Association schemes on general measure spaces

# Association schemes on general measure spaces and zero-dimensional Abelian groups

Alexander Barg  and  Maxim Skriganov
July 5, 2019
###### Abstract.

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues).

Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of -adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group.

Next we focus on the connection between schemes on zero-dimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood-Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis.

We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.

Date: October 4, 2013, revised May 1, 2014.
Institute for Systems Research, University of Maryland, College Park, MD 20742, and Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia. Email: abarg@umd.edu. Research supported in part by NSA grant H98230-12-1-0260 and NSF grants DMS1101697, CCF1217245, and CCF1217894.
St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences, nab. Fontanki 27, St. Petersburg, 191023, Russia. Email: maksim88138813@mail.ru.
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## 1. Introduction

### 1-A. Motivation of our research

Association schemes form a fundamental object in algebraic combinatorics. They were defined in the works of Bose and his collaborators [5, 6] and became firmly established after the groundbreaking work of Delsarte [12]. Roughly speaking, an association scheme is a partition of the Cartesian square of a finite set into subsets, or classes, whose incidence matrices generate a complex commutative algebra, called the adjacency algebra of the scheme Properties of the adjacency algebra provide numerous insights into the structure of combinatorial objects related to the set such as distance regular graphs and error correcting codes, and find applications in other areas of discrete mathematics such as distance geometry, spin models, experimental design, to name a few. The theory of association schemes is presented from several different perspectives in the books by Bannai and Ito [2], Brouwer et al. [7], and Godsil [25]. A recent survey of the theory of commutative association schemes was given by Martin and Tanaka [37]. Applications of association schemes in coding theory are summarized in the survey of Delsarte and Levenshtein [13]. The approach to association schemes on finite groups via permutation groups and Schur rings is discussed in detail by Evdokimov and Ponomarenko [19].

Classically, association schemes are defined on finite sets. Is it possible to define them on infinite sets? In the case of countable discrete sets, such a generalization was developed by Zieschang [55]. However, this extension does not include an important part of the classical theory, namely, duality of association schemes. Indeed, our results imply existence of translation-invariant association schemes on Abelian groups that are countable, discrete and periodic. Such schemes can of course be described in usual terms [55]. However their duals are defined on uncountable, compact and zero- dimensional groups, so the classical definition of association schemes does not apply: in particular, the intersection numbers of the dual scheme are not well defined. The above discussion shows that the notion of association schemes on infinite sets cannot be restricted to the case of countable discrete sets. Such a theory would not include the important concept of the dual scheme and would therefore be incomplete.

Association schemes on arbitrary measure spaces are also important in applications, notably, in harmonic analysis and approximation theory. In particular, we became interested in these problems while discussing combinatorial aspects of papers [48, 49] devoted to the theory of uniformly distributed point sets. The connection of association schemes to harmonic analysis is well known in the finite case: in particular, there are classes of the so-called - and -polynomial association schemes, i.e., schemes whose eigenvalues coincide with the values of classical orthogonal polynomials of a discrete variable [2]. The most well-known example is given by the Hamming scheme for which the polynomials belong to the family of Krawtchouk polynomials; see [12]. Generalizing this link to the infinite case is another motivation of this work. Of course, these generalizations rely on general methods of harmonic analysis: for instance, the Littlewood-Paley theory, martingale theory, and the theory of Haar-like wavelets arise naturally while studying association schemes on measure spaces.

### 1-B. Overview of the paper

In an attempt to give a general definition of the association scheme, we start with a measure space (a set equipped with some fixed -additive measure) and define intersection numbers as measures of the corresponding subsets. It is possible to deduce several properties of such association schemes related to their adjacency algebras. These algebras are generated by bounded commuting operators whose kernels are given by the indicator functions of the blocks of Common eigenspaces of these operators play an important role in the study of the parameters and properties of the scheme. The set of projectors on the eigenspaces in the finite case forms another basis of the adjacency algebra, and provides a starting point for the study of duality theory of schemes. At the same time, in the general case, proving that the projectors are contained in the algebra and computing the associated spectral parameters of the scheme becomes a difficult problem.

Specializing the class of spaces considered, we focus on the case of translation association schemes defined on topological Abelian groups. A scheme defined on a group is said to have the translation property if the partition of into classes is invariant under the group operation. Translation schemes on Abelian groups come in dual pairs that follow the basic duality theory for groups themselves. Formally, the definition of the dual scheme in the general case is analogous to the definition for the finite Abelian group, see [12, 7]. At the same time, unlike the finite case, for infinite topological Abelian groups the structure of the group and the structure of its group of characters can be totally different. This presents another obstacle in the analysis of the adjacency algebras and their spectral parameters. To overcome it, we define association schemes in terms of “spectrally dual partitions” of and the dual group Roughly speaking, a spectrally dual partition is a partition of and into blocks such that the Fourier transform is an isomorphism between the spaces of functions on and that are constant on the blocks. In the finite case such partitions constitute an equivalent language in the description of translation schemes on Abelian groups [56], [57]. We show that in the general case, spectrally dual partitions form a sufficient condition for the projectors to be contained in the adjacency algebra. Using the language of partitions, we develop a theory of translation association schemes in the general case of infinite, possibly uncountable topological Abelian groups. Of course, in the finite or countably infinite case with the counting measure, our definition coincides with the original definition of the scheme.

While the above discussion motivates the general definitions given in the paper, the main question to be answered before moving on is whether this generalization is of interest, i.e., whether there are informative examples of generalized association schemes on uncountable sets. We prove that spectrally dual partitions do not exist if either or is connected. This observation suggests that one should study totally disconnected (zero-dimensional) Abelian groups. Indeed, we construct a large class of examples of translation association schemes on topological zero-dimensional Abelian groups with respect to their Haar measure. These schemes occur in dual pairs, including in some cases self-dual schemes.

In classical theory, many well-known examples of translation schemes, starting with the Hamming scheme, are metric, in the sense that the partitions of the group are defined by the distance to the identity element. In a similar way, we construct classes of metric schemes defined by the distance on zero-dimensional groups. The metric on such groups is non-Archimedean, which gives rise to some interesting general properties of the metric schemes considered in the paper. We also construct classes of nonmetric translation schemes on zero-dimensional groups and compute their parameters.

One of the important results of classical theory states that a finite association scheme is metric if and only if it is -polynomial, i.e., if its eigenvalues coincide with values of orthogonal polynomials of a discrete variable; see [2, Sect. 3.1], [7, Sect. 2.7]. This result establishes an important link between algebraic combinatorics and harmonic analysis and is the source of a large number of fundamental combinatorial theorems. In the finite case the metric on is a graphical distance, which implies that the triangle inequality can be satisfied with equality. This condition can be taken as an equivalent definition of the metric scheme. At the same time, in the non-Arcimedean case this condition is not satisfied because of the ultrametric property of the distance, and so the schemes are not polynomial (an easy way to see this is to realize that the coefficients in the three-term relation for the adjacency matrices turn into zeros). Therefore we are faced with the question of describing the functions whose values coincide with eigenvalues of metric schemes with non-Archimedean distances. We note that even in the finite case this question is rather nontrivial; see, e.g., [36, 3] for more about this. At the same time, the characterization of metric schemes is of utmost importance for our study because zero-dimensional groups are metrizable precisely by non-Archimedean metrics.

In order to resolve this question, we note that the chain of nested subgroups of defines a sequence of increasingly refined partitions of the group. Projection operators on the spaces of functions that are constant on the blocks of a given partition play an important role in our analysis: namely, we show that eigenvalues of the scheme on coincide with the values of the kernels of these operators. This enables an interpretation of the eigenvalues in terms of the Littlewood-Paley theory [18], connecting the eigenvalues of metric schemes and orthogonal systems known as Littlewood-Paley wavelets [11, p.115]. We also discuss briefly an interpretation of these results in terms of martingale theory.

Another observation in the context of harmonic analysis on zero-dimensional topological groups relates to the uncertainty principle. We note that the Fourier transforms of the indicator functions of compact subgroups of are supported on the annihilator subgroups which are compact as well. Developing this observation, we note that there exist functions on that “optimize” the uncertainty principle, in stark contrast to the Archimedean case.

Perhaps the most interesting result in this part concerns eigenvalues of nonmetric schemes on zero-dimensional groups. We observe that the eigenvalues coincide with the values of orthogonal functions on zero-dimensional groups defined in terms of multiresolution analyses, a basic concept in wavelet theory [54, 39]. We introduce a new class of orthogonal functions on zero-dimensional groups, calling them Haar-like wavelets. We also isolate a sufficient condition for these wavelets to have self-similarity property. While there is a large body of literature on self-similar wavelets on zero-dimensional groups, e.g., [32, 4, 33], to the best of our knowledge their connection to algebraic combinatorics so far has not been observed. Concluding this discussion, we would like to stress that the choice of zero-dimensional groups and the associated wavelet-like functions is naturally suggested by the logic of our study and is by no means arbitrary. This construction arises naturally as the main example of the abstract theory developed in the paper.

Outline of the paper: We begin with the definition of an association scheme on a general measure space. In Section 3 we derive simple properties of the adjacency (Bose-Mesner) algebra of the scheme. Then in Section 4 we consider translation schemes on topological Abelian groups. Assuming existence of spectrally dual partitions of the group and its dual group we prove the main results of duality theory for schemes, including the fact that orthogonal projectors on common eigenspaces are contained in the adjacency algebra, and perform spectral analysis of the adjacency operators. The main results of this part of the paper are contained in Section 5 where we show that spectrally dual partitions and dual pairs of translation schemes exist for the case of compact and locally compact Abelian zero-dimensional groups with the second countability axiom such as the additive group of -adic integers or groups of the Cantor type (countable direct products of cyclic groups). In Sect. 6 we study metric schemes from the geometric viewpoint and prove that they are nonpolynomial. In Sect. 7 we construct classes of nonmetric schemes. The question of characterizing the adjacency algebras of the constructed schemes turns out to be nontrivial. It is addressed in Section 8 where we construct these algebras as algebras of functions closed with respect to multiplication and convolution (Schur rings), addressing both the metric and nonmetric schemes. Section 9 offers several different viewpoints of the eigenvalues of the schemes constructed in the paper in the framework of harmonic analysis. In Section 10 we consider analogs of some basic results of coding theory related to the theory of association schemes. To make the paper accessible to a broad mathematical audience, we have included some background information on zero-dimensional Abelian groups; see Sect. 5-A.

Further directions: Further problems related to the theory developed in this paper include in particular, a general study of infinite association schemes in terms of Gelfand pairs and spherical functions on homogeneous spaces, an extension of the construction of the paper to noncommutative zero-dimensional groups, a study of the connection with (inductive and projective limits) of Schur rings outlined in Section 8, and a more detailed investigation of the new classes of orthogonal bases constructed in the paper.

Remarks on notation and terminology: Throughout the paper we denote by a second-countable topological space that is endowed with a countably additive measure A partition of is written as where the denote the blocks (classes) of the partition. An association scheme on defined by is denoted by For a subset we denote by the indicator function of in , and use the notation as a shorthand for the indicators of the classes. The notation refers to nonnegative integers. The cardinality of a finite set is denoted by or

Constructing schemes on groups, we consider compact and locally compact Abelian groups. When the group is compact, we explicitly say so, reserving the term “locally compact” for noncompact locally compact groups.

## 2. Association schemes on measure spaces

In this section we define association schemes on an arbitrary set with a measure. For reader’s convenience we begin with the standard definition in the finite case [12, 2, 7].

### 2-A. The finite case

###### Definition 0.

Let where is some positive integer. Let be a finite set and let be a family of disjoint subsets that have the following properties:
()
()
() where and
() For any and let

 pij(x,y)=card{z∈X:(x,z)∈Ri,(z,y)∈Rj}.

For any , the quantities are constants that depend only on Moreover,

The configuration is called a commutative association scheme. The quantities are called the intersection numbers, and the quantities are called the valencies of the scheme. If , then is called symmetric.

The adjacency matrices of an association scheme are defined by

 (Ai)xy={1if (x,y)∈Ri0 otherwise.

The definition of the scheme implies that

 (i)A0=I,(ii)d∑i=1Ai=J,(iii)ATi=Ai′,(iv)AiAj=d∑k=0pkijAk, (2.1)

where is the all-one matrix. The matrices form a complex -dimensional commutative algebra called the adjacency (Bose-Mesner) algebra [6]. The space decomposes into common eigenspaces of of multiplicities This algebra has a basis of primitive idempotents given by projections on the eigenspaces. of the matrices We have The adjacency algebra is closed with respect to matrix multiplication as well as with respect to the element-wise (Schur, or Hadamard) multiplication . We have

 Ei∘Ej=1card(X)∑k∈ΥqkijEk (2.2)

where the real numbers are called the Krein parameters of If two association schemes have the property that the intersection numbers of one are the Krein parameters of the other, then the converse is also true. Two such schemes are called formally dual. A scheme that is isomorphic to its dual is called self-dual. In the important case of schemes on Abelian groups, there is a natural way to construct dual schemes. This duality will be the subject of a large part of our work.

Finally, since for all , we have

 Ai =∑j∈Υpi(j)Ej,i∈Υ (2.3) Ej =∑i∈Υqj(i)Aij∈Υ (2.4)

(we have changed the normalization slightly from the standard form of these relations). The matrices and are called the first and the second eigenvalue matrices of the scheme. They satisfy the relations

An association scheme is called metric if it is possible to define a metric on so that any two points satisfy if and only if for some strictly monotone function Equivalently, is metric if for some ordering of its classes we have only if Metric schemes have the important property that their eigenvalues are given by (evaluations of) some discrete orthogonal polynomials; see [12, 2, 7].

An association scheme is noncommutative if it satisfies Definition 2-A without the condition If the definition is further relaxed so that the diagonal is a union of some classes then is called a coherent configuration [30].

Before moving to the general case of uncountable sets we comment on the direction of our work. Once we give the definition of the scheme (Def. 1 below), it is relatively easy to construct the corresponding adjacency algebra. The main problem arises in describing duality, in particular, in finding conditions under which the relations (2.3) can be inverted to yield relations of the form (2.4). While a general answer proves elusive, we find classes of schemes for which this can be accomplished, thereby constructing an analog of the classical theory in the infinite case.

### 2-B. The general case

Let us extend the above definition to infinite, possibly uncountable sets with a measure.

###### Definition 1.

Let be an arbitrary set equipped with a -additive measure and let \glsI be finite or countably infinite set. Consider the direct product with measure Let \glscR be a collection of measurable sets in Assume that the following conditions are true.

(i) For any and any the set

 {y∈X:(x,y)∈Ri} (2.5)

is measurable, and its measure is finite. If then the last condition can be omitted.

 R0:={(x,x):x∈X}∈R (2.6)

The set forms a partition of i.e.,

 X×X=∪i∈ΥRi,Ri∩Rj=∅ if i≠j (2.7)

where and is the transpose of

For any and let

 pij(x,y)=μ({z∈X:(x,z)∈Ri,(z,y)∈Rj}). (2.8)

For any the quantities are constants that depend only on . Moreover,

The configuration is called a commutative association scheme (or simply a scheme) on the set with respect to the measure . The sets are called classes of the scheme and the nonnegative numbers are called intersection numbers of the scheme. The notion of symmetry is unchanged from the finite case.

We will not devote special attention to the noncommutative schemes and coherent configurations restricting ourselves to the remark that the corresponding definitions carry over to the general case without difficulty. It is also straightforward to define metric schemes, which will be studied in more detail in Sect. 6 below.

For the time being it will be convenient not to specialize the index set , leaving it to be an abstract set. We note that any scheme can be symmetrized by letting to be a scheme with The intersection numbers of can be expressed via the intersection numbers of ; see [2, p.57].

Define the numbers

 μi=p0ii′=μ({y∈X:(x,y)∈Ri}),i∈Υ. (2.9)

These quantities are finite because of condition (i) and do not depend on because of (v). Call the valency of the relation Clearly

 \glsmui=μi′and∑i∈Υμi=μ(X). (2.10)

The intersection numbers and valencies of finite schemes satisfy a number of well-known relations; see [2, Prop.2.2] or [7, Lemma 2.1.1]. All these relations can be established for schemes on sets with a measure without difficulty. In particular, the following statement, which is analogous to [2, Prop.2.2(vi)], will be used below in the paper.

###### Lemma 2.1.
 μkpkij=μipikj′=μjpji′k. (2.11)

For symmetric schemes this means that the function

 σ(i,j,k)=μkpkij (2.12)

is invariant under permutations of its arguments.

###### Proof.

Let us prove the first equality in (2.11). Let and consider the measurable subsets and :

 Ekij ={(z,y)∈X×X:(x,z)∈Ri,(y,z)∈Rj,(x,y)∈Rk} (2.13) Ek ={y∈X:(x,y)∈Rk} (2.14)

and let and be their indicator functions. The definition of the scheme implies that

 ∫Xχ[Ekij;(y,z)]dμ(z) =pkijχ[Ek;y] (2.15) ∫Xχ[Ekij;(y,z)]dμ(y) =pikj′χ[Ei;z] (2.16)

as well as (see (2.9))

 ∫Xχ[El;y]dμ(y)=μl. (2.17)

Since the indicator function of the subset is nonnegative and measurable, we can use the Fubini theorem to write

 ∬X×Xχ[Ekij;(y,z)]dμ(y)dμ(z) =∫Xdμ(y)∫Xχ[Ekij;(y,z)]dμ(z) =∫Xdμ(z)∫Xχ[Ekij;(y,z)]dμ(y). (2.18)

Substituting in this equation expressions (2.15) and (2.16) and using (2.17) with and , we obtain the first equality in (2.11). The remaining equalities can be proved by a very similar argument or derived from the first one using commutativity.

Note the following important difference between schemes on countable and uncountable sets. Consider the valency of the diagonal relation It equals the measure of a point: and is the same for all Thus, if then is at most countably infinite and while if then is uncountable and the measure is non-atomic.

In accordance with the above we can introduce the following

Classification of association schemes

1. and In this case is a classical scheme on the finite set given by Definition 2-A [5, 12]. This is the most studied case.

2. and In this case is a scheme on a countable discrete set. Such schemes are studied in [55].

3. and Examples of such schemes can be constructed on uncountable compact zero-dimensional Abelian groups, see Sect. 5-B. Note that their duals are schemes of type

4. and Schemes of this kind can be constructed on locally compact zero-dimensional Abelian groups. Examples will be considered in Sect. 5-B below. We note that in this case, similarly to the case , a scheme can be self-dual.

Generalization of the Bose-Mesner algebras to the infinite case is nontrivial. Here we consider only the main features of such generalized adjacency algebras that follow directly from the definition of the scheme on an arbitrary measure set. For a given scheme consider the indicator functions of its relations:

 \glschii={1if (x,y)∈Ri0otherwise . (3.1)

Definition 1 immediately implies the following properties of the indicators:

###### Lemma 3.1.

(i) For a fixed , the functions are measurable and integrable functions of

(ii) and for each , is the indicator of a single point

(iii) and if is symmetric then

(iv) For any

 ∑i∈Υχi(x,y)=j(x,y),

where for all

(v) The following equality holds true:

 ∫Xχi(x,z)χj(z,y)dμ(z)=∑k∈Υpkijχk(x,y). (3.2)

In particular, we have

 ∫Xχi(x,z)χj(z,y)dμ(z)=∫Xχj(x,z)χi(z,y)dμ(z).

(vi) The valencies of satisfy the relation

 μi=∫Xχi(x,y)dμ(y)=∫Xχi(x,y)dμ(x). (3.3)

These properties parallel the finite case; see the relations in (2.1).

Consider the set \glsfrakA of finite linear combinations

 a(x,y)=∑i∈Υciχi(x,y), (3.4)

where for all This is a linear space of functions piecewise constant on the classes . We have Multiplication on this space can be introduced in two ways. Clearly, is closed with respect to the usual product of functions (because ). Define the convolution of functions and as follows:

 (a∗b)(x,y)=∫Xa(x,z)b(z,y)dμ(z). (3.5)

By (3.2) and the commutativity condition of , convolution is commutative on We conclude that linear space is a complex commutative algebra with respect to the product of functions. By (3.2) this algebra is also closed with respect to convolution. It is called the adjacency algebra of the scheme

The question of multiplicative identities of with respect to each of the product operations deserves a separate discussion.

In the classical case the adjacency algebra contains units for both operations. For the usual product of functions (Schur product of matrices) the identity is the function while for convolution (matrix product) the identity is given by Clearly, both these functions are contained in

In the case the identity for convolution is given by however, the identity for the usual multiplication is not contained in because the convolution is not well defined.

In the case there is an identity for the usual multiplication, but no identity for convolution (this should be an atomic measure, viz., the Dirac delta-function but the product cannot be given any meaning).

Finally, in the case the algebra generally contains neither the usual multiplicative identity nor the identity for convolution.

Remark: Note that if an algebra has only one multiplication operation, we can always adjoin to it its identity element. At the same time, if there is more than one multiplication, it is generally impossible to adjoin several identities in a coordinated manner.

### 3-A. Operator realizations of adjacency algebras

Consider the space of square-integrable functions on . Given a measurable function , define the integral operator with the kernel

 Af(x)=∫Xa(x,y)f(y)dμ(y). (3.6)

Let be an association scheme on . With every class associate an integral operator with the kernel (3.1) defined by (3.6):

 \glsAif(x)=∫Xχi(x,y)f(y)dμ(y). (3.7)

Linear combinations (3.4) give rise to operators of the form

 A=∑i∈ΥciAi. (3.8)

Operators of this kind will be used to describe association schemes and their adjacency algebras, therefore we will devote some space to the study of their basic properties.

###### Lemma 3.2.

For any scheme operators (3.6) with are bounded in

###### Proof.

According to the Schur test of boundedness, if the kernel of an integral operator satisfies the conditions

 α1 :=ess\,supx∈X∫X|a(x,y)|dμ(y)<∞ α2 :=ess\,supy∈X∫X|a(x,y)|dμ(x)<∞

then the operator is bounded in and its norm [28, p. 22]. For the operators we obtain, on account of (3.3),

 α1=∫Xχi(x,y)dμ(y)=μi,

and . We conclude that

 ∥Ai∥≤μifor all i. (3.9)

Since the sums in (3.8) are finite, the proof is complete.

In fact, integral operators (3.6) with kernels belong to a special class of operators called Carleman operators [28]. Recall that is called a Carleman operator if

 ξ(A,x)=(∫X|a(x,y)|2dμ(y))\nicefrac12<∞

almost everywhere on . For operators in (3.8) we obtain

 ξ(A,x) =(∑i∈Υ|ci|2μi)\nicefrac12

where the last equality is obtained using (3.3). Thus, for finite sums in (3.8) these functions are finite constants.

If in addition then operators (3.6) with are compact Hilbert-Schmidt. Indeed, the Hilbert-Schmidt norm of is estimated as follows:

 ∥Ai∥2HS=∬X×X|χi(x,y)|2dμ(x)dμ(y)=∫Xdμ(x)∫Xχi(x,y)dμ(y)=μ(X)μi, (3.10)

where we have used Fubini’s theorem and (3.3).

Let us introduce some notation. Define the set

 \glsI0={i∈Υ:μi>0} (3.11)

and note that only if Let and let be an integral operator in with kernel , and let be the orthogonal projector on the subspace of constants. Let us list basic properties of the operators .

###### Lemma 3.3.

(i) , where is the identity operator in . In particular, for schemes of type and is the zero operator.

(ii)

 tAi=A∗i=Ai′ (3.12)

where is the transposed operator and is the adjoint operator of .

(iii) in particular . Thus, the operators are normal, and if the scheme is symmetric, they are self-adjoint.

(iv) Let and let be the orthogonal projector on constants. Then

 ∑i∈Υ0Ai =\glsJ=μ(X)P (3.13) AiAj =∑k∈Υ0pkijAk (3.14)

where both the series converge in the operator norm.

Proof: Part (i) is immediate from the definitions, Part (ii) follows from Lemma 3.1(iii), Part (iii) follows from (3.5), and equations (3.13) and (3.14) are implied by parts (iv) and (v) of Lemma 3.1, respectively. The convergence of the series in (3.13) follows from (2.10) and (3.9):

 ∥∥∑i∈Υ0Ai∥∥≤∑i∈Υ0∥Ai∥≤∑i∈Υ0μi=μ(X)

As for the series in (3.14), Eq. (2.8) implies that and so

 ∥∥∑k∈Υ0pkijAk∥∥≤μ(X)2.

Remark: Let us make an important remark about the definition of the adjacency algebras. Since the adjacency algebra of the scheme generally is infinite-dimensional, the notion of the basis as well as relations of the form (3.7), (3.14) become rigorous only upon defining a topology on the algebra that supports the needed convergence of the series. So far we have understood convergence in sense of the operator (Hilbert-Schmidt) norm, but this norm is generally not closed with respect to the Schur product of operators: for instance, in part (iv) of the previous lemma we need the compactness assumption for convergence. Thus, in general, our arguments in this part are of somewhat heuristic nature. We make them fully rigorous for the case of association schemes on zero-dimensional groups; see Sect. 8.

### 3-B. Spectral decomposition

In the previous subsection we established that the operators are bounded in commuting normal operators (in the symmetric case, even self-adjoint). By the spectral theorem [16, p.895], they can be simultaneously diagonalized. The analysis of spectral decomposition of is simple in the case In this case, all the operators are compact Hilbert-Schmidt, and the situation resembles the most the classical case of finite sets when the s are finite-dimensional matrices. Namely, if then the space contains a complete orthonormal system of functions that are simultaneous eigenfunctions of all viz.

 Aifj=λi(j)fj;

here is some set of indices. For every the nonzero eigenvalues have finite multiplicity. The sequence has at most one accumulation point . (These two statements follow from general spectral theory, e.g. [16], Cor. X.4.5.) Moreover,

 ∑j∈Υ1|λi(j)|2 =μ(X)μi,i∈Υ0 (3.15) ¯¯¯¯¯¯¯¯¯¯¯λi(j) =λi′(j). (3.16)

Indeed, Eq. (3.15) follows from (3.10), and Eq. (3.16) is a consequence of (3.12).

Our next goal is to define the minimal idempotents (cf. (2.2)). Let

 L2(X,μ)=⨁j∈Υ2Vj (3.17)

be the expansion of into an orthogonal direct sum of common eigenspaces of all the operators so that

 Aif=pi(j)ffor all f∈Vj (3.18)

where is the maximal eigenspace in the sense that for any there exists an operator such that Now let be the orthogonal projectors on the subspaces Then we can write

 Ai=∑j∈Υ2pi(j)Ej,i∈Υ0, (3.19)

where are the eigenvalues of the operators on the subspace (cf. Eq.(2.3)). Call the quantities

 \glsmj=dimVj,j∈Υ2 (3.20)

the multiplicities of the scheme In accordance with (3.15)-(3.16), taking into account the multiplicities, we have

 ∑j∈Υ2mj|pi(j)|2 =μ(X)μi (3.21) ¯¯¯¯¯¯¯¯¯¯¯pi(j) =pi′(j). (3.22)

These relations have their finite analogs; see e.g., [2, pp.59,63].

The projectors clearly satisfy the relation

 ∑j∈Υ2Ej=I, (3.23)

where is the identity operator in Recall that for schemes of type , the operator is not an integral operator.

We will return to relations (3.21)-(3.23) in the next section in the context of duality theory. This theory is well developed in the finite case, where relations (3.19) can be inverted so that the projectors are written in terms of the adjacency operators (2.4) [2, p.60]. These relations and their corollaries form one of the main parts of the classical theory of association schemes. Unfortunately, in the general case of measure spaces we did not manage to prove the invertibility of relations (3.19) even in the case In the case of the situation becomes even more complicated because the operators can have continuous spectrum. Thus, in the general case it is not known whether the spectral projectors are contained in the adjacency algebra

In the next section we show that relations (3.19) can be inverted in the case of schemes on topological Abelian groups, leading to relations (2.4). This will enable us to introduce Krein parameters of schemes and develop a duality theory in the infinite case.

## 4. Association schemes and spectrally dual partitions on topological Abelian groups

### 4-A. Harmonic analysis on topological Abelian groups

We begin with reminding the reader the basics about topological Abelian groups. Details can be found, e.g., in [43, 29].

Let be a second countable topological compact or locally compact Abelian group written additively. Let be the character group of (the group of continuous characters of ) written multiplicatively. Just as , is a topological compact or locally compact Abelian group. By Pontryagin’s duality theorem [43, Thm. 39], its character group is topologically canonically isomorphic to

For any and we have

 ϕ(x+y)=ϕ(x)ϕ(y),(ϕ⋅ψ)(x)=ϕ(x)ψ(x)¯¯¯¯¯¯¯¯¯¯¯ϕ(x)=ϕ(−x). (4.1)

Let and be the Haar measures on and , respectively. Define the Fourier transforms and as follows:

 \glsFsim:f(x)→˜f(ξ)=∫Xξ(x)f(x)dμ(x),ξ∈^X (4.2) \glsFhat:g(ξ)→g♮(x)=∫^X¯¯¯¯¯¯¯¯¯¯ξ(x)g(ξ)d^μ(ξ),x∈X. (4.3)

Assume for the moment that and similarly, It is known that the Haar measures can be normalized to fulfill the Parseval identities and equalities for the inner products

 ∫X|f(x)|2dμ(x) =∫^X|˜f(ξ)|2d^μ(ξ),∫^X|g(ξ)|2d^μ(ξ)=∫X|g♮(x)|2dμ(x) (4.4) ∫Xf1(x)¯¯¯¯¯¯¯¯¯¯¯¯f2(x)dμ(x)=∫^X˜f1(ξ)¯¯¯¯¯¯¯¯¯¯¯¯¯˜f2(ξ)d^μ(ξ)∫^Xg1(ξ)¯¯¯¯¯¯¯¯¯¯¯g2(ξ)d^μ(ξ)=∫Xg♮1(x)¯¯¯¯¯¯¯¯¯¯¯¯g♮2(x)dμ(x) (4.5)

In what follows we always assume that equalities (4.4), (4.5) are fulfilled. The corresponding normalizations of the Haar measures and will be given below. Equalities (4.4), (4.5) imply that the mappings and can be extended to mutually inverse isometries of the Hilbert spaces and that preserve inner products:

 F∼F♮=I,F♮F∼=^I,

where and are the identity operators in and respectively, and so

Recall the formulas for convolutions and their Fourier transforms. Let

 (f1∗f2)(x)=∫Xf1(x−y)f2(y)dμ(y),x∈X (4.6) (g1∗g2)(ϕ)=∫^Xg1(ϕξ−1)g2(ξ)d^μ(ξ)ϕ∈^X (4.7)

According to the Young inequality [17, p.157], for any functions

 ∥f1∗f2∥r≤∥f1∥p∥f2∥q, (4.8)

where and Thus, the convolutions (4.6), (4.7) are in if one of the functions is in and the other in We have

 (˜f1∗f2)(ξ)=˜f1(ξ)˜f2(ξ),ξ∈^X(g1∗g2)♮(x)=g♮1(x)g♮2(x),x∈X. (4.9)

These formulas are useful for spectral analysis of integral operators that commute with translations on the groups and Consider the integral operators on the spaces and given by

 Af(x) =∫Xa(x−y)f(y)dμ(y) Bg(ξ) =∫^Xb(ξη−1)g(η)d^μ(η),

where and are kernels. We have

 ˜Af(ξ) =˜a(ξ)˜f(ξ) (Bg)♮(x) =b♮(x)g♮(x),

which shows that the operators and are diagonal (i.e., are multiplication operators). Their spectra are given by the values of the functions and In particular, if is an infinite compact group, then the spectrum of is discrete and the spectrum of of continuous. If is locally compact, then both the spectra of and are continuous. This happens, for instance, when the graph of the function has constant segments supported on sets of positive measure, which corresponds to the continuous spectrum in spectral theory. We will encounter this situation later in the paper.

### 4-B. Translation association schemes

###### Definition 2.

Let be an Abelian group. A scheme