Reproducing Kernel Hilbert Algebras

• RKHAs are reproducing kernel Hilbert spaces augmented with a bounded comultiplication that enables controlled pointwise multiplication.
• They extend classical harmonic analysis and Fourier methods by incorporating algebraic operations in non-commutative and quantum contexts.
• Applications include weighted Fourier algebras and quantum paragrassmann algebras, providing new tools for spectral analysis and function approximation.

Reproducing kernel Hilbert algebras (RKHAs) generalize the structure of reproducing kernel Hilbert spaces (RKHSs) by introducing an algebraic operation compatible with the Hilbert space norm. This fusion of harmonic analysis, operator theory, and Banach algebra theory allows pointwise multiplication to become a bounded operation in suitably weighted function spaces, and extends to non-commutative and quantum settings. RKHAs have well-defined spectral properties, categorical frameworks, and deep connections to Fourier analysis, convolution semigroups, and function approximation theory.

1. Structure and Definition of Reproducing Kernel Hilbert Algebras

Let $X$ be a set and $\mathcal{H} \subset \mathbb{C}^X$ be an RKHS with reproducing kernel $K : X \times X \to \mathbb{C}$ and inner product $\langle f, g \rangle_{\mathcal{H}}$. Feature maps $k_x(\cdot) = K(\cdot, x)$ allow the identification $f(x) = \langle f, k_x \rangle_{\mathcal{H}}$, guaranteeing continuity of evaluation.

A reproducing kernel Hilbert algebra is an RKHS with a bounded linear comultiplication $$\Delta: \mathcal{H} \to \mathcal{H} \otimes \mathcal{H}$$, which extends the rule $\Delta(k_x) = k_x \otimes k_x$ for any $x \in X$. Its adjoint, $$\Delta^*: \mathcal{H} \otimes \mathcal{H} \to \mathcal{H}$$, implements pointwise multiplication:

$\Delta^* (f \otimes g)(x) = f(x) g(x)\,,$

with the bound:

$$\|fg\|_{\mathcal{H}} = \| \Delta^*(f \otimes g) \|_{\mathcal{H}} \leq \|\Delta\| \|f\|_{\mathcal{H}} \|g\|_{\mathcal{H}}\,.$$

If $1 \in \mathcal{H}$, $\mathcal{H}$ is unital and multiplication operators $f \mapsto M_f$ identify $\mathcal{H}$ with its multiplier algebra (Giannakis et al., 2 Jan 2024).

2. Subconvolutivity and Algebraic Closure

Concrete constructions arise on locally compact abelian groups $G$, where weights $\lambda \in L^1(\widehat{G}) \cap C_0(\widehat{G})$ determine the algebraic structure. The critical condition is subconvolutivity:

$$\lambda * \lambda(\gamma) = \int_{\widehat{G}} \lambda(\alpha) \lambda(\gamma - \alpha) \, d\hat{\mu} (\alpha) \leq C \lambda(\gamma) \,,$$

the necessary and sufficient criterion for bounded comultiplication and closure under pointwise multiplication (Giannakis et al., 2 Jan 2024, Das et al., 2019).

In the canonical example,

$$\mathcal{H}_\lambda = \widehat{\mathcal{F}} \left[ L^2_\omega (\widehat{G}) \right],\qquad \omega = \lambda^{-1/2},$$

with kernel

$$K(x, y) = \int_{ \widehat{G} } \lambda(\gamma)\, \overline{\gamma(x)} \gamma(y) \, d\hat{\mu} (\gamma)\,.$$

$\Delta$ diagonalizes on the orthonormal basis $\psi_\gamma = \sqrt{\lambda(\gamma)} \gamma$:

$$\Delta(\psi_\gamma) = \sum_{\alpha + \beta = \gamma} \sqrt{ \frac{ \lambda(\alpha) \lambda(\beta) }{ \lambda(\gamma) } } \psi_\alpha \otimes \psi_\beta\,,$$

with $$\|\Delta(\psi_\gamma)\|^2 = (\lambda * \lambda)(\gamma) / \lambda(\gamma) \leq C$$ (Giannakis et al., 2 Jan 2024).

For compact abelian groups $G$ ($\widehat{G}$ discrete), the condition $\omega^{-1} \in \ell^1(\widehat{G})$ and $\omega^{-1}$ subconvolutive ensures $\mathcal{H}_\omega(G)$ is a unital symmetric Banach *-algebra dense in $C(G)$ (Das et al., 2019).

3. Example Constructions and Embeddings

For $G = \mathbb{Z}^n$, $\widehat{G} = \mathbb{T}^n$, and weights $\lambda(k) = e^{ -\tau |k|_p^p }$, $\lambda$ is strictly positive and subconvolutive. The corresponding RKHA:

$$\mathcal{H}_\lambda = \left\{ f = \sum_{ k \in \mathbb{Z}^n } \widehat{f}(k) e^{2\pi i k \cdot x} : \sum | \widehat{f}(k) |^2 \lambda(k) < \infty \right\}$$

has kernel $$K(x, y) = \sum_{ k \in \mathbb{Z}^n } \lambda(k) e^{2\pi i k \cdot (y-x)}$$. When $G=\mathbb{R}^n$, $\lambda(\xi) = e^{ -\tau |\xi|_p^p }$ yields a non-unital RKHA (Giannakis et al., 2 Jan 2024).

On compact groups, weighted Fourier spaces $L^2_\omega(\widehat{G})$ yield RKHSs and Banach *-algebras when $\omega^{-1}$ is subconvolutive. For weights $$w_{s, r}(\gamma) = \prod_{j=1}^d (1 + |\gamma_j|)^{s/r}$$, algebras of dominating mixed smoothness arise, with continuous embeddings between associated RKHA and Fourier-Wermer algebras (Das et al., 2019).

4. Tensor Products, Functoriality, and Pullbacks

RKHAs are closed under Hilbert space tensor product: for $(\mathcal{H}_1, \Delta_1)$ and $(\mathcal{H}_2, \Delta_2)$, the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$ inherits a bounded comultiplication:

$$\Delta_{12} = (\mathrm{id} \otimes \tau \otimes \mathrm{id}) \circ (\Delta_1 \otimes \Delta_2),$$

with $$\Delta_{12}(k_x \otimes k_y) = (k_x \otimes k_y) \otimes (k_x \otimes k_y)$$, giving a bounded multiplication on $\mathcal{H}_1 \otimes \mathcal{H}_2$ (Giannakis et al., 2 Jan 2024).

Given any map $\varphi : S \to X$ and RKHA $(\mathcal{H}, K, \Delta)$, the pullback $$\varphi^* \mathcal{H} = \{ f \circ \varphi : f \in \mathcal{H} \} \subset \mathbb{C}^S$$ remains an RKHA with induced comultiplication:

$$\Delta_{\text{pull}} = (T_\varphi^* \otimes T_\varphi^*) \circ \Delta \circ T_\varphi,$$

where $T_\varphi$ is the isometry $T_\varphi(k_{\varphi(s)}) = k_s^{\text{pull}}$ (Giannakis et al., 2 Jan 2024).

5. Categorical Framework and the Spectrum Functor

The category RKHA comprises objects (unital or nonunital RKHAs) and morphisms $T : \mathcal{H}_1 \to \mathcal{H}_2$ intertwining reproducing kernels: $T(k^1_x) = k^2_{F(x)}$ for an underlying map $F$. The monoidal product is the Hilbert space tensor $\otimes$, with unit $\mathbb{C}$ as the one-point RKHA.

The spectrum functor $\operatorname{sp}: \text{RKHA} \to \text{Top}$ assigns to $\mathcal{H}$ the set of characters:

$$\sigma(\mathcal{H}) = \{ \chi: \mathcal{H} \to \mathbb{C}\;\text{multiplicative, non-zero} \}$$

with the weak-* topology. For unital RKHAs, there is a natural homeomorphism:

$$\Phi : \sigma(\mathcal{H}_1) \times \sigma(\mathcal{H}_2) \to \sigma(\mathcal{H}_1 \otimes \mathcal{H}_2),\qquad \Phi(\xi_1, \xi_2) = \xi_1 \otimes \xi_2$$

making $(\operatorname{sp}, \Phi)$ a monoidal functor to compact Hausdorff spaces (Giannakis et al., 2 Jan 2024).

For weighted Fourier RKHAs on compact abelian groups, every nonzero multiplicative linear functional is evaluation at a point of $G$, and $\operatorname{Spec}(\mathcal{H}) \cong G$ (Das et al., 2019).

6. Spectral Realization and Function Approximation

Given a weight $\lambda$ symmetric, strictly positive, subconvolutive, and satisfying the Gelfand–Raikov–Shilov condition

$$\lim_{n\to\infty} \lambda(n\gamma)^{1/n} = 1, \qquad \forall \gamma \in \widehat{G}$$

the Gelfand map

$$\Gamma: G \to \sigma( \mathcal{H}_\lambda ), \qquad \Gamma(x) = k_x$$

is a homeomorphism. For $\widehat{G} = \mathbb{Z}^n$, $\sigma(\mathcal{H}_\lambda) \cong \mathbb{T}^n$; for $\widehat{G} = \mathbb{R}^n$, $\sigma(\mathcal{H}_\lambda) \cong \mathbb{R}^n$ (one-point compactified if nonunital). By pullback, the spectrum functor can realize all compact subspaces of $\mathbb{R}^n$ (Giannakis et al., 2 Jan 2024).

Weighted Wiener-type algebras $A_w(G)$ embed continuously into RKHA analogues, and the RKHA spaces can serve as Banach algebras for high-dimensional function approximation and analysis of mixed smoothness (Das et al., 2019).

7. Noncommutative and Quantum Examples

Paragrassmann algebras $\mathrm{PG}_{\ell,q}$, with nilpotent generators and $q$-commutation relations, exemplify non-function RKHAs. Despite lacking isomorphisms to algebras of functions, Segal–Bargmann subalgebras $\mathcal{B}_H$ of $\mathrm{PG}_{\ell,q}$ admit Hilbert space structures and reproducing kernels:

$$K_{SB}(\theta, \eta) = \sum_{j=0}^{\ell-1} \frac{1}{w_j} \bar{\theta}^j \otimes \eta^j$$

with the reproducing property $$f(\theta) = \langle K_{SB}(\theta, \cdot), f(\cdot) \rangle_w$$ for any $f \in \mathcal{B}_H$ (Sontz, 2012).

For the full (noncommutative) algebra $\mathrm{PG}_{\ell,q}$, the Gram matrix of the anti-Wick basis is invertible and provides a reproducing kernel $K_{PG}$ satisfying $$f(\theta, \bar{\theta}) = \langle K_{PG}(\theta, \bar{\theta}; \cdot), f(\cdot) \rangle_w$$ for elements $f \in \mathrm{PG}_{\ell,q}$. The notion of "evaluation" is interpreted by substitution homomorphisms ($\eta \mapsto \theta$) rather than point evaluation.

Quantum RKHAs retain kernel properties such as uniqueness and Hermitian symmetry, but operator inequalities rather than scalar pointwise bounds govern norms, reflecting the noncommutative structure (Sontz, 2012). Such examples illustrate the extension of RKHA theory beyond classical function spaces and establish new paradigms for reproducing kernels in quantum and algebraic analysis.