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C τ G > τ ∗ aetesrcueof structure the have G tspaces. rt . agba osrce by constructed -algebras ∗ pe ihteuniform the with ipped = k agbastructure -algebra 0, S : G 1 H X hs spaces these , p analogously , τ ∈ × sdnein dense is X C X X [1 ( sacompact a is X , saBanach a is ∞ → .If ). H ly ,adin and ], τ C X The . sa is r 2 SUDDHASATTWADASANDDIMITRIOSGIANNAKIS and the evaluation functional f f(x) is bounded analogously to C-spaces. If X is a topological space, or a Cr manifold, equipped7→ with a Borel measure µ, there is a rich theory on embeddings of RKHSs into Lp(µ) and/or Cr(X) spaces [1], which in some ways parallels the classical embedding theorems for Sobolev spaces. RKHSs have found widespread use in the mathematical theory of learning [2, 3], and more recently in ergodic theory [4]. Yet, despite their attractive properties, RKHSs generally lack the algebra structure enjoyed by Cr and L∞ spaces. Intuitively, an RKHS induces a notion of regularity of functions (depending on the kernel), which need not be preserved under products of functions. While there are important examples of Hilbert spaces of functions with an algebra structure, such as Sobolev algebras of sufficient regularity on Rn [5], these spaces rely on the existence of an underlying differentiable structure. Other examples of Sobolev algebras include Banach spaces induced from abstract “Carr´e du Champ” operators [6], or positive characters on Lie groups [7]. More generally, in situations where an RKHS fails to be closed under multiplication, one can employ weak product spaces with a Banach algebra structure [8, 9]. The main contribution of this work is to construct a class of function spaces which are simulta- neously RKHSs and unital Banach ∗-algebras under the pointwise product of functions. We refer to such spaces as reproducing kernel Hilbert algebras (RKHAs). Our focus is on the setting where X has the structure of a compact abelian group, which is relevant in a variety of signal processing and ergodic theory applications such as analysis of periodic or quasiperiodic signals. However, some of our results also hold for more general locally compact abelian groups. We now state more precisely the algebraic properties required of an RKHA, starting from the notion of a Hilbert ∗-algebra, or H∗-algebra, introduced by Ambrose [10]. Definition 1. An H∗-algebra is a Hilbert space H over the complex numbers, equipped with a continuous binary multiplication operation making it a Banach algebra. That is, we require that fg C f g , f, g H, k kH ≤ k kH k kH ∀ ∈ where C is a constant and H the norm induced by the inner product , H of H. Additionally, we require that for every f k·kH there exists an adjoint element f ∗ H,h· such·i that ∈ ∈ fg,h = g,f ∗h = f, hg∗ , g, h H. (1) h iH h iH h iH ∀ ∈ Equation (1) implies that for each f H, the operations of left and right multiplication by f is a norm-preserving, ∗-homomorphism∈ between H and B(H), the C∗-algebra of bounded linear maps on H. Note that this notion of a Hilbert algebra is somewhat different from other definitions [11, 12], mainly by the requirement of a Banach algebra property. A classical example of a Hilbert ∗-algebra is the L2(X) convolution algebra of a compact topological group X equipped with a Haar measure. In what follows, we will work with a weaker notion than the H∗-algebras in Definition 1, obtained by relaxing the adjoint condition in (1), while requiring an isometric antilinear involution as in Banach ∗-algebras. We refer to the resulting algebraic structure as a semi-H∗-algebra. Definition 2. A semi-H∗-algebra is a Hilbert space H over the complex numbers, equipped with a continuous binary multiplication operation and an antilinear, norm-preserving involution ∗ : H H, making it a Banach ∗-algebra. That is, we require that → fg C f g , f ∗ = f , (fg)∗ = g∗f ∗, f, g H, (2) k kH ≤ k kHk kH k kH k kH ∀ ∈ where C is a constant and the norm of H. k·kH The RKHAs constructed in this work are unital semi-H∗-algebras of complex-valued functions. Moreover, we employ pointwise function multiplication and complex conjugation as the algebraic product and involution, respectively, akin to C∗-algebras of functions. 3
Definition 3. An RKHS of complex-valued functions on a set X is said to be a reproduc- ing kernel Hilbert algebra (RKHA)H if it is a (commutative) semi-H∗-algebra when endowed with the pointwise product and complex conjugation of functions as the multiplication operation and antilinear involution, respectively. If is unital, we will use the symbol 1 to denote the unit of , i.e., the function equal to 1 at H X H every point in X. Note that we do not require that the norm of 1X is equal to 1. Remark 1. In this paper, we have opted to state the Banach algebra condition fg C f g k kH ≤ k kHk kH allowing a general constant C, as opposed to the more conventional definition fg H f H g H. This choice does not affect any of the results presented below, as C can be absorbedk k in≤ a redefinition k k k k of the reproducing kernel k of to a scaled kernel k˜ := C2k. The corresponding RKHA, ˜, has the same elements as , and satisfiesH fg f g , so we can view as an equivalentH H k kH˜ ≤ k kH˜ k kH˜ k·kH˜ norm to H. In the present work, it is natural to allow a general C, as this enables a direct identificationk·k the reproducing kernels of certain RKHAs with Markov transition kernels without having to employ additional normalization factors. That being said, it should be kept in mind that a number of standard results on Banach algebras require appropriate modification when C 1. For example, the fact that a state on a unital Banach ∗-algebra with isometric involution has≥ unit operator norm does not necessarily hold when C 1. ≥ Locally compact abelian groups. For the rest of the paper, G X will be assumed to be a locally compact abelian group (LCA), equipped with a Haar measure≡ µ. If G is compact, we will assume that µ is normalized to a probability measure. We let Gˆ denote the dual group of G, i.e., the abelian group of continuous homomorphisms γ : G S1, equipped with its dual measure,µ ˆ → [13, 14]. We identify each element of Gˆ with a continuous, complex-valued function on G, taking values in the unit circle S1 C, and acting on C multiplicatively as a unitary character. The trivial ˆ ⊂ character in G will be denoted by 0Gˆ. When there is no risk of confusion with scalar multiplication of functions, the inverse of γ Gˆ will be denoted by γ. We use (G) to denote the set of Borel ∈E − P probability measures on G and ν( )= G( ) dν the expectation operator with respect to ν (G). All vector spaces are over the complex· numbers.· ∈ P We present a general scheme for constructingR continuous, positive-definite kernels and RKHSs on LCAs. Our first result is for the case when G is a compact, connected Lie group of dimension d, in which case we can assume without loss of generality that G = Td. The dual group Gˆ is then isomorphic to Zd, and can be explicitly characterized as Gˆ = φ : j Zd , φ (x) := eij·x, (3) { j ∈ } j where φj are the standard Fourier functions. The duality between G and Gˆ is seen from the fact that φ φ = φ , φ∗ = φ , i, j Zd, i j i+j j −j ∀ ∈ ∗ d where denotes complex conjugation of functions. For any 0 Definition 4. A collection τ : τ > 0 of RKHSs is said to be a nested family if the following hold for each τ ′ > 0. {H } (i) ′ is a dense subspace of with 0 <τ<τ ′. Hτ Hτ (ii) For all f ′ , τ (0,τ ′] f is a continuous, increasing function. ∈Hτ ∈ 7→ k kHτ With this notation and definitions, we have: 4 SUDDHASATTWADASANDDIMITRIOSGIANNAKIS Theorem 1 (RKHAs on Td). Fix p (0, 1) and τ > 0. Then, for every x,y Td, the series ∈ ∈ −τ|j|p kτ (x,y) := λτ,jφj(x)φj(y), λτ,j := e , (4) Zd jX∈ ∞ d converges uniformly and absolutely to a C , strictly positive-definite kernel kτ on T , inducing a sequence of RKHSs with orthonormal bases λ1/2φ : j Zd . Moreover, : τ > 0 is a Hτ { τ,j j ∈ } {Hτ } nested family of unital RKHAs, satisfying 1 = 1, and each is a subspace of C∞(Td) lying k GkHτ Hτ dense in C(Td). A general framework. Theorem 1 gives an explicit construction of RKHAs on Td. We now construct kernels and RKHAs in a more general setting. In what follows, C0(G) (resp. C0(Gˆ)) will denote the Banach space of complex-valued, continuous functions on an LCA G (resp. Gˆ) vanishing at infinity, equipped with the uniform norm. Moreover, : L1(G) C (Gˆ) and ˆ : L1(Gˆ) F → 0 F → C0(G) will denote the Fourier and inverse Fourier transforms, i.e., f(γ) := f(x)γ( x) dµ(x), ˆfˆ(x)= fˆ(γ)γ(x) dµˆ(γ). F − F ˆ ZG ZG We also use and ⋆ to denote the convolution (algebraic product) and antilinear involution opera- tions on the∗ group algebra L1(G), respectively, i.e., (f g)(x)= f(x y)g(y) dµ(y), f ⋆(x)= f( x), ∗ − − ZG and a similar notation for the corresponding operations on L1(Gˆ). We will furthermore need the following notion of uniform continuity on LCAs, which is equivalent to the corresponding notion for uniform spaces [15, Definition 7.6]. Definition 5. A function f : G C on an LCA is said to be uniformly continuous if for every ǫ > 0 there is a neighborhood U→of the identity element of G, such that for every x G and y x + U, f(x) f(y) <ǫ. ∈ ∈ | − | Note that every function in C0(G) is uniformly continuous, and every uniformly continuous function is continuous. With this notation and definitions, we now consider a one-parameter family of functions λτ on the dual group which, depending on the context, will be required to satisfy one or more of the following properties. Assumption 1. The following hold for every τ > 0: 1 (i) λτ L (Gˆ) C0(Gˆ) is a continuous, absolutely integrable, positive-valued function on the dual∈ group. ∩ (ii) For each γ Gˆ, λ (γ) is a continuous, decreasing function of τ. ∈ τ (iii) λτ is strictly positive-valued, and there exists a constant Cτ > 0 such that (λ λ )(γ) C λ (γ), γ G.ˆ (5) τ ∗ τ ≤ τ τ ∀ ∈ 1 ˆ ⋆ (iv) λτ is self-adjoint as an element of the L (G) convolution algebra, i.e., λτ = λτ . (v) λτ is normalized such that λτ (0Gˆ) = 1. Given such a family λτ , we will be interested in the following translation-invariant kernels kτ : G G C on the group, defined as × → k (x,y)= l (x y), l := ˆλ C (G). (6) τ τ − τ F τ ∈ 0 1 By Bochner’s theorem for LCAs [13, Section 1.4.2], since λτ is a positive-valued function in L (Gˆ), k is a continuous, positive-definite kernel on G, and thus induces an RKHS of continuous τ Hτ 5 functions. Moreover, if Assumption 1(iv) holds, then lτ and kτ are real. The following proposition summarizes the main properties of the RKHSs and integral operators associated with kτ . Proposition 2 (Kernels on LCAs). With notation as above and under Assumption 1(i), the fol- lowing hold for every τ > 0. (i) kτ is uniformly continuous, and τ is a subspace of C0(G). (ii) The integral operator H Kτ : f kτ ( ,x)f(x) dµ(x) 7→ G · ∞ Z 2 maps L (G) into the Banach space of bounded functions on G. Moreover, Kτ maps L (G) and L1(G) into the space of uniformly continuous functions on G. (iii) For every x G and γ Gˆ, ∈ ∈ Kτ γ(x)= λτ (γ)γ(x). If, in addition, G is compact, we have: 2 ∗ (iv) Kτ is a compact operator from L (G) into τ with dense range. Moreover, τ := Kτ Kτ is a self-adjoint, Hilbert-Schmidt operator on HL2(G). Its eigenvalues are λ (γK) , and the { τ }γ∈Gˆ characters γ Gˆ are corresponding eigenfunctions forming an orthonormal basis of L2(G). ∈ (v) If λτ is strictly positive-valued, then kτ is strictly positive-definite, and τ is dense in C(G). Moreover, R : (G) with H τ P →Hτ R (ν)= k (x, ) dν(x) τ τ · ZG is an injective map from Borel probability measures into linearly independent RKHS func- tions, satisfying E f = R (ν),f , f . ν h τ iHτ ∀ ∈Hτ Furthermore, restricted to Radon probability measures in (G), R is weakly continuous. P τ A positive-definite kernel k on a locally compact Hausdorff space X whose corresponding RKHS lies dense in C0(X) is known as C0-universal [16], or C-universal if X is compact [17]. If the map HR : (X) defined as in Proposition 2(v) is injective, the kernel k is called characteristic [18]. ForP a characteristic→H kernel, the so-called feature map F : X with F (x) = k(x, ) is injective, and has linearly independent range [17]. Using these results,→H we construct RKHAsh assuming· that G is compact. Recall that if G is compact, then Gˆ has a discrete topology. Theorem 3 (RKHAs on compact abelian groups). With notation as above, if G is compact and Assumption 1 holds, τ : τ > 0 is a nested family of unital RKHAs satisfying 1G Hτ = 1. Moreover, every element{H of this} family is a dense subspace of C(G), and the correspondingk k Hτ kernel kτ is characteristic. The following corollary of Theorem 3 is a consequence of the theory of Markov semigroups. We recall that a strongly continuous semigroup M of operators on L2(G) is a Markov semigroup { τ }τ≥0 if for every τ 0, Mτ is positivity-preserving (i.e., Mτ f 0, µ-a.e., whenever f 0, µ-a.e.), ≥ 2 ≥ ≥ Mτ 1G = 1G, and G Mτ f dµ = G f dµ for all f L (G). Moreover, Mτ τ≥0 is said to be ergodic if each M has a simple eigenvalue at 1. ∈ { } τ R R Corollary 4. With the assumptions and notation of Theorem 3, if λτ further satisfies ′ λ (γ) < 1, λ (γ)λ ′ (γ)= λ ′ (γ), τ,τ > 0, γ Gˆ 0 , τ τ τ τ+τ ∀ ∀ ∈ \ { Gˆ} then each corresponding kernel kτ is a transition probability density kernel with respect to Haar measure, i.e., k (x, ) 0, k (x, ) dµ = 1, τ > 0, x G. τ · ≥ τ · ∀ ∀ ∈ ZG 6 SUDDHASATTWADASANDDIMITRIOSGIANNAKIS Moreover, Id is an ergodic Markov semigroup on L2(G). {Kτ }τ>0 ∪ { } In particular, the kernels on Td from Theorem 1 have the Markov property stated in Corollary 4. 1 Fractional diffusions on the circle. In the case G = S , Gˆ = Z, the kernel kτ in (4) can be recognized as the transition kernel of the fractional diffusion semigroup on L2(G) generated by the power ∆p/2 of the (positive-definite) Laplacian on S1. That is, ∆p/2 is the self-adjoint operator on L2(G) uniquely characterized by its action on the Fourier functions, ∆p/2φ = j pφ , j | | j and it follows directly from the definition of kτ that τ is the time-τ fractional heat operator of order p/2, i.e., K p/2 = e−τ∆ , τ> 0. Kτ This concludes the statement of our main results. The remainder of the paper is organized as follows. Proposition 2 is proved in Section 2. Section 3 contains the proofs of Theorem 3 and Corollary 4. In Section 4, we prove Theorem 1 by invoking Theorem 3 for the choice λτ (φj) = λτ,j, which will be shown to satisfy Assumption 1. In Section 5, we study properties of RKHAs stemming from the existence of continuous pointwise evaluation functionals. In particular, in Theorem 10, we show that under appropriate conditions on the kernel, a compact abelian group G is homeomorphic to the spectrum of a unital RKHA defined on it, analogously to the corresponding result for the C∗-algebra of continuous functions onHG. We also describe a ∗-compatibility property (Proposition 12) for a set of “classical” states of B( ) induced by Borel probability measures on G, under evaluation on the regular representatives ofH in B( ). Section 6 contains the proof of Theorem 10. The paper ends in Section 7 with a conclusoryH discH ussion. 2. Proof of Proposition 2 In what follows, Sx will denote the shift operator by group element x G; i.e., Sxf(y)= f(x+y) x∈ for any element y G and function f : G C. The collection S = S x∈G then forms a strongly ∈ → p { } continuous group of isometries on any of the spaces L (G) and C0(G). We also recall the standard property of Fourier transforms that (Sxf)(γ)= γ(x)( f)(γ), f L1(G), x G, γ G.ˆ (7) F F ∀ ∈ ∀ ∈ ∀ ∈ Reusing notation, we shall let : L2(G) L2(Gˆ) denote the unitary extension of the Fourier operator on L1(G). F → As further auxiliary results, we will need the following standard facts from Mercer theory [1, 19, 20], which we state without proof. Lemma 5. Let X be a compact Hausdorff space, µ a finite Borel measure with full support in X, and k : X X C a positive-definite, continuous kernel with associated RKHS . Then, the following hold.× → H .i) is a subspace of C(X), and the inclusion ֒ C(X) is compact) H 2 H → (ii) K : L (G) with Kf = X k( ,x)f(x) dµ(x) is a well-defined, compact integral operator with dense→H range. · (iii) The adjoint K∗ : L2(GR) is equal to the restriction of the inclusion map C(X) ֒ L2(G) on ; that is, K∗H→f = f, µ-a.e. → (iv) :=H K∗K is a positive, self-adjoint Hilbert-Schmidt operator on L2(G), with eigenvalues K λ0 λ1 0 and a corresponding orthonormal basis φ0, φ1,... of eigenfunctions. Moreover,≥ ≥···ց if k is positive-valued-valued, is of trace class. { } K (v) The set ψ = λ−1/2Kφ : λ > 0 is an orthonormal basis of satisfying K∗ψ = λ1/2φ . { j j j j } H j j j 7 (vi) The kernel admits the Mercer series expansion k(x,y)= ψj(x)ψj(y), j:Xλj>0 which converges uniformly for (x,y) X X. ∈ × We now proceed to prove the claims of the proposition. Claim (i). To show that kτ is uniformly continuous, note first that the kernel shape function lτ ′ lies in C0(G), and is thus uniformly continuous. As a result, for every ǫ > 0 and (x,x ) G G there exists a neighborhood U of the identity element of G such that ∈ × l (x x′) l (z) < ǫ, z (x x′)+ U. | τ − − τ | ∀ ∈ − Therefore, defining the open neighborhood V = (y,y′) G G : y y′ U of the identity of G G, we get { ∈ × − ∈ } × l (x x′) l (y y′) = k (x,x′) k (y,y′) < ǫ, (y,y′) (x,x′)+ V, | τ − − τ − | | τ − τ | ∀ ∈ which proves that kτ is uniformly continuous. To show that is a subspace of C (G), note that every f is the -norm limit of finite Hτ 0 ∈Hτ Hτ linear combinations of kernel sections of the form f = n−1 c k (x , ), where k (x , ) lies in n j=0 j τ j · τ j · C (G). Moreover, proceeding similarly to [1, Lemma 2.1], we have 0 P f(x) = kτ (x, ),f H kτ (x, ) f = kτ (x,x) f lτ f H , | | |h · i τ | ≤ k · kHτ k kHτ k kHτ ≤ k kC0(G)k k τ and thus f l f . k ksup ≤ k τ kC0(G) k kHτ The above implies that the Cauchy sequence f C (G) converging to f is also Cauchy n ∈Hτ ∩ 0 ∈Hτ with respect to C0(G) norm, so f lies in C0(G). This proves Claim (i). 1 Claim (ii). By Assumption 1(i), lτ is an element of L (G) C0(G), which implies that for every x x ∩ ∞ ∈ G, kτ (x, )= S lτ also lies in this space. Therefore, for every f L (G), Kτ f(x)= G kτ (x, )f dµ exists for· every x G, and we have ∈ · ∈ R x K f(x) = k (x, )f dµ S l 1 f ∞ = l 1 f ∞ . | τ | τ · ≤ k τ kL (G)k kL (G) k τ kL (G)k kL (G) ZG ∞ This shows that Kτ is well-defined as a bounded linear map from L (G) to the space of bounded functions on G, as claimed. 2 Suppose now that f is an element of L (G). Then, we can express Kτ f(x) as the inner product x K f(x)= k (x, ),f 2 = S l ,f 2 , τ h τ · iL (G) h τ iL (G) 1 2 where we have used the fact that kx lies in C0(G) L (G), and thus in L (G) irrespective of whether G is compact or non-compact. Also, note that for∩ every u C (G) L1(G), ∈ 0 ∩ 2 u 2 u u 1 . k kL (G) ≤ k kC0(G)k kL (G) Using these facts and the Cauchy-Schwartz inequality on L2(G), we get 2 x y x y 2 K f(x) K f(y) S l S l S l S 1 f 2 | τ − τ | ≤ k τ − τ kC0(G)k τ − τ kL (G)k kL (G) x y 2 2 S l S l l 1 f 2 , ≤ k τ − τ kC0(G)k τ kL (G)k kL (G) for all x,y G. The uniform continuity of Kτ f then follows by the strong continuity of S, using a neighborhood∈ of the identity of G G analogous to V in the proof of Claim (i). × 1 Finally, the uniform continuity of Kτ f for f L (G) (which is a stronger result than the f L2(G) case if G is compact) follows from a similar∈ argument using the bound ∈ K f(x) K f(y) k (x, ) k (y, ) f 1 . | τ − τ | ≤ k τ · − τ · kC0(G)k kL (G) 8 SUDDHASATTWADASANDDIMITRIOSGIANNAKIS This completes the proof of Claim (ii). Claim (iii). The claim is a direct consequence of (7) and the definition of kτ in (6), viz. K γ(x)= k (x,y)γ(y) dµ(y)= Sxl ( y)γ(y) dµ(y)= Sxl (y)γ( y) dµ(y) τ τ τ − τ − ZG ZG ZG = (Sxl )(γ)= γ(x)( l )(γ)= γ(x)( −1λ )(γ)= γ(x)λ (γ), F τ F τ F τ τ where we have used the fact that l = ˆλ = −1λ since λ lies in C (Gˆ). τ F τ F τ τ 0 Claim (iv). If G is compact, µ is a finite Borel measure with full support. The well-definition 2 and compactness of Kτ : L (G) τ , as well as the facts that Kτ has dense range and τ is Hilbert-Schmidt, are direct consequences→ H of Lemma 5. K In addition, since Gˆ now has the discrete topology, Claim (iii) holds for every γ Gˆ. As a ∈ result, the quantities λτ (γ) are the eigenvalues of τ , and the characters γ are the corresponding eigenfunctions. The characters also form an orthonormalK basis of L2(G) by standard properties of compact abelian groups. Claim (v). To verify that the range of Kτ is dense in C(G) whenever kτ is strictly positive-definite, observe that ran Kτ includes the span of all characters whenever λτ is strictly positive-valued. It is known that the linear space spanned by the group of characters separates points on an LCA [21, Theorem 4.2.2], and is therefore dense in C0(G) by the Stone-Weierstrass theorem. We therefore conclude that ran Kτ is dense in C(G), i.e., kτ is C-universal, if λτ is strictly positive-valued. Next, it is known that a C-universal kernel on a compact Hausdorff space is strictly positive- definite and characteristic [22, 19]. If νn is a sequence of Radon measures in (G) converging to a ∗ P (Radon) measure p (G) in the weak- topology, then for every f τ C(G), Rτ (νn),f Hτ = E ∈E P ∈H ⊆ h i νn f converges to νf = Rτ (ν),f Hτ . Thus, Rτ is weakly continuous on Radon probability measures, proving Claim (iv)h and completingi the proof of the proposition. 3. Proof of Theorem 3 and Corollary 4 We begin by fixing some notation. First, the norms and inner products in the space τ will be denoted by and , respectively. Moreover, by Proposition 2(iii), we identify theH eigenfunc- k·kτ h· ·iτ tions φ of the integral operator with the characters of G, i.e., elements of the discrete set Gˆ. j Kτ Then, using Λ Gˆ to denote the set γ Gˆ : λ (γ) > 0 , and defining ξ (γ) := λ (γ), we let τ ⊆ { ∈ τ } τ τ 1 p ψτ,γ := Kτ γ = ξτ (γ)γ, γ Λτ , (8) ξτ (γ) ∈ be the orthonormal basis functions of constructed as per Lemma 5. We also let Sˆγ be the shift Hτ operator on Gˆ, defined analogously to Sx on G. The convolution of any two elementsu, ˆ vˆ L2(Gˆ) can then be expressed as ∈ ⋆ γ (ˆu vˆ)(γ)= uˆ , Sˆ vˆ 2 . ∗ h iL (Gˆ) Section 3.1 below contains two auxiliary lemmas aiding the proof of the theorem. Theorem 3 is proved in Section 3.2 by invoking these results and Proposition 8. The latter is proved in Section 3.3. Section 3.4 contains the proof of Corollary 4. 3.1. Auxiliary results. First, note that, by Lemma 5, the elements of τ can be explicitly char- acterized as H = f = fˆ γ = fˆ ψ /ξ (γ) : fˆ 2/λ (γ) < . Hτ γ γ τ,γ τ | γ| τ ∞ γX∈Λτ γX∈Λτ γX∈Λτ 9 In the above, the coefficients fˆγ coincide with the values of the Fourier transform of the continuous −1 −1 1 function f τ C(G) on Λτ . That is, we have fˆγ = ˆ f(γ), where ˆ : C(G) L (Gˆ) is the ∈H ⊆ F F →2 inverse of the Fourier operator on the dual group. Moreover, the condition fˆγ /λτ (γ) < γ∈Λτ | | ∞ is equivalent to the statement that the functionu ˆ : Gˆ C with → P fˆ /ξ (γ), γ Λ , uˆ(γ)= γ τ ∈ τ (0, otherwise, lies in L2(Gˆ). Together, these facts imply: Lemma 6. Under Assumption 1(i), the following statements are equivalent: (i) f is an element of τ . 2H −1 (ii) There exists uˆ L (Gˆ) with f = uˆ 2 such that ˆ f = ξ uˆ. ∈ k kτ k kL (Gˆ) F τ Moreover, uˆ is unique, and can be explicitly constructed as uˆ = ξ+ ˆ−1f, where τ F 1/ξ (γ), γ Λ , ξ+(γ)= τ ∈ τ τ 0, γ Gˆ Λ . ( ∈ \ τ In addition, we have: Lemma 7 (Nested structure of the τ ). Under Assumptions 1(i, ii), the following hold for every τ ′ > 0. H ′ (i) For each τ (0,τ ], τ ′ is a subspace of τ . Moreover, if Λτ ′ = Λτ , the inclusion τ τ ′ is dense. ∈ H H H ⊆H (ii) For every f ′ , the map τ f is continuous and monotonically increasing on (0,τ ′]. ∈Hτ 7→ k kτ Proof. Claim (i). First, observe that under Assumptions 1(i, ii), τ Λτ is a decreasing sequence of + ′ ′ ′ ′ 7→ ′ + ′ sets, so ξτ ξτ ξτ = ξτ whenever τ <τ . Moreover, for every τ > 0, : (0,τ ] ξτ ξτ is a sequence of increasing functions, pointwise-continuous with respect to τ. LetT now f be7→ an arbitrary element of ′ . By Lemma 6 and the observation just made, there existsu ˆ L2(Gˆ) such that Hτ ∈ −1 + ′ ˆ f = ξ ′ uˆ = ξ ξ ξ ′ , τ (0,τ ]. F τ τ τ τ ∀ ∈ Thus, definingv ˆ = ξ+ξ ′ uˆ L2(Gˆ), we get ˆ−1f = ξ vˆ , and we conclude that f lies in by τ τ τ ∈ F τ τ Hτ ′ ′ ′ ′ Lemma 6. Supposing now that Λτ = Λτ , ψτ ,γ γ∈Λτ is an orthonormal basis of τ , whose span { } ′ H in τ is equal to the span of the orthonormal basis ψτ,γ γ∈Λτ . Thus, τ lies dense in τ , proving ClaimH (i). { } H H Claim (ii). With notation as above, we have + f = v 2 = ξ ξ ′ uˆ 2 , k kτ k τ kL (Gˆ) k τ τ kL (Gˆ) and the claim follows from the fact that τ (0,τ ′] ξ+ξ ′ is a sequence of increasing functions. ∈ 7→ τ τ ∗ 3.2. The Banach -algebra structure. We now show that Assumption 1(iii) leads to τ being closed under the pointwise product of functions. For that, we need the following importantH result, which will be proved in Section 3.3. Proposition 8. Let Assumptions 1(i–iii) hold. Then, for every τ > 0 and f, g τ , the pointwise product fg lies in , and satisfies fg C f g . ∈H Hτ k kτ ≤ τ k kτ k kτ Assuming that Proposition 8 holds, it follows that τ is an RKHS which is also an abelian Banach algebra under the pointwise product of functions.H 10 SUDDHASATTWA DAS AND DIMITRIOS GIANNAKIS Next, we show that under the additional Assumption 1(iv), f ∗(x) = f(x) is an isometric, anti- linear involution on . Since λ is both self-adjoint and positive-valued, we have λ (γ)= λ ( γ) H τ τ τ − for every γ Gˆ, and the orthonormal basis elements ψ from (8) satisfy ∈ τ,γ 1/2 1/2 1/2 −1 ψ (x)= λτ (γ)γ(x)= λ ( γ)γ(x)= λ ( γ)γ (x)= ψ (x), τ,γ τ − τ − τ,−γ ∗ ∗ so ψτ,γ τ = ψτ,−γ τ = 1. Therefore, preserves the norm of orthonormal basis vectors of τ . k k k k ∗ H Moreover, it is clearly antilinear and involutive, so τ is a Banach -algebra equipped with this operation and pointwise function multiplication. His also unital and satisfies 1 = 1 since, Hτ k Gkτ by Assumption 1(v), the unit basis vector ψτ,0 is equal to the trivial character in Gˆ, and thus everywhere equal to 1 on G, i.e., ψτ,0 = λτ (0Gˆ)0Gˆ = 0Gˆ = 1G. Put together, the facts above show for each τ > 0, τ is a unital RKHA satisfying 1G τ = 1, as claimed. That : τ > 0 is a nested family thenH follows from Lemma 7 and thek factk that {Hτ } λ is strictly positive-valued (which implies that Λ = Gˆ for all τ > 0). Finally, that each lies τ τ Hτ dense in C(G), and kτ is characteristic, follows from Propositions 2(iv, v), respectively, again in conjunction with the strict positivity of λτ . This completes the proof of the theorem. 3.3. Proof of Proposition 8. Let τ > 0 and f, g be arbitrary. By Lemma 6, to show that ∈Hτ the continuous function fg lies in τ it is enough to show that the functionw ˆ : Gˆ C defined as + ˆ−1 2 ˆ H + ˆ−1 + ˆ−1 → 2 ˆ wˆ = ξτ (fg) lies in L (G). To that end, lettingu ˆ = ξτ f andv ˆ = ξτ g be the L (G) representativesF of f and g, respectively, from Lemma 6, we obtainF F −1 −1 −1 ⋆ γ ˆ (fg)(γ) = ( ˆ f ˆ g)(γ) = ((ξ uˆ) (ξ vˆ))(γ)= (ξ uˆ) , Sˆ (ξ vˆ) 2 . F F ∗ F τ ∗ τ h τ τ iL (Gˆ) Then, using standard properties of shift operators and L2 inner products, as well as the fact that 1 ξτ is real and L (Gˆ)-self-adjoint, we get −1 2 ⋆ γ 2 ⋆ γ γ 2 ˆ (fg)(γ) = (ξ uˆ) , Sˆ (ξ vˆ) 2 = ξ uˆ , (Sˆ ξ )(Sˆ vˆ) 2 |F | |h τ τ iL (Gˆ)| |h τ τ iL (Gˆ)| γ ∗ γ 2 γ γ ∗ γ ∗ γ = ξ S ξ , uˆ Sˆ vˆ 2 ξ S ξ ,ξ S ξ 2 uˆ Sˆ v, uˆ Sˆ v 2 |h τ τ iL (Gˆ)| ≤ h τ τ τ τ iL (Gˆ)h iL (Gˆ) γ ⋆ 2 γ 2 ⋆ 2 2 = λ , Sˆ λ 2 uˆ ,S vˆ 2 = [(λ λ )(γ)][( uˆ vˆ )(γ)] h τ τ iL (Gˆ)h| | | | iL (Gˆ) τ ∗ τ | | ∗ | | C λ (γ)[( uˆ⋆ 2 vˆ 2)(γ)], ≤ τ τ | | ∗ | | where we used Assumption 1(iii) to arrive at the last line. Thus, since λτ is strictly positive-valued, we have ξ+(γ) = 1/√λ (γ), and wˆ(γ) 2 C ( uˆ⋆ 2 vˆ 2)(γ). Therefore, τ τ | | ≤ τ | | ∗ | | ⋆ 2 2 wˆ 2 C uˆ vˆ 1 , k kL (Gˆ) ≤ τ k| | ∗ | | kL (Gˆ) and we conclude thatw ˆ lies in L2(Gˆ) since uˆ⋆ 2 vˆ 2 is the convolution of the L1(Gˆ) elements uˆ⋆ 2 and vˆ 2. | | ∗ | | | | | | 3.4. Proof of Corollary 4. Since λ λ ′ = λ ′ , we have l l ′ = l ′ , and thus ′ = τ τ τ+τ τ ∗ τ τ+τ Kτ Kτ ′ . Moreover, by Proposition 2(iii), the eigenvalues of each are given by λ (γ), and are Kτ+τ Kτ τ therefore contained in the interval (0, 1]. In particular, 1 = λτ (0Gˆ ) is a simple eigenvalue of each 1/n τ corresponding to the constant eigenfunction 0Gˆ = 1G. We also have λτ/n = λτ , for any n> 0, K + so as τ 0 , λτ converges pointwise to 1. Consequently, the operators τ converge pointwise to the → 2 2 K identity on L (G), i.e., lim + K f = f for all f L (G). Moreover, 2 = λ (0 ) = 1, τ→0 τ ∈ kKτ kB(L (G)) τ Gˆ so we conclude that := τ τ>0 Id is a strongly continuous contraction semigroup, consisting of self-adjoint compactK operators.{K } ∪ By Hille-Yosida theory for strongly continuous, contraction semigroups, there exists a positive, −τD self-adjoint operator such that, for all τ > 0, τ = e . This operator is diagonal in the character basis of L2(DG), i.e., γ = η(γ)γ, where ηK(γ)= τ −1 log λ for any τ > 0. In particular, D − τ 11 has a simple eigenvalue η(0Gˆ) = 0 corresponding to the constant eigenfunction 1G. It then followsD from results on Markov semigroups (e.g., [23, Chapter 14, Theorem 2]) that is the infinitesimal generator of an ergodic Markov semigroup, e−τD . This semigroup is identical−D to { }τ≥0 , proving the corresponding claim in the corollary. That kτ (x, ) is a transition probability density withK respect to Haar measure then follows immediately from t·he fact that it is the kernel of the integral operator , which was just shown to be Markovian on L2(G). Kτ 4. Proof of Theorem 1 d When G = T , the characters are the standard Fourier functions φj, given by (3). Moreover, the coefficients λτ,j in (4) lead to the function λτ (φj) := λτ,j. We begin by showing that λτ satisfies Assumption 1. d First, λτ is clearly positive-valued. Moreover, it is continuous since Gˆ = Z is discrete, and −τnp vanishes at infinity by definition of the λτ,j. Defining an = e for n N, we have a2n/an = 2p−1 ∈ (an) and a2n+1/an a2n/an, and thus lim n→∞a2n/an = limn→∞ a2n+1/an = 0. It therefore ≤ ∞ follows from the second ratio test for convergence of series that S := n=0 an is finite, so −τ|j|p d λτ dµˆ = e = (2S 1) P ˆ| | − G Zd Z jX∈ 1 is also finite. This shows that λτ lies in L (Gˆ) C0(Gˆ), and Assumption 1(i) is satisfied. The continuity and monotonicity properties in Assumption∩ 1(ii), the strict positivity property in As- sumption 1(iii), the self-adjointness property in Assumption 1(iv), and the normalization condition in Assumption 1(iv) follow immediately from (4). Next, to check that (5), and thus Assumption 1(iii), hold, we put (λ λ ) (φ ) λ λ 1/2 τ ∗ τ m = R ,R := τ,m−j τ,j , λ (φ ) m m λ τ m Zd τ,m jX∈ and show that sup Rm Cτ , (9) m∈Zd ≤ which is equivalent to (5). We first consider the case d = 1, and then extend the proof to higher dimensions. p p p (1) −τu (j) The case d = 1. Defining u (t) := m t + t m , we have R = Rm Z e m . m | − | | | − | | m ≡ j∈ (1) Z We show that Rm is uniformly bounded over m . As one can readily verify, um(Pt)= u−m( t), so we can assume without loss of generality that ∈m> 0. In that case, we get − 0 m ∞ (1) −τum(j) −τum(j) −τum(j) −τum(j) Rm = e = e + e + e . (10) Z Xj∈ j=X−∞ Xj=1 j=Xm+1 Now, the first and third sums in the right-hand side of (10) are equal, and have the uniform upper bound 0 ∞ ∞ ∞ p p p p e−τum(j) = e−τum(j) = e−τ((m+n) +n −m ) < e−τn < . ∞ n=0 n=0 j=X−∞ j=Xm+1 X X Moreover, for m > 0 and t [0, m/2] we have u (m/2+ t) = u (m/2 t), so the second sum is ∈ m m − ⌈m/2⌉ −τum(j) less than or equal to 2 j=1 e . It thus remains to establish that P ⌈m/2⌉ −τum(j) sup Sm < , Sm := e . (11) m∈Z ∞ Xj=1 12 SUDDHASATTWA DAS AND DIMITRIOS GIANNAKIS Note that for any 0 Proof. The sum in (11) can be split as m∗ ⌈m/2⌉ −τum(j) −τum(j) Sm = Am + Bm, Am := e , Bm := e . Xj=1 j=Xm∗+1 We show that the assumptions in the lemma lead to uniform bounds for the sums Am and Bm. First, by strict concavity of um on [0,m], we have j um(j) um(m∗) Uj, 1 j m∗, ≥ m∗ ≥ ≤ ≤ where U is any strictly positive number less than lim infm→∞ um(m∗)/m∗. This leads to the uniform bound m∗ m∗ A = e−τum(j) e−τUj < . m ≤ ∞ Xj=1 Xj=1 Next, since um is monotonically increasing on [0, m/2], we have u (j) u (m ), m < j m/2, m ≥ m ∗ ∗ ≤ leading again to a uniform bound, ⌈m/2⌉ ⌈m/2⌉ B = e−τum(j) e−τum(m∗) = ( m/2 m )e−τum(m∗) < . m ≤ ⌈ ⌉− ∗ ∞ j=Xm∗+1 j=Xm∗+1 It remains to identify an m∗ satisfying the conditions in Lemma 9. To that end, observe that u (t) t p t p t m = 1 + 1= u . mp − m m − 1 m Moreover, choosing m∗ := sm for any fixed s (0, 1/2), there exist constants M1, M2 > 0 such that ⌈ ⌉ ∈ u (m ) M u (s)mp, m ∗ ≥ 1 1 and thus u (m )/m M u (s)/s > 0, m ∗ ∗ ≥ 2 1 so the first condition in Lemma 9 is satisfied. In addition, there exists a constant M3 such that m/2 m M m, so ⌈ ⌉− ∗ ≤ 3 p ( m/2 m )e−τum(m∗) M me−τu1(s)m , ⌈ ⌉− ∗ ≤ 3 which is bounded over m N. Thus, the conditions in the lemma are satisfied, and we conclude that (9) holds for d = 1. We∈ now verify the claim for general d> 1. 13 The case d> 1. Letting j ,...,j and m ,...,m denote the components of the vectors j, m Zd, 1 d 1 d ∈ the terms Rm in the left-hand side of (9) can be written as d d R = exp τ u (j ) = R(1). m − mi i mi Zd ! jX∈ Xi=1 Yi=1 d (1) The terms R are thus uniformly bounded over m Z since each of the factors Rm is uniformly m ∈ i bounded. We therefore conclude that λτ satisfies Assumption 1, and thus, by Theorem 3, that τ Td Td H is a nested family of unital RKHAs on , lying dense in C( ), and satisfying 1G Hτ = 1. ∞ d k k It remains to verify the claim that the τ are subspaces of C (T ). For that, using elementary tools from analysis, it can be shown that theH Mercer series in (4) series converges absolutely and in r ∞ C norm to kτ for every r> 0; see, e.g., [24, Lemma 17]. Thus, the kernel itself is C , and by results in [1, Section 6], the RKHS is a subspace of C∞. This completes the proof of Theorem 1. Hτ 5. Spectra and states of RKHAs In general, an RKHA on a compact abelian group G (including the τ from Theorems 1 and 3) does not satisfy theHC∗ identity, f ∗f = f 2 , holding for theH algebra of continuous k kC(G) k kC(G) functions on G, nor does it satisfy the H∗-identity in (1) enjoyed by the L2(G) convolution algebra. Failure to meet, in particular, the last property means that the regular representation of into B( ) is not a ∗-representation. Yet, by virtue of their RKHS structure, RKHAs possess continuousH evaluationH functionals δ : C at every x G, x H→ ∈ δ f = f(x)= k(x, ),f , δ ′ = k(x,x), (12) x h · iH k xkH satisfying p δ (fg) = (δ f)(δ g), δ f ∗ = δ f, f, g , x x x x x ∀ ∈H where ′ is the operator norm of functionals in the dual space ′. Every nonzero evaluation k·kH H functional δx is an element of the spectrum of , i.e., the set of nonzero homomorphisms of into C, denoted by σ( ). In addition, as we will seeH below, under appropriate conditions on the kernel,H H the δx provide an abundance of states on , and also induce a set of states on the non-abelian C∗-algebra B( ). Throughout this sectionH we make the following standing assumption. H Assumption 2. is an RKHA on a compact abelian group G with a translation-invariant, re- H producing kernel k(x,y)= l(x y) with l = ˆλ and λ L1(Gˆ). In addition: − F ∈ (i) λ(0Gˆ ) = 1. (ii) λ(γ) > 0 for all γ Gˆ. ∈ Under Assumption 2(i), is unital with 1 = 1, and every δ is nonzero with operator H k kH x norm equal to δx H′ = l(0G). Moreover, under Assumption 2(ii) is dense in C(G) (by Proposition 2(v)),k andk k is a characteristic kernel. Note that the RKHAsH in Theorems 1 and 3 p τ satisfy these conditions. H Recall now that for a compact Hausdorff space G, the spectrum of the C∗-algebra C(G) consists precisely of the evaluation functionals δ at every x G [12]. Moreover, the map β : G x ∈ → σ(C(G)) with β(x)= δx and the Gelfand transform Γ : C(G) C(σ(C(G))) with (Γf)(δx)= f(x) are homeomorphisms with respect to the weak-∗ topology on→ σ(C(G)). The following theorem characterizes analogously the spectra of RKHAs satisfying Assumption 2 and the associated Gelfand transforms. Theorem 10. Under Assumption 2, the following hold. ∗ (i) The map βH : G σ( ) with βH(x)= δx is a homeomorphism with respect to the weak- topology on σ( )→inheritedH as a subset of ′. H H 14 SUDDHASATTWA DAS AND DIMITRIOS GIANNAKIS (ii) Under the identification G σ(C(G)) induced by β, the Gelfand transform ΓH : C(σ( )) with (Γ f)(δ ) =≃f(x) coincides with the inclusion map ι : ֒ C(G)H. →In H H x H → particular, the operator norm of ΓH is equal to l(0G). A proof of Theorem 10 is included in Section 6. Thep following is an immediate corollary of Claim (ii). Corollary 11. Under Assumption 2, the Gelfand transform ΓH has trivial kernel. As a result, is semisimple. H Next, we consider the state space, ( ), of a unital RKHA , i.e., the set of (automatically S H H continuous) positive functionals ϕ : C, normalized such that ϕ(1G) = 1. By (12), for a unital RKHA with reproducing kernel kH→, each nonzero evaluation functional is a state with operator H norm equal to k(x,x). It should be noted that because we allow continuity constants C different from 1 in our definition of Banach algebras in (2), the elements of ( ) need not have unit operator p norm (which would be the case if C = 1). S H Suppose now that the evaluation functional δx at every x G is nonzero. Then, viewing δx as a Dirac probability measure in (G) leads to the identity ∈ P δ = R(δ ), , x G, (13) x h x ·iH ∀ ∈ where R : (G) is the RKHS-valued map on Borel probability measures defined as in Propo- sition 2(v).P By continuity→H of the assignment x k(x, ) as a map from G into , (13) extends to a map P : (G) ( ) such that 7→ · H P → S H (Pν)f = δ f dν(x)= R(ν), . (14) x h ·iH ZG Similarly, to each Dirac probability measure δx (G), we can assign a state ρx (B( )) of the C∗-algebra B( ) given by ∈ P ∈ S H H ρ = tr(Π ), (15) x x· where Π : is the rank-1 projection operator x H→H k(x, ),f k(x, ) f(x)k(x, ) Π f = h · iH · = · . x k(x,x) k(x,x) The assignment δ ρ in (15) then extends to a map Q : (G) (B( )) with x 7→ x P → S H (Qν)A = ρxA dν(x). ZG Intuitively, we think of states of B( ) in the range of Q as “classical” states induced by Borel probability measures on G. Letting Hπ : B( ) denote the regular representation of with π(f)g = fg, the following proposition justifiesH → the interpretationH of states in ran Q as classicalH states, in the sense of acting consistently on regular representatives of with expectation operators for Borel probability measures. H Proposition 12. With notation as above and under Assumption 2 the following hold. (i) The maps P and Q are injective. Moreover, their restrictions to the Radon probability measures in (G) are weakly continuous. (ii) For every ν P (G) and f the compatibility relations ∈ P ∈H Eνf = P (ν)(f)= Q(ν)(π(f)) hold. As a result, we have Q(ν)(π(f ∗)) = Q(ν)(π(f)), even though π need not be a ∗- homomorphism. 15 Proof. By Assumption 2, is a dense subspace of C(G), so its reproducing kernel is C-universal and thus characteristic (seeH the proof of Proposition 2(v)). It follows that R is injective, and therefore so is P since R(ν) is the Riesz representative of P (ν) according to (14). Letting now ψγ = λ(γ)γ : γ Gˆ be the orthonormal basis of , constructed analogously to (8), for every {ν (G) and f ∈ we} get H ∈ P p ∈H f(x) ψγ , k(x, ) H k(x, ), ψγ H Q(ν)(π(f)) = tr(Πxπ(f)) dν(x)= h · i h · i dν(x) G G k(x,x) Z Z γX∈Gˆ ˆ ψγ (x)ψγ (x) = f(x) γ∈G dν(x)= f(x) dν(x)= E f = P (ν)(f). k(x,x) ν ZG P ZG This proves the compatibility relations in Claim (ii), and also implies that Q is injective by the injectivity of P . We also have ∗ ∗ Q(ν)(π(f )) = Eνf = Eνf = Q(ν)(π(f)), verifying the ∗-compatibility result in Claim (ii). The weak continuity of P and Q restricted to Radon probability measures follows analogously to the proof of Proposition 2(v). 6. Proof of Theorem 10 We begin with the following observation about maximal ideals of unital RKHAs. Lemma 13. With the assumptions and notation of Theorem 10, every maximal ideal in is H orthogonal to the unit 1G. Proof. Let 1 = u + v with u I and v I⊥. Since I is a proper, closed subspace of , the unit G ∈ ∈ H 1G does not lie in I, and v is nonzero, i.e., 0 < v H 1. We claim that, in fact, v H = 1. To verify this, by rescaling the kernel of k,k we≤ assume without loss of generalityk k that the multiplicative constant C in Definition 1 is equalH to 1 (see Remark 1). Next, following standard techniques for unital Banach algebras, we equip with an equivalent norm, , induced from the operator norm of B( ) and the regular representationH π, viz. |||·||| H f = π(f) . ||| ||| k kB(H) This norm is a Banach algebra norm satisfying fg f g , 1 = 1, f f . ||| ||| ≤ ||| |||||| ||| ||| G||| ||| ||| ≤ k kH In particular, v v H, so if v H < 1, then u = 1 v. The latter, implies that I contains an invertible element,||| ||| ≤ k contradictingk k k the fact that it is a ma− ximal, and thus proper, ideal in . It follows that v = 1 and u = 0, proving that 1 lies in I⊥. H k kH G We now continue with the proof of Theorem 10. Claim (i). For every ψ σ( ), x G, and net (x ) converging to x, we have ∈ H ∈ i β (x )(ψ)= ψ(x ) ψ(x)= β (x)(ψ), H i i → H ∗ so βH is weak- continuous. In addition, βH is injective since δx = k(x, ), H, and k(x, ) is the image of x under the feature map F : G (which is injective sinceh k· is·i characteristic· by → H Assumption 2). Therefore, since G and σ( ) are compact Hausdorff spaces, to show that βH is a homeomorphism it suffices to show that itH is surjective. To prove the latter by contradiction, suppose that there exists ψ σ( ) ran β . Then, ∈ H \ H I := ker ψ is a maximal ideal in , which is distinct from ker δx for all x G. We claim that I is a dense subspace of C(G). To verifyH this claim, observe first that the closure∈ I¯ of I in C(G) is an ideal. Indeed, if that were not the case, there would exist f C(G) and g I¯ such that fg / I¯. But since is a dense subspace of C(G), there exists a sequence∈ f converging∈ to f in C∈(G) H n ∈H 16 SUDDHASATTWA DAS AND DIMITRIOS GIANNAKIS norm, and similarly there exists g I such that g g in C(G). Defining h = f g , it follows n ∈ n → n n n that hn is a sequence in I (since fn , gn I, and I is an ideal in ) with a C(G)-norm limit h I¯. The latter is equal to fg, contradicting∈ H ∈ the assertion that fgH / I¯. Now suppose that I¯ were∈ contained in a maximal ideal in C(G). Since the maximal ideal space∈ of C(G) is in bijective correspondence with the spectrum σ(C(G)), there would exist an x G such that I¯ ker δ ∈ ⊆ x (with δx understood here as an evaluation functional on C(G)), contradicting the fact that I is distinct from the kernels of all evaluation functionals on . We therefore conclude that I¯ is an ideal distinct from any maximal ideal of C(G), so it mustH be equal to the whole space C(G). We have thus verified that I is dense in C(G). Next, since I is a maximal ideal in , every f I is -orthogonal to 1G by Lemma 13. Moreover, H 2 ∈ H by Assumption 2, the integral operator K : L (G) associated with k satisfies K1G = 1G, and we get →H ∗ 0= 1 ,f = K1 ,f = 1 , K f 2 . h G iH h G iH h G iL (G) As a result, 2 ∗ 2 ∗ 2 1 f 1 K f 2 =1+ K f 2 1, k G − kC(G) ≥ k G − kL (G) k kL (G) ≥ which contradicts the assertion that I is dense in C(G), proving Claim (i). Claim (ii). The fact that ΓH coincides with the inclusion map ι follows directly from the definition of the former and the fact that is a subspace of C(G), viz., H (ΓHf)(δx)= f(x) = (ιf)(x). To verify the claim on the operator norm of Γ , we use the reproducing property of to get H H (Γ f)(δ ) = f(x) = k(x, ),f k(x, ) f = k(x,x) f = l(0 ) f . | H x | | | |h · iHk ≤ k · kHk kH k kH G k kH Setting f to the unit vector f = k(x, )/ k(x, ) H (which is wellp defined since δxpis nonzero at any x G) then saturates the inequality,· provingk · thek claim and the theorem. ∈ 7. Discussion The role of strict concavity. A key property of the functions p employed in the definition of |·| d the coefficients λτ,j in (4) is strict concavity for p (0, 1) on the upper half-space of Z (i.e., on the d ∈ subset (j1,...,jd) Z with j1,...,jd 0). This property ensures that the convolution condition { }⊂ ≥ d in Assumption 1(iii) holds, allowing one to show that the RKHSs τ on T from Theorem 1 are H d actually RKHAs by invoking Theorem 3. Let p, p 1, denote the p-norm on vectors in Z . As a k·k ≥ −τkjk2 simple demonstration of the important role of concavity, we now show that the choice λτ,j = e 2 associated with the strictly convex 2-norm (and the eigenvalues of the heat operator e−τ∆), does not lead to an RKHS with a Banach algebra structure under the pointwise product of functions. Letting ψ = e−τkjk2/2φ be the orthonormal basis functions of the RKHS associated with τ,j j Hτ this choice of λτ,j, we have 1/2 ψτ,iψτ,j = (λτ,iλτ,j/λτ,i+j) ψτ,i+j. Therefore, the product of two functions f = f ψ and g = ψ in is formally given by i i τ,i j τ,j Hτ 1/2 1/2 fg = figj(λτ,iλτ,j/λτ,i+j) ψτ,i+j = P hmψτ,m, hm :=P figm−i(λτ,iλτ,m−i/λτ,m) . Zd Zd i,jX∈ mX∈ Xi Setting d = 1 and f = ψτ,1, we get −τ(k1k2+km−1k2−kmk2)/2 τm hm = gm−1e 2 2 2 = gm−1e . 2 Now the sequence (gm) could be an arbitrary ℓ sequence, and there are choices of such sequences for which (h ) does not lie in ℓ2. Thus, is not closed under pointwise multiplication for p = 2. m Hτ 17 As a side note, we mention that in dimension d = 1, the kernel kτ associated with the limiting case p = 1 (corresponding to the 1-norm on Z which is concave, but not strictly concave) can be simplified as follows: ∞ ∞ k (x,y)= e−|j|τ eij(x−y) = e−jτ eij(x−y) + e−jτ e−ij(x−y) 1 τ − Xj Xj=0 Xj=0 −1 −1 sinh(τ) = 1 e−τ+i(x−y) + 1 e−τ−i(x−y) 1= . − − − cosh(τ) cos(x y) h i h i − − Unfortunately, despite the availability of this closed-form expression for the kernel, the method of proof in Section 4 fails to yield the appropriate uniform bounds required in Assumption 1(iii), and indeed one can construct counter-examples as in the p = 2 case above showing that τ is not closed under the pointwise product of functions. H Applications to learning problems. RKHSs are widely employed in learning theory and tech- niques [2, 3]; e.g., as hypothesis spaces in supervised learning and feature spaces in unsupervised learning. In these and other settings, orthonormal RKHS basis functions (analogous to ψτ,γ in this work) are constructed by eigendecomposition of kernel matrices representing integral oper- 2 ators (analogous to τ ) on finite-dimensional L spaces associated with the empirical sampling measure of the trainingK data. The performance of such schemes depends crucially on the ability to obtain high-quality approximations of large numbers of eigenfunctions of τ from eigenvectors of N N kernel matrices, where N is the number of training samples. BesidesK the considerable com- putational× cost involved, at fixed N, the approximation error from empirical eigenvectors rapidly increases as the corresponding eigenvalue decreases (i.e., as increasingly oscillatory features are being sought), posing significant practical limitations to the performance of techniques utilizing kernel eigenfunctions. RKHAs can potentially alleviate such limitations by virtue of their Banach ∗-algebra structure, which allows generating large numbers of basis functions by taking products and ∗-conjugates of elements of a set of generating eigenfunctions. For example, in the case of data sampled on Td, the full RKHA τ from Theorem 1 can be generated by d appropriate basis functions ψτ,j, the computation ofH which is a significantly more feasible task than the d eigenfunctions required in a typical supervised learning problem. When the data is generated≫ by a measure-preserving, ergodic dynamical dynamical system with non-trivial point spectrum, techniques based on delay- coordinate maps [25] allow one to construct empirical approximations of the characters φj (which, in this case correspond to eigenfunctions of the Koopman operator on the L2 space associated with the invariant measure) without prior knowledge of the coordinates of the samples in Td, allowing empirical construction of RKHS basis vectors from arbitrary observation modalities of the system. In such settings, the RKHAs from Theorem 1 should provide rich function spaces with useful properties for both forecasting and identification of temporally coherent observables. Acknowledgments. This research was supported by NSF grant DMS 1854383, ONR MURI grant N00014-19-1-242, and ONR YIP grant N00014-16-1-2649. References [1] J. C. Ferreira and V. A. Menegatto. Positive definiteness, reproducing kernel Hilbert spaces and beyond. Ann. Funct. 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