamma \in {\hat{G}}, \end{aligned}$$

for a constant ${\tilde{C}} >0$, then ${\mathcal {H}}_\lambda$ is a reproducing kernel Hilbert space (RKHS; also known as a harmonic Hilbert space in this context [3, 5]) that lies dense in C(G), and is additionally a unital Banach $^*$-algebra with respect to pointwise function multiplication and complex conjugation.

In more detail, the reproducing kernel $k: G \times G \rightarrow {\mathbb {C}}$ of ${\mathcal {H}}_\lambda$ is given by $k(x,y) = (\hat{{\mathcal {F}}} \lambda )(x,y)$, where $\hat{{\mathcal {F}}}: L^1({\hat{G}}) \rightarrow C(G)$ is the Fourier operator on the dual group, $\hat{{\mathcal {F}}} {\hat{f}}:= \sum _{\gamma \in {\hat{G}}} \gamma {\hat{f}}(\gamma )$. Moreover, the Banach $^*$-algebra property for ${\mathcal {H}}_\lambda$ means that (i) for every $f, g \in \mathcal H_\lambda$ the pointwise product fg and complex conjugate $f^*$ lie in ${\mathcal {H}}_\lambda$; (ii) the norm conditions

$$\begin{aligned} \Vert fg\Vert _{{\mathcal {H}}_\lambda } \le C \Vert f\Vert _{{\mathcal {H}}_\lambda } \Vert g\Vert _{{\mathcal {H}}_\lambda }, \quad \Vert f^*\Vert _{{\mathcal {H}}_\lambda } = \Vert f\Vert _{{\mathcal {H}}_\lambda } \end{aligned}$$

(1)

hold for a constant $C \ge 1$; and (iii) the function $1_G: G \rightarrow {\mathbb {C}}$ equal everywhere to 1 is an element of $\mathcal H_\lambda$ that acts as its unit. The paper [2] called such spaces ${\mathcal {H}}_\lambda$ reproducing kernel Hilbert algebras (RKHAs). In what follows, $\langle \cdot , \cdot \rangle _{{\mathcal {H}}_\lambda }$ will denote the inner product of ${\mathcal {H}}_\lambda$.

We emphasize that the multiplicative constant C in (1) is allowed to be greater than 1, in contrast to conventional definitions of Banach $^*$-algebras that require $C = 1$. While cases with $C > 1$ can be reduced to the standard case by scaling the kernel k by a factor of $C^{-2}$ (leading to an equivalent RKHA $\tilde{{\mathcal {H}}}_\lambda$ satisfying $\Vert fg \Vert _{\tilde{{\mathcal {H}}}_\lambda } \le \Vert f \Vert _{\tilde{{\mathcal {H}}}_\lambda } \Vert g\Vert _{\tilde{{\mathcal {H}}}_\lambda }$; see [2, Remark 1]), a number of Banach algebra results that assume $C=1$ do not necessarily hold for $C>1$. In particular, elements $f \in {\mathcal {H}}_\lambda$ satisfying $\Vert 1_G - f \Vert _{{\mathcal {H}}_\lambda } < 1$ are not necessarily invertible (which would be the case if (1) holds for $C = 1$). The latter, turned out to be a source of error in one of the main results of [2], namely, Theorem 6 on the Gelfand spectra of RKHAs, which we now correct.

2 Gelfand Spectra of RKHAs

Let $\sigma ({\mathcal {H}}_\lambda )$ be the spectrum of an RKHA ${\mathcal {H}}_\lambda$, i.e., the set of unital homomorphisms $\chi : {\mathcal {H}}_\lambda \rightarrow {\mathbb {C}}$. It is a direct consequence of the fact that ${\mathcal {H}}_\lambda$ is an RKHS that the evaluation functionals $\delta _x: {\mathcal {H}}_\lambda \rightarrow {\mathbb {C}}$, $\delta _x f = f(x)$, at any $x \in G$ are elements of $\sigma ({\mathcal {H}}_\lambda )$. This implies that the mapping $\beta _\lambda : G \rightarrow \sigma ({\mathcal {H}}_\lambda )$ with $\beta _\lambda (x) = \delta _x$ is a weak-$^*$ continuous inclusion of the group into the RKHA spectrum.

In fact, since ${\mathcal {H}}_\lambda$ is a Hilbert space, every unital homomorphism $\chi$ has, by virtue of being continuous, an associated vector $u \in {\mathcal {H}}_\lambda$ such that $\chi = \langle u, \cdot \rangle _{{\mathcal {H}}_\lambda }$. By the reproducing property of ${\mathcal {H}}_\lambda$, the evaluation functionals $\delta _x$ are of this form with $u = k_x:= k(x,\cdot )$, i.e.,

$$\begin{aligned} \delta _x f = f(x) = \langle k_x, f\rangle _{{\mathcal {H}}_\lambda }. \end{aligned}$$

In light of the above, it is natural to ask whether the spectrum $\sigma ({\mathcal {H}}_\lambda )$ of an RKHA contains other elements besides the evaluation functionals. Theorem 6 of [2] made a claim that for a strictly positive, symmetric function $\lambda \in L^1({\hat{G}})$ (not necessarily subconvolutive) the answer to that question is negative; that is, it was claimed $\beta _\lambda$ is a homeomorphism between G and $\sigma ({\mathcal {H}}_\lambda )$ (the latter, equipped with the weak-$^*$ topology as usual). Such a property has important consequences, including [2, Corollary 7](i) Every non-vanishing function $f \in {\mathcal {H}}_\lambda$ has a multiplicative inverse in ${\mathcal {H}}_\lambda$; (ii) every strictly positive function $f \in {\mathcal {H}}_\lambda$ has a square root in ${\mathcal {H}}_\lambda$; and (iii) the spectrum of any element $f \in {\mathcal {H}}_\lambda$ is equal to the range of f.

Unfortunately, the proof of [2, Theorem 6] made use of an erroneous intermediate result, Lemma 10, which claimed that every maximal ideal of ${\mathcal {H}}_\lambda$ is orthogonal to the unit of $1_G$. The error in Lemma 10 stemmed from the possible non-invertibility of elements $f \in {\mathcal {H}}_\lambda$ satisfying $\Vert 1_G - f \Vert _{{\mathcal {H}}_\lambda } < 1$ when C in (1) is strictly greater than 1 (which is the case for all non-trivial examples of RKHAs known to us). Below, we state a corrected version of [2, Lemma 10], which, even though weaker, gives a geometrical characterization of maximal ideals of RKHAs.

Lemma 1

(Correction of [2, Lemma 10]) Every element f of a maximal ideal of ${\mathcal {H}}_\lambda$ satisfies $|\langle 1_G, f\rangle _{{\mathcal {H}}_\lambda } |\le \sqrt{\Vert 1_G\Vert _{{\mathcal {H}}_\lambda }^2- C^{-2}} \Vert f\Vert _{{\mathcal {H}}_\lambda }$, where $C \ge 1$ is the optimal multiplicative constant in (1).

Proof

See Sect. 4. $\square$

As it turns out, [2, Theorem 6] does not hold without making additional assumptions on the function $\lambda$. We give counter-examples in section 3 below. The following theorem establishes the Gelfand–Raikov–Shilov (GRS) condition as necessary and sufficient for $\sigma (\mathcal H_\lambda )$ to contain no other elements than the evaluation functionals. This is a Hilbert space analog of Wiener’s lemma for Banach algebras under pointwise multiplication known as Fourier–Beurling algebras or Fourier–Wermer algebras, which are modeled after $L^1$ (see, e.g., [2, Sect. 7], [6, §19], [7, Theorem 6.4], [8, Sect. 2.8], [9, Sect. 3.7]). Sufficient conditions for Fourier–Wermer algebras to have isomorphic Gelfand spectra to the group G include the Beurling–Domar condition [4] (which implies the GRS condition), and subadditivity of the weight function when G is compact [1] (see [2, Sect. 7.1] for further details on the compact case). Tchamitchian [10, 11] studies Fourier images of weighted $L^2$ spaces on ${\mathbb {R}}^d$ with Banach algebra structure under pointwise multiplication, associated with rapidly-decaying, radial weights satisfying a logarithmic integrability condition. They establish the existence of compactly supported elements and characterize the associated multiplier algebras, among other results.

In what follows, we will use additive notation for the dual group ${\hat{G}}$, and for $\gamma \in {\hat{G}}$, $n\gamma = \gamma + \cdots + \gamma$ is the n-fold sum of $\gamma$. We also note that ${\hat{G}}$ can be identified with a subset of ${\mathcal {H}}_\lambda$ by $\gamma \mapsto \phi _\gamma$ where $\phi _\gamma (x):= \gamma (x)$. Moreover, the set $\{ \psi _\gamma \}_{\gamma \in {\hat{G}}}$, where $\psi _\gamma = \sqrt{\lambda (\gamma )}\phi _\gamma$, is an orthonormal basis of ${\mathcal {H}}_\lambda$.

Theorem 2

(Correction of [2, Theorem 6]) Let $\lambda \in L^1({\hat{G}})$ be a strictly positive function on the dual group such that ${\mathcal {H}}_\lambda$ is an RKHA. Then, the spectrum $\sigma ({\mathcal {H}}_\lambda )$ consists entirely of the pointwise evaluation functionals, i.e.,

$$\begin{aligned} \sigma ({\mathcal {H}}_\lambda ) = \left\{ \delta _x = \langle k_x,\cdot \rangle _{{\mathcal {H}}_\lambda }:\; x \in G \right\} , \end{aligned}$$

if and only if $\lambda$ satisfies the GRS condition

Proof

Fix $\chi \in \sigma ({\mathcal {H}}_\lambda )$ and observe that ${\hat{G}} \subset {\mathcal {H}}_\lambda$ implies $\chi |_{{\hat{G}}} :{\hat{G}} \rightarrow {\mathbb {C}}^\times$ is a group homomorphism. There exists $u=\sum _{\gamma \in {\hat{G}}} u_\gamma \psi _\gamma \in {\mathcal {H}}_\lambda$ such that $\chi =\langle u,\cdot \rangle _{{\mathcal {H}}_\lambda }$, and so $\chi (\phi _\gamma )={u_\gamma }/{\sqrt{\lambda (\gamma )}}$. Since $(u_\gamma )_{\gamma \in {\hat{G}}} \in \ell ^2({\hat{G}})$,

$$\begin{aligned} 1 \ge \limsup _{n\rightarrow \infty } |u_{n\gamma }|^{2/n} = \limsup _{n\rightarrow \infty }\lambda (n\gamma )^{1/n} |\chi (\phi _\gamma )|^2=|\chi (\phi _\gamma )|^2, \end{aligned}$$

where we have used condition (GRS) to obtain the last equality. Thus, we have $|\chi (\phi _{\gamma })| \le 1$ for every $\gamma \in {\hat{G}}$, and so $|\chi (\phi _{\gamma })| = 1$ and $\chi |_{{\hat{G}}} \in \hat{\hat{G}}$. By Pontryagin duality, there exists $x \in G$ such that $\chi |_{{\hat{G}}} = \delta _x$. By density of the span of ${\hat{G}}$ in ${\mathcal {H}}_\lambda$, $\chi =\delta _x$.

To show necessity, suppose that $\lambda$ violates the (GRS) condition. Since $\lambda$ is bounded, this means that for some $\gamma \in {\hat{G}}$,

$$\begin{aligned} \limsup _{n \rightarrow \infty } \lambda (n\gamma )^{1/n} = e^{-\alpha } <1. \end{aligned}$$

(2)

We will show that ${\mathcal {H}}_\lambda$ contains non-vanishing, non-invertible functions.

By (2), $\lambda (n\gamma ) \le e^{-\alpha n/2}$ for some $n > n_0$. For $0< \delta < \alpha /4$, define $f = 1-e^{-\delta } \phi _\gamma \in {\mathcal {H}}_\lambda$ and observe that $|f(x)| \ge 1-e^{-\delta } >0$ and $\frac{1}{f(x)}=\sum _{n=0}^\infty e^{-n\delta } \phi _{n\gamma }(x).$ This implies that $\frac{1}{f} \notin {\mathcal {H}}_\lambda$ as

$$\begin{aligned} \left\Vert \frac{1}{f}\right\Vert _{{\mathcal {H}}_\lambda }^2 =\sum _{n=0}^\infty \frac{e^{-2n\delta }}{\lambda (n\gamma )} \ge \sum _{n=n_0}^\infty e^{n(\alpha /2 - 2\delta )} = \infty . \end{aligned}$$

Therefore, $f{\mathcal {H}}_\lambda$ is a nontrivial ideal, and there exists a nontrivial character $\chi$ such that $\chi f=0$. This character cannot be a point evaluation character as $0 \notin {{\,\textrm{ran}\,}}f$. $\square$

Remark 3

We have stated the GRS condition using the function $\lambda$ that determines the reproducing kernel k of ${\mathcal {H}}_\lambda$ (via Fourier transform), rather than the corresponding weight function $w = \lambda ^{-1/2}$ commonly used in the definition of Beurling algebras. Note that the assumption that ${\mathcal {H}}_\lambda$ is an RKHA implies that w is submultiplicative (as required in formulations of the GRS condition in terms of w; see, e.g., [7, Definition 1]). In particular, we have $w(\gamma +\gamma ') \le C w(\gamma ) w(\gamma ')$ for the constant C from (1). Indeed, since $\Vert \phi _\gamma \Vert ^{-2}_{{\mathcal {H}}_\lambda } = \lambda (\gamma )$ and $\Vert \phi _{\gamma +\gamma '}\Vert _{{\mathcal {H}}_\lambda } \le C^2 \Vert \phi _\gamma \Vert _{{\mathcal {H}}_\lambda }^2 \Vert \phi _{\gamma '}\Vert _{{\mathcal {H}}_\lambda }^2$, we have $\lambda (\gamma )\lambda (\gamma ') \le C^2 \lambda (\gamma +\gamma ')$ and thus $w(\gamma +\gamma ') \le C w(\gamma ) w(\gamma ')$.

3 RKHAs with Non-evaluation Unital Characters

The (GRS) condition also provides a way to build examples of RKHAs with non-evaluation unital characters.

Let $\lambda \in L^1({\mathbb {Z}})$ be a strictly positive, symmetric, and subconvolutive function inducing an RKHA $\mathcal H_\lambda$ on ${\mathbb {T}}^1$. For $0<r<1$, $\nu (k)=\lambda (k)r^{|k|}$ is also strictly positive, symmetric, and subconvolutive, and thus induces an RKHA ${\mathcal {H}}_\nu$. However, $\nu$ violates (GRS) since $\limsup _{n\rightarrow \infty } \nu (nk)^{1/n} \le r^{|k|} < 1$ whenever $k \ne 0$. Since $(\sqrt{\nu (n)}\rho ^n)_{n \in {\mathbb {Z}}} \in \ell ^2({\mathbb {Z}})$ for $\frac{1}{\sqrt{r}}<\rho < \sqrt{r}$, then for every complex number z in the annulus $A_r:= \{ z \in {\mathbb {C}} \mid \frac{1}{r}<|z|^2<r \}$ the vector $u_z=\sum _{n \in {\mathbb {Z}}} \sqrt{\nu (n)} \overline{z}^n \psi _n \in \mathcal H_\nu$ defines a unital character of ${\mathcal {H}}_\nu$, $\chi _z=\langle u_z, \cdot \rangle _{{\mathcal {H}}_\nu }$ (since $\langle u_z, \phi _\gamma \phi _{\gamma '} \rangle _{\mathcal H_\nu } = \langle u_z, \phi _\gamma \rangle _{\mathcal H_\nu } \langle u_z, \phi _{\gamma '} \rangle _{\mathcal H_\nu }$). These characters provide an identification of ${\mathcal {H}}_\nu$ as a Hilbert space of functions that are analytic in $A_r$ by $f(z)=\langle u_z, f \rangle _{{\mathcal {H}}_\lambda }$. The involution on ${\mathcal {H}}_\nu$ then becomes $f^*=\overline{f \circ \alpha }$ where $\alpha$ is the automorphism of the annulus given by $\alpha (z)=\overline{1/z}$.

4 Proof of Lemma 1

We equip ${\mathcal {H}}_\lambda$ with an equivalent norm, ${\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| \cdot \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }$, induced from the operator norm of $B(\mathcal H_\lambda )$ and the regular representation $\pi$, viz.

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| f \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }= \Vert \pi f \Vert _{B({\mathcal {H}}_\lambda )}. \end{aligned}$$

This norm is a Banach algebra norm satisfying

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| fg \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } \le {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| f \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| g \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }, \quad {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| 1_G \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } = 1. \end{aligned}$$

As a result, any element $f \in {\mathcal {H}}_\lambda$ such that ${\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| 1_G -f \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } < 1$ is invertible. Moreover, one readily verifies that

$$\begin{aligned} \Vert f\Vert _{{\mathcal {H}}_\lambda } \le {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| f \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| } \le C \Vert f\Vert _{{\mathcal {H}}_\lambda }, \quad \forall f \in {\mathcal {H}}_\lambda , \end{aligned}$$

so any $f \in {\mathcal {H}}_\lambda$ satisfying $\Vert 1 - f \Vert _{{\mathcal {H}}_\lambda } < C^{-1}$ is invertible.

Next, suppose that $I \subset {\mathcal {H}}_\lambda$ is a maximal ideal, and let $1_G = u + v$ with $u \in I$, $v \in I^\perp$, and $\Vert u\Vert _{{\mathcal {H}}_\lambda }^2 + \Vert v \Vert _{{\mathcal {H}}_\lambda }^2 = \Vert 1_G\Vert _{\mathcal H_\lambda }^2$. Since u is an element of a maximal ideal, it is not invertible, and from the above it follows that $\Vert v\Vert _{{\mathcal {H}}_\lambda } = \Vert 1_G - u \Vert _{\mathcal H_\lambda } \ge C^{-1}$, giving $\Vert u\Vert _{{\mathcal {H}}_\lambda } \le \sqrt{\Vert 1_G\Vert _{{\mathcal {H}}_\lambda }^2- C^{-2}}$. As a result, for every $f \in I$, we have

$$\begin{aligned} |\langle 1_G, f\rangle _{{\mathcal {H}}_\lambda }|= |\langle u,f\rangle _{{\mathcal {H}}_\lambda } |\le \sqrt{\Vert 1_G\Vert _{{\mathcal {H}}_\lambda }^2- C^{-2}} \Vert f\Vert _{{\mathcal {H}}_\lambda }. \end{aligned}$$

$\square$