Supercharacters of discrete algebra groups
aa r X i v : . [ m a t h . R T ] J a n SUPERCHARACTERS OF DISCRETE ALGEBRA GROUPS
CARLOS A. M. ANDR´E AND JOCELYN LOCHON
Abstract.
The concept of a supercharacter theory of a finite group was intro-duced by Diaconis and Isaacs in [15] as an alternative to the usual irreduciblecharacter theory, and exemplified with a particular construction in the case offinite algebra groups. We extend this construction to arbitrary countable dis-crete algebra groups, where superclasses and indecomposable supercharactersplay the role of conjugacy classes and indecomposable characters, respectively.Our construction can be understood as a cruder version of Kirillov’s orbitmethod and a generalisation of Diaconis and Isaacs construction for finite al-gebra groups. However, we adopt an ergodic theoretical point of view. Thetheory is then illustrated with the characterisation of the standard superchar-acters of the group of upper unitriangular matrices over an algebraic closedfield of prime characteristic. Introduction
This paper mainly deals with the unitary representation theory of countablediscrete algebra groups. Let k be an arbitrary field, and let A be an associative nil k -algebra; we recall that an associative k -algebra A is said to be nil if every elementof A is nilpotent (in particular, A does not have an identity). Let G = 1 + A bethe set of formal objects of the form 1 + a where a ∈ A ; then, G is easily seen to bea group with respect to the multiplication defined by (1 + a )(1 + b ) = 1 + a + b + ab for all a, b ∈ A . (In fact, G is a subgroup of the group of units of the k -algebra A = k · A .) Following [24], a group G constructed in this way will be referredto as an algebra group over k ; by the way of example, if A = u n ( k ) is the k -algebra consisting of all strictly uppertriangular n × n matrices over k , then thecorresponding algebra group G = 1 + A is (isomorphic to) the upper unitriangulargroup U n ( k ). Henceforth, we view G as a subgroup of the group of units of the k -algebra k · A ; however, we observe that A is not assumed to have finite dimensionover k .In general, countable discrete algebra groups are not tame (or of type I), mean-ing that the decomposition of an arbitrary unitary representation (on a separableHilbert space) as a direct integral of irreducible representations might not be unique; Date : January 28, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Amenable countable discrete algebra group; supercharacter; ergodicmeasure.This research was made within the activities of the Group for Linear, Algebraic and Com-binatorial Structures of the Center for Functional Analysis, Linear Structures and Applications(University of Lisbon, Portugal), and was partially supported by the Portuguese Science Founda-tion (FCT) through the Strategic Projects UID/MAT/04721/2013 and UIDB/04721/2020. Thesecond author was partially supported by the Lisbon Mathematics PhD program (funded by thePortuguese Science Foundation). The major part of this work is included in the second authorPh.D. thesis. indeed, this is not the case if and only if the group in question has an abelian sub-group of finite index, as shown by E. Thoma in [33, Satz 6] (see also [36]). Forthis reason, we will approach the representation theory of a discrete algebra groupthrough its character theory.More generally, let G be an arbitrary topological group. A complex-valued con-tinuous function ϕ : G → C is said to be positive definite if the following twoconditions are satisfied:(i) ϕ ( g − ) = ϕ ( g ) for all g ∈ G .(ii) For every g , . . . , g m ∈ G , the Hermitian matrix (cid:2) ϕ ( g i g − j ) (cid:3) ≤ i,j ≤ m is non-negative, that is, P ≤ i,j ≤ n ϕ ( g i g − j ) z i z j ≥ z , . . . , z n ∈ C . A continuous function ϕ : G → C is said to be central (or a class function ) if it isconstant on the conjugacy classes of G , that is, if ϕ ( ghg − ) = ϕ ( h ) for all g, h ∈ G ,and it is said to be normalised if ϕ (1) = 1. We denote by Ch( G ) the set consistingof all normalised, central, positive definite continuous functions ϕ : G → C . If ϕ, ψ ∈ Ch( G ), then for every real number 0 ≤ t ≤ − t ) ϕ + tψ isalso an element of Ch( G ), which means that Ch( G ) is a convex set. An element ofa convex set is said to be extreme if it is not contained in the interior of any intervalwhich is entirely contained in the set; we denote by Ch + ( G ) the subset of Ch( G )consisting of extreme elements of Ch( G ). We refer to an element of Ch( G ) as a character of G and to an element of Ch + ( G ) as an indecomposable character of G .If G is a compact group (in particular, a finite group), then Ch + ( G ) consists of all normalised irreducible characters of G ; the character of an irreducible (complex)representation π of the group G is the function χ : G → C given by the trace χ ( g ) = tr( π ( g )) for all g ∈ G , while the normalised irreducible character is given by b χ = χ (1) − χ . This definition makes sense because every irreducible representationof a compact group is finite-dimensional, but it does not make sense in general fornon-compact groups.In the case where G is a discrete countable group, then Ch( G ) is a Choquetsimplex, with respect to the topology of pointwise convergence, such that Ch + ( G )is a G δ -set (as proved by Thoma in [33, 35]). Consequently, every character of G isuniquely representable as a convex mixture of indecomposable characters (see, forexample, [22] or [29]): for every ϕ ∈ Ch( G ), there is a unique probability measure µ ϕ on Ch + ( G ) such that ϕ ( g ) = Z Ch + ( G ) χ ( g ) dµ ϕ , g ∈ G ;we refer to µ ϕ as the Choquet (or spectral) measure associated with ϕ . (Noticethat the uniqueness of this representation of a character with respect to a Choquetmeasure mirrors the classical decomposition of characters for finite groups as a sumof irreducible characters; moreover, it allows to view Ch + ( G ) as a dual space for G ,in contrast to the usual dual space consisting of all classes of irreducible representa-tions which for non-tame groups, not only does not provide unique decomposition,but also lacks good topological features; we refer to [16] fore more details.)The main tool to study characters of discrete groups is the Gelfand-Naimark-Segal construction which associates a unitary cyclic representation with every char-acter in such a way that indecomposable characters are paired with factor repre-sentations of finite von Neumann type. Let G be an arbitrary discrete group, andlet ϕ : G → C be a character of G . Let C [ G ] denote the complex group algebra of G which is naturally equipped with the involution ∗ : C [ G ] → C [ G ] defined bythe rule (cid:0) P g ∈ G a g g (cid:1) ∗ = P g ∈ G a g g − where a g ∈ C for g ∈ G ; in particular, C [ G ]ia a ∗ -algebra. By abuse of notation, we denote the linear extension of ϕ to C [ G ]also by ϕ ; since ϕ is positive-definite, the formula h a, b i = ϕ ( b ∗ a ), for a, b ∈ C [ G ].defines a positive semi-definite sesquilinear form on C [ G ]. Thus, via separation andcompletion, we obtain a Hilbert space H ϕ equipped with a natural map G → H ϕ which sends an element g ∈ G to the characteristic function δ g of { g } . It is astandard fact that the left action of G on C [ G ] induces a unitary representation π ϕ : G → U ( H ϕ ) such that ϕ ( g ) = h π ( g ) δ , δ i for all g ∈ G ; following the termi-nology of [11] , we will refer to ( π ϕ , H ϕ ) as the GNS representation of G associatedwith ϕ . Let L ( G, ϕ ) denote the von Neumann algebra generated by π ϕ ( G ); hence, L ( G, ϕ ) = π ϕ ( G ) ′′ is the bicommutant of π ϕ ( G ). Since ϕ is a character, the map-ping x ϕ ( x ) = h xδ , δ i defines a faithful normal unital trace on L ( G, ϕ ), andthus L ( G, ϕ ) is a finite von Neumann algebra. Finally, we mention that the centraldecomposition of L ( G, ϕ ) as a direct integral of factor representations correspondsuniquely to the Choquet decomposition of ϕ ∈ Ch( G ) as an integral of indecom-posable characters; in particular, ϕ ∈ Ch + ( G ) if and only if L ( G, ϕ ) is a factor. Formore details on von Newmann algebras, we refer to Dixmier’s book [17]; see also[16].Indecomposable characters have been explicitly classified for concrete groups (seefor example [18, 23, 14, 39, 38]). However, the set Ch + ( G ) may be too large or toocomplicated to describe (even in the case of a finite group), and thus it might be ofinterest to consider smaller and more manageable families of characters which arestill rich enough to provide some relevant information about their representations.This question was first addressed in the context of finite groups by P. Diaconis andI.M. Isaacs in the foundational paper [15] where the notion of supercharacter theory was formalised (motivated by previous work of C. Andr´e [2, 3, 4, 5, 6] and of N.Yan [41] on the character theory of the unitriangular groups over finite fields). Bya supercharacter theory of a finite group G we mean a pair ( K , E ) where K is apartition of G , and E is an orthogonal set of characters of G satisfying:(i) |K| = |E| ,(ii) every character ξ ∈ E takes a constant value on each member K ∈ K , and(iii) each irreducible character of G is a constituent of one of the characters ξ ∈ E .We refer to the members K ∈ K as superclasses and to the characters ξ ∈ E as indecomposable supercharacters of G . Notice that superclasses of G are alwaysunions of conjugacy classes; moreover, { } is a superclass and the trivial character1 G is always a supercharacter of G . The superclasses and the indecomposablesupercharacters of a particular supercharacter theory exhibit much of the sameduality as the conjugacy classes and the irreducible characters of the group do (andthus a supercharacter theory can be interpreted as an approximation of the classicalcharacter theory); indeed, the usual character theory of G is a trivial example of a It is worth to mention that, here and throughout the paper, we have chosen to use a ter-minology which differs from the most common one; indeed, in [15] the term “supercharacter” isreserved to the elements of E (that is, to the indecomposable supercharacters). We prefer to usethe term “supercharacter” with the meaning of a “character which is constant on superclasses”;this is in fact consistent with the fact that an arbitrary character is not necessarily an irreducible(or, indecomposable) character. supercharacter theory where K is the set Cl( G ) consisting of all conjugacy classesof G and E is the set Irr( G ) consisting of all irreducible characters of G .The standard supercharacter theory of an arbitrary finite algebra group is de-scribed in [15], and the main goal of this paper is to extend this construction to thecase of an arbitrary countable discrete algebra group. Prototype examples of count-able discrete algebra groups are the unitriangular groups defined over an arbitrarycountable discrete field k . On the one hand, for every n ∈ N , let U n ( k ) denote the unitriangular group over k consisting of all n × n upper-triangular matrices over k with all entries in the main diagonal equal to 1; notice that U n ( k ) = 1+ u n ( k ) where u n ( k ) is the nil k -algebra consisting of all n × n upper-triangular matrices with ze-roes on the main diagonal. On the other hand, let U ∞ ( k ) denote the locally finitedimensional unitriangular group over k consisting of all infinite upper-triangularsquare matrices over k with all diagonal entries equal to 1 and such that everyelement has only a finite number of non-zero entries above the main diagonal. Forevery n ∈ N , the unitriangular group U n ( k ) may be naturally identified as the sub-group of U n +1 ( k ) consisting of all matrices x ∈ U n +1 ( k ) which satisfy x i,n +1 = 0for all 1 ≤ i ≤ n , and thus U ∞ ( k ) may be realised as the direct limit U ∞ ( k ) = lim −→ n ∈ N U n ( k ) = [ n ∈ N U n ( k ) . We also note that U ∞ ( k ) = 1 + u ∞ ( k ) where u ∞ ( k ) = S n ∈ N u n ( k ) is the locallyfinite dimensional nil k -algebra consisting of all infinite upper-triangular squarematrices over k with zeroes on the main diagonal; notice also that u ∞ ( k ) is naturallyisomorphic to the direct limit lim −→ n ∈ N u n ( k )..The superclasses of an arbitrary algebra group G = 1 + A are easy to describe;since there is no danger of ambiguity, we will abbreviate the terminology and referto a standard superclass simply as a superclass of G . Indeed, the group Γ = G × G acts naturally on the left of A by the rule( g, h ) · a = gah − , g, h ∈ G, a ∈ A . Then, the k -algebra A is partitioned into Γ-orbits Γ · a = GaG for a ∈ A , and thisdetermines a partition of the algebra group G = 1+ A into subsets 1+Γ · a = 1+ GaG for a ∈ A ; these are precisely what we define as the superclasses of G . We use thenotation SCl( G ) to the denote the set consisting of all superclasses of G ; noticethat { } ∈ SCl( G ) and that every K ∈ SCl( G ) is a union of conjugacy classes.We next define the set of supercharacters of G ; as in the case of superclasses,by a supercharacter of G we will always understand a standard supercharacter. A(continuous) function ϕ : G → C is said to be supercentral (or a superclass function )if it is constant on the superclasses, that is, if ϕ (1 + gah ) = ϕ (1 + a ) for all g, h ∈ G and all a ∈ A . We denote by SCh( G ) the set consisting of all normalised,supercentral, positive definite continuous functions defined on G ; it is clear thatSCh( G ) ⊆ Ch( G ). As in the case of characters, SCh( G ) is a convex set (in fact, aChoquet simplex); we denote by SCh + ( G ) the subset of SCh( G ) consisting of allextreme elements of SCh( G ). We refer to an element of SCh( G ) as a supercharacter of G , and to an element of SCh + ( G ) as an indecomposable supercharacter of G .In the case where G = 1 + A is a finite algebra group, the indecomposablesupercharacters are parametrised by the orbits of the contragradient action of Γ = G × G on the Pontryagin dual A ◦ of the additive group A + of A (hence, A ◦ consistsof all unitary characters ϑ : A + → C × ). For each of the natural actions of G on A , there is a corresponding contragradient action of G on A ◦ : given ϑ ∈ A ◦ and g ∈ G ,we define gϑ, ϑg ∈ A ◦ by the formulas ( gϑ )( a ) = ϑ ( g − a ) and ( ϑg )( a ) = ϑ ( ag − )for all a ∈ A ; thus, for every ϑ ∈ A ◦ , we have a left G -orbit Gϑ , a right G -orbit ϑG , and also a (left) Γ-orbit Γ · ϑ = GϑG (notice that the left and right G -actionson A ◦ commute). For every Γ-orbit O ⊆ A ◦ , we define the function ξ O : G → C × by the rule(1.1) ξ O ( g ) = 1 |O| X ϑ ∈O ϑ ( g − , g ∈ G. By [15, Theorem 5.6]), we know that SCh + ( G ) = { ξ O : O ∈ Ω } where Ω denotesthe set consisting of all Γ-orbits on A ◦ .The formula above may be interpreted in measure theoretical terms: the valueof an indecomposable supercharacter ξ O ( g ) at an element g ∈ G is given by the in-tegral over O evaluated at g − ∈ A with respect to the unique ergodic Γ-invariantmeasure on A ◦ supported on O ; for an arbitrary countable discrete algebra groupa similar phenomenon occurs. More precisely, in Theorem 2.5 we establish a one-to-one correspondence between SCh + ( G ) and the set consisting of all Γ-invariantmeasures on A ◦ which are ergodic with respect to the (contragradient) action ofΓ on A ◦ (and this provides an alternative proof of [15, Theorem 5.6]); as a con-sequence, we conclude that SCh( G ) is indeed a Choquet simplex. While it is truethat every ergodic Γ-invariant measure must be supported on the closure of someΓ-orbit (Proposition 2.6), and that, under the assumption that G is amenable, theclosure of an arbitrary Γ-orbit must be the support of some ergodic Γ-invariantmeasure (Proposition 2.7), in general we can not guarantee that every indecompos-able supercharacter is in one-to-one correspondence with the closure of a Γ-orbit(this is because the closure of a Γ-orbit may support distinct ergodic Γ-invariantmeasures).Another consequence is discussed in Section 3 where we consider the regularcharacter of G , and we obtain (Theorem 3.1) a necessary and sufficient conditionfor the regular character to be an indecomposable supercharacter of G ; furthermore,the ergodic correspondence allows to establish (Proposition 3.8) a supercharacteranalogue for Thoma’s Plancherel formula for countable discrete groups (see [34,Satz 1]). We should mention that these properties are the natural supercharactergeneralisation of well-known properties of the regular character.Finally, in Section 4 we consider approximately finite algebra groups G (that is,direct limits of finite algebra groups), and use Lindenstrauss’ pointwise ergodic the-orem ([27, Theorem 1.3]) to approximate supercharacters of G by supercharactersof finite algebra groups (Theorem 4.1); we illustrate this method by describing thesupercharacters of the infinite unitriangular group U n ( k ), where k is the algebraicclosure of a finite field of prime characteristic.2. Supercharacters of algebra groups
Throughout this section, we let G = 1 + A be an arbitrary discrete countablealgebra group associated with a nil algebra A over a (countable discrete) field k ;furthermore, we consider the group Γ = G × G acting naturally on the left of A andon the Pontryagin dual A ◦ of the additive group A + of A (via the contragradientaction). Our main goal is to parametrise the supercharacters of G (as defined inthe introduction) in terms of Γ-invariant probability measures on A ◦ , in such a way that the indecomposable supercharacters of G correspond to those measures whichare ergodic with respect to the Γ-action on A ◦ .We equip A ◦ with the topology induced by convergence on compact sets, whichis nothing else than the topology of pointwise-convergence (because A + is discrete);we note that, since A + is abelian, A ◦ is in fact the set consisting of all indecom-posable characters of A . Furthermore, A ◦ has a structure of an abelian topologicalcompact group (see for example [21, Proposition 4.35]) which we write additivelyand where the sum ϑ + ϑ ′ ∈ A ◦ of two characters ϑ, ϑ ′ ∈ A ◦ is determined bythe pointwise product of functions, that is, ( ϑ + ϑ ′ )( a ) = ϑ ( a ) ϑ ′ ( a ) for all a ∈ A ;the zero element of A ◦ is the trivial character A which is constantly equal to 1(accordingly, we sometimes write = A ).We consider A ◦ equipped with its Borel σ -algebra of measurable sets, and denoteby M ( A ◦ ) the vector space consisting of all finite complex regular Borel measureson A ◦ ; we equip M ( A ◦ ) with the topology of weak*-convergence. A measure µ ∈ M ( A ◦ ) is said to be Γ -invariant if µ ( γ · B ) = µ ( B ) for all γ ∈ Γ and allBorel subset B of A ◦ ; we denote by M Γ ( A ◦ ) the vector space consisting of allΓ-invariant measures on A ◦ . On the other hand, let M + ( A ◦ ) denote the subsetof M ( A ◦ ) consisting of all Borel probability measures, and let M +Γ ( A ◦ ) denotethe subset of M + ( A ◦ ) consisting of all Γ-invariant probability measures. Noticethat M +Γ ( A ◦ ) (and hence M Γ ( A ◦ )) is non-empty because the Dirac measure δ A supported on the trivial character A ∈ A ◦ is clearly a Γ-invariant probabilitymeasure. Furthermore, M +Γ ( A ◦ ) is a Choquet simplex whose extreme elements arethe ergodic measures in M +Γ ( A ◦ ) (see [29, Section 12] and [37, Theorem 3.1]); werecall that a Γ-invariant probability measure µ ∈ M +Γ ( A ◦ ) is said to be ergodic if, for every Γ-invariant Borel subset B of A ◦ , either µ ( B ) = 0 or µ ( B ) = 1(equivalently, a measure µ ∈ M +Γ ( A ◦ ) is ergodic if and only if every Γ-invariantfunction f ∈ L ( A ◦ , µ ) is constant µ -almost everywhere; see, for example, [40,Theorem 1.6]).The main purpose of this section is to establish the existence of an affine home-omorphism between SCh( G ) and M +Γ ( A ◦ ) (see Theorem 2.5 below); in particular,we deduce that the indecomposable supercharacters of G are in one-to-one corre-spondence with the ergodic measures in M +Γ ( A ◦ ). In order to prove it, we firstassume a more general situation where N is an arbitrary countable discrete group(not necessarily abelian), and Γ is a group which acts on the left of N via continu-ous automorphisms; we write γ · n = γ ( n ) for all γ ∈ Γ and all n ∈ N . We defineCh Γ ( N ) to be the set of all characters of N which are constant on every Γ-orbiton N ; notice that Ch Γ ( N ) is precisely the set of Γ-invariant characters in Ch( N )with respect to the usual contragradient Γ-action on Ch( N ). As before, Ch Γ ( N ) isclearly a convex set, and thus we may consider the subset Ch +Γ ( N ) of Ch Γ ( N ) con-sisting of all extreme elements of Ch Γ ( N ). Following the previous terminology, werefer to an element of Ch Γ ( N ) as a Γ -character of N , and to an element of Ch +Γ ( N )as an indecomposable Γ -character of N . Let C ∗ ( N ) denote the group C ∗ -algebra of N (see, for example, [16, Section 13.9]); it follows from [16, Proposition 6.8.7] thatCh( N ) is a (compact convex) subset of the unit ball B ∗ of the topological dual of C ∗ ( N ), and hence [20, Proposition 3.101] implies that the Choquet simplex Ch( N )is metrisable (notice that in our situation the topology of weak*-convergence coin-cides with the topology of pointwise-convergence). If Ch + ( N ) denotes the set ofextreme elements of Ch( N ) (that is, the indecomposable characters of N ), then we know that every ϕ ∈ Ch( N ) is the barycenter of a unique Borel probability mea-sure µ ϕ supported on Ch + ( N ) (see [29] for an exhaustive course on the theory ofChoquet simplices; see also [1]). Therefore, if C (Ch( N )) denotes the vector spaceconsisting of all complex-valued continuous functions defined on Ch( N ), then f ( ϕ ) = Z Ch + ( N ) f ( ϑ ) dµ ϕ ( ϑ ) , f ∈ C (Ch( N ));in particular, if we choose the evaluation map at n ∈ N , then we easily deduce that(2.1) ϕ ( n ) = Z Ch + ( N ) ϑ ( n ) dµ ϕ ( ϑ ) , n ∈ N (hence, µ ϕ is the Choquet measure on Ch + ( N ) associated with ϕ ). The mapping ϕ µ ϕ defines a bijection Ch( N ) → M + (Ch + ( N )) where M + (Ch + ( N )) denotesthe set consisting of all Borel probability measures supported on Ch + ( N ); for every µ ∈ M + (Ch + ( N )), we denote by ϕ µ the character ϕ ∈ Ch( N ) such that µ ϕ = µ ,so that(2.2) ϕ µ ( n ) = Z Ch + ( N ) ϑ ( n ) dµ ( ϑ ) , n ∈ N. By a well-known theorem of Bauer (see for example [29, Proposition 11.1]; see also[1, Theorem II.4.1]), the inverse bijection M + (Ch + ( N )) → Ch( N ) (given by themapping µ ϕ µ ) is affine and continuous, and it is a homeomorphism if and onlyif Ch + ( N ) is a closed subset of Ch( N ), in which case Ch( N ) is a Bauer simplex.(We mention that every metrisable Choquet simplex is affinely homeomorphic tothe intersection of a decreasing sequence of metrisable Bauer simplices; see [19,Theorem 9].) In particular, this holds in the case where N is abelian where Ch + ( N )equals the Pontryagin dual N ◦ of N , and hence is a compact subset of Ch( N )(recall that N is discrete). The following result is essentially a consequence of[1, Corollary II.4.2]; we note that, since Ch +Γ ( N ) is the subset consisting of Γ-fixedelements of Ch + ( N ), it is closed (and hence compact) whenever Ch + ( N ) is a closedsubset of Ch( N ). Proposition 2.1.
Let N be a countable discrete group such that Ch + ( N ) is a closedsubset of Ch( N ) , and let Γ be a group consisting of continuous automorphisms of N .Then, the mapping ϕ µ ϕ defines an affine homeomorphism between Ch Γ ( N ) and M +Γ (Ch + ( N )) with inverse given by the mapping µ ϕ µ . In particular, for every ϕ ∈ Ch +Γ ( N ) , the Choquet measure µ ϕ ∈ M +Γ (Ch + ( N )) is ergodic; conversely, if µ ∈ M +Γ (Ch + ( N )) is ergodic, then the Γ -character ϕ µ ∈ Ch Γ ( N ) is indecomposable.Proof. Let ϕ ∈ Ch( N ) and γ ∈ Γ be arbitrary. Since the Choquet measures µ ϕ and µ γ · ϕ are uniquely determined by the characters ϕ and γ · ϕ , Eq. (2.1) impliesthat γ · ϕ = ϕ if and only if µ ϕ = µ γ · ϕ . On the other hand, for every n ∈ N , weevaluate ( γ · ϕ )( n ) = ϕ ( γ − · n ) = Z Ch + ( N ) ϑ ( γ − · n ) dµ ϕ = Z Ch + ( N ) ( γ · ϑ )( n ) dµ ϕ = Z Ch + ( N ) ϑ ( n ) d ( γ − · µ ϕ ) , and thus µ γ · ϕ = γ − · µ ϕ . Therefore, γ · ϕ = ϕ if and only if µ ϕ = γ − · µ ϕ , andthis clearly implies that ϕ ∈ Ch Γ ( N ) if and only if µ ϕ ∈ M +Γ (Ch + ( N )). Conversely, let µ ∈ M +Γ (Ch + ( N )) be arbitrary, and consider the character ϕ µ ∈ Ch( N ) defined as in Eq. (2.2). Since µ is Γ-invariant, we easily deduce that( γ · ϕ µ )( n ) = ϕ µ ( n ) for all γ ∈ Γ and all n ∈ N , and hence ϕ µ ∈ Ch Γ ( N ). Thiscompletes the proof of the first assertion of the proposition.Finally, we know that a measure in M +Γ (Ch + ( N )) is extreme if and only if itis ergodic (see [29, Section 12]), and thus µ ϕ ∈ M +Γ (Ch + ( N )) is ergodic for all ϕ ∈ Ch +Γ ( N ). (cid:3) We now return to the previous situation where G = 1+ A be an arbitrary discretecountable algebra group associated with a nil algebra A over a (countable discrete)field k ; furthermore, we let Γ = G × G , and consider the natural action of Γ on A ◦ .For every g ∈ G , let e g : A ◦ → C denote the evaluation map at g − ∈ A , that is, e g ( ϑ ) = ϑ ( g −
1) for all ϑ ∈ A ◦ . As a consequence of Pontryagin duality theorem (see[21, Theorem 4.31]), we easily conclude that { e g : g ∈ G } is a linearly independentset of C ( A ◦ ) (the vector space consisting of all complex continuous functions on A ◦ ), which is closed under pointwise multiplication (indeed, e a e b = e a + b ,for all a, b ∈ A ); moreover, in virtue of the Stone-Weierstrass theorem (see [16,Theorem 11.3.1]), the C -linear span e ( G ) of { e g : g ∈ G } is a dense C ∗ -subalgebraof C ( A ◦ ).The following elementary result will be useful. Lemma 2.2.
For every µ ∈ M +Γ ( A ◦ ) and every g, h ∈ G , we have Z A ◦ e g e h dµ = Z A ◦ e h − g dµ. Proof.
Let g, h ∈ G be arbitrary, and let a, b ∈ A be such that g = 1 + a and h = 1 + b . A straightforward calculation shows that e h ( hϑ ) = e h − ( ϑ ) for all ϑ ∈ A ◦ (recall that hϑ = ( h, · ϑ for all ϑ ∈ A ◦ ), and thus Z A ◦ e g ( ϑ ) e h ( ϑ ) dµ = Z A ◦ e g ( ϑ ) e h − ( h − ϑ ) dµ = Z A ◦ e g ( hϑ ) e h − ( ϑ ) dµ (because µ is Γ-invariant). Since e g ( hϑ ) e h − ( ϑ ) = e h − g ( ϑ ) for all ϑ ∈ A ◦ , weconclude that Z A ◦ e g ( hϑ ) e h − ( ϑ ) dµ = Z A ◦ e h − g ( ϑ ) dµ, and this completes the proof. (cid:3) Now, let ξ ∈ SCh( G ) be an arbitrary supercharacter of G . Then, the mapping e g ξ ( g ) extends by linearity to a C -linear map Φ : e ( G ) → C ; furthermore, since e ( G ) is a dense C ∗ -subalgebra of C ( A ◦ ), Φ extends by continuity to a continuouslinear function Φ : C ( A ◦ ) → C . Therefore, by the Riesz-Markov-Kakutani repre-sentation theorem ([32, Theorem 6.19]), there is a unique measure µ ξ ∈ M ( A ◦ )such that ξ ( g ) = Z A ◦ e g ( ϑ ) dµ ξ , g ∈ G ;moreover, the fact that ξ ∈ SCh( G ) ensures that µ ξ ∈ M Γ ( A ◦ ). Proposition 2.3. If ξ ∈ SCh( G ) is an arbitrary supercharacter of G , then µ ξ ∈M Γ ( A ◦ ) is a probability measure (hence, µ ξ ∈ M +Γ ( A ) ). Proof.
Let B be an arbitrary Borel subset of A ◦ , and let I B ∈ L ( A ◦ , µ ξ ) be thecorresponding indicator function. Since e ( G ) is a dense subalgebra of C ( A ◦ ), itsimage in L ( A ◦ , µ ξ ) is also dense, and hence there are families { α i,n ∈ C : 1 ≤ i ≤ n, n ∈ N } and { g i,n ∈ G : 1 ≤ i ≤ n, n ∈ N } such that I B ( ϑ ) = lim n →∞ X ≤ i ≤ n α i,n e g i,n ( ϑ ) , for µ -almost all ϑ ∈ A ◦ .Since I B ( ϑ ) = I B ( ϑ ) I B ( ϑ ) for all ϑ ∈ A ◦ , we deduce that µ ξ ( B ) = lim n →∞ X ≤ i,j ≤ n α i,n α j,n Z A ◦ e g i,n ( ϑ ) e g j,n ( ϑ ) dµ ξ , and thus Lemma 2.2 implies that µ ξ ( B ) = lim n →∞ X ≤ i,j ≤ n α i,n α j,n Z A ◦ e g − j,n g i,n ( ϑ ) dµ ξ = lim n →∞ X ≤ i,j ≤ n α i,n α j,n ξ ( g − j,n g i,n ) . Since ξ ∈ SCh( G ) is a positive-definite function, we know that X ≤ i,j ≤ n α i,n α j,n ξ ( g − j,n g i,n ) ≥ , n ∈ N , and thus µ ξ ( B ) ≥
0. Finally, since e ( ϑ ) = ϑ (0) = 1, we see that µ ξ ( A ◦ ) = Z A ◦ e ( ϑ ) dµ ξ = ξ (1) = 1 , and so µ is a probability measure, as required. (cid:3) In virtue of Proposition 2.1, we conclude that, for every supercharacter ξ ∈ SCh( G ), there is a unique Γ-character ζ ∈ Ch Γ ( A ) such that ζ ( a ) = Z A ◦ ϑ ( a ) dµ ξ , a ∈ A , and thus µ ξ = µ ζ is the Choquet measure which is associated with ζ ; in particular,we see that ξ ( g ) = ζ ( g −
1) for all g ∈ G . Conversely, for every µ ∈ M +Γ ( A ◦ ),we consider the Γ-character ζ µ ∈ Ch Γ ( A ) and define the function ξ µ : G → C by ξ µ ( g ) = ζ µ ( g −
1) for all g ∈ G ; hence, ξ µ ( g ) = Z A ◦ ϑ ( g − dµ = Z A ◦ e g ( ϑ ) dµ, g ∈ G. It is clear that the function ξ µ is constant on the superclasses of G , and that itis normalised (that is, ξ µ (1) = 1); thus, in order to conclude that ξ µ ∈ SCh( G ),it remains to trove that ξ µ is definite positive. We recall that, according to theGelfand-Naimark-Segal construction, a function ϕ : G → C is a character if andonly if there is a unitary representation ( π, H ) and a cyclic vector v ∈ H such that ϕ ( g ) = h π ( g ) v | v i , g ∈ G, where h· | ·i denotes the inner product of H (for details, we refer to [16, Proposi-tion 2.4.4]). Proposition 2.4.
For every µ ∈ M +Γ ( A ◦ ) , the function ξ µ : G → C is a super-character of G .Proof. In order to see that ξ µ is definite positive, we show that it is the characterof G afforded by an explicit cyclic representation of G . Let H µ denote the Hilbertspace L ( A ◦ , µ ); for every f ∈ H µ and every g ∈ G , we define the linear operator T µ ( g ) : H µ → H µ by( T µ ( g ) f )( ϑ ) = e g ( ϑ ) f ( g − ϑ ) , ϑ ∈ A ◦ . We claim that, for every g ∈ G , the operator T µ ( g ) is unitary. To check this,let g ∈ G be arbitrary, and compute the adjoint operator of T µ ( g ): for every f , f ∈ H µ , we evaluate hT µ ( g ) f | f i = Z A ◦ e g ( ϑ ) f ( g − ϑ ) f ( ϑ ) dµ = Z A ◦ e g ( gϑ ) f ( ϑ ) f ( gϑ ) dµ = Z A ◦ f ( ϑ ) e g − ( ϑ ) f ( gϑ ) dµ = h f | T µ ( g − ) f i (in the second equality, we took into account that the measure µ is Γ-invariant),and thus the adjoint operator of T µ ( g ) is T µ ( g − ). On the other hand, we clearlyhave T µ ( g ) T µ ( g − ) = T µ ( g − ) T µ ( g ) = id H µ where id H µ : H µ → H µ denotes theidentity operator, and hence T µ ( g ) is unitary, as claimed. Furthermore, it is easyto see that the mapping g
7→ T µ ( g ) defines a unitary representation of G on H µ .Finally, since e ( G ) is a dense subalgebra in C ( A ◦ ), its image in H µ is also dense;moreover, we have e g = T µ ( g ) e for all g ∈ G , and this implies that e is a cyclicvector of H µ , and that the representation ( T µ , H µ ) affords the character given bythe formula hT µ ( g ) e | e i = Z A ◦ e g ( ϑ ) dµ = ξ µ ( g ) , g ∈ G, as required. (cid:3) For every µ ∈ M +Γ ( A ◦ ), we refer to the representation ( T µ , H µ ), as defined inthe previous proof, as the super-representation of G associated with µ .The following result is now a clear consequence of Proposition 2.1 (and also ofPropositions 2.3 and 2.4); for the last assertion, we note that a supercharacter ξ ∈ SCh( G ) is indecomposable if and only if the corresponding Γ-character ζ ∈ Ch Γ ( A )is indecomposable. Theorem 2.5.
Let G = 1+ A be a countable discrete algebra group (associated witha nil k -algebra A ). Then, the mapping ξ µ ξ defines an affine homeomorphismbetween SCh( G ) and M +Γ ( A ◦ ) with inverse given by the mapping µ ξ µ . In par-ticular, the indecomposable supercharacters of G are in one-to-one correspondencewith the ergodic measures in M +Γ ( A ◦ ) . Consequently, a parametrisation of the ergodic measures in M +Γ ( A ◦ ) yields aparametrisation of the set SCh + ( G ) consisting of all indecomposable supercharac-ters of the algebra group G = 1 + A . In the case where G is finite, every Γ-invariantmeasure on A ◦ is supported on a unique Γ-orbit and, conversely, every Γ-orbit on A ◦ supports a unique Γ-invariant measure (hence, the Γ-orbits on A ◦ provide acomplete description of SCh + ( G ). However, if G = 1 + A is infinite, then an orbitΓ · ϑ supports a Γ-invariant measure if and only if Γ · ϑ is finite; we recall that the support supp( µ ) of a probability measure µ on A ◦ is defined to be the set consisting of all ϑ ∈ A ◦ such that µ ( U ) ∈ R + for every open neighbourhood U ⊆ A ◦ of ϑ (equivalently, supp( µ ) the smallest closed subset C of A ◦ such that µ ( A ◦ \ C ) = 0).Nevertheless, the following is true; henceforth, for every ϑ ∈ A ◦ , we denote by O ϑ the closure Γ · ϑ in A ◦ of the Γ-orbit Γ · ϑ . Proposition 2.6.
Let G = 1 + A be a countable discrete algebra group (associatedwith a nil k -algebra A ). Then, for every ergodic measure µ ∈ M +Γ ( A ◦ ) , there is atleast one ϑ ∈ A ◦ such that supp( µ ) = O ϑ .Proof. Let µ ∈ M +Γ ( A ◦ ) be ergodic, and consider supp( µ ) equipped with the sub-space topology; hence, in particular, supp( µ ) is second countable. Since supp( µ ) isclearly Γ-invariant, µ can be though naturally as an ergodic Γ-invariant measureon supp( µ ) having full support. Let { U n : n ∈ N } be a topological basis of supp( µ );hence, µ ( U n ) ∈ R + for all n ∈ N . Since Γ · U n is a Γ-invariant set of positive measureand µ is ergodic, we must have µ (Γ · U n ) = 1; moreover, the family { Γ · U n : n ∈ N } is also a topological basis of supp( µ ). Let V = T n ∈ N Γ · U n , and note that µ ( V ) = 1(because V is an intersection of sets with measure 1), and this clearly implies that V is non-empty; furthermore, for every ϑ ∈ V and every n ∈ N , the intersectionΓ · ϑ ∩ U n is non-empty. Since { U n : n ∈ N } be a topological basis of supp( µ ), weconclude that Γ · ϑ is a dense subset of supp( µ ), which means that O ϑ = supp( µ ),as stated. (cid:3) In the case where the group G is amenable, the previous result may be slightlyimproved; we recall that a discrete group G is amenable if it admits a Følnersequence , that is, a family { F n : n ∈ N } of finite subsets of G such that, for every g ∈ G , lim n →∞ | F n △ ( gF n ) || F n | = 0where △ denotes the symmetric difference of sets (for the definition and a detailedexposition on amenable groups, we refer to [30]; see also [12, Appendix G]). It isclear that G is amenable if and only if Γ is amenable. Proposition 2.7.
Let G = 1 + A be an amenable countable discrete algebra group(associated with a nil k -algebra A ). Then, for every ϑ ∈ A ◦ , there is at least oneergodic measure µ ∈ M +Γ ( A ◦ ) such that supp( µ ) = O ϑ .Proof. Let ϑ ∈ A ◦ , and let { F n : n ∈ N } be a Følner sequence for Γ; notice that G is amenable if and only if Γ is amenable. Since (cid:12)(cid:12)(cid:12)(cid:12) | F n | X γ ∈ F n e g ( γ · ϑ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | F n | X γ ∈ F n | e g ( γ · ϑ ) | = 1 , g ∈ G, and since { e g : g ∈ G } is a countable dense subset of C ( A ◦ ), [31, Theorem I.24](applied twice) implies that there is a subsequence { F n k : k ∈ N } of { F n : n ∈ N } such that the limit lim k →∞ | F n k | X γ ∈ F nk f ( γ · ϑ )exists for all f ∈ C ( A ◦ ). Therefore, we may define a measure µ on A ◦ by the rule Z A ◦ f dµ = lim k →∞ | F n k | X γ ∈ F nk f ( γ · ϑ ) , f ∈ C ( A ◦ ); since { F n k : k ∈ N } is a Følner sequence for Γ, it is straightforward to check that µ is a Γ-invariant probability measure (that is, µ ∈ M +Γ ( A ◦ )). Finally, for everyBorel subset B of A ◦ with µ ( B ) = 0, we have µ ( B ) = lim k →∞ | F n k | (cid:12)(cid:12) { γ ∈ F n k : γ · ϑ ∈ B } (cid:12)(cid:12) , and thus supp( µ ) is contained in the closure O ϑ of Γ · ϑ . Furthermore, if B is aΓ-invariant Borel subset of A ◦ with µ ( B ) >
0, then B ∩ Γ · ϑ is non empty, and thusΓ · ϑ ⊆ B , which implies that { γ ∈ F n k : γ · ϑ ∈ B } = F n k for all k ∈ N , and hence µ ( B ) = 1. It follows that the measure µ ∈ M +Γ ( A ◦ ) is ergodic, and this concludesthe proof. (cid:3) We remark that, in the case where G = 1 + A is a finite algebra group, theprevious results assure that every indecomposable supercharacter of G corresponduniquely to the closure of some Γ-orbit on A ◦ (see Eq. (1.1)); however, it is notclearly deduced that a similar result holds in the more general case of an infiniteamenable discrete countable algebra group (in fact, the closure of a given Γ-orbitmay support different ergodic Γ-invariant measures, and hence the the indecompos-able supercharacters of G are not necessarily parametrised by the closures of theΓ-orbits on A ◦ ). 3. The regular representation
As before, we let G = 1 + A be a discrete countable algebra group associatedwith a nil algebra A over a (countable discrete) field k , and consider the usual left regular representation ( π, L ( G, κ )) of G on the Hilbert space L ( G, κ ) where κ is the counting measure on G (which can be taken as a Haar measure on G );recall that, for every g ∈ G and every f ∈ L ( G, κ ), the map π ( g ) f ∈ L ( G, κ ) isdefined by π ( g ) f ( x ) = f ( g − x ) for all x ∈ G . For every g ∈ G , let δ g ∈ L ( G, κ )be the Dirac function supported on { g } , and note that the C -linear span of theset { δ g : g ∈ G } is dense in L ( G, κ ). Since π ( g ) δ = δ g for all g ∈ G , we seethat the function δ is a cyclic vector for the representation ( π, L ( G, κ )), and thusthe regular representation affords a character of G to which we refer as the regularcharacter of G and denote by λ G ; hence, λ G ( g ) = h π ( g ) δ | δ i = δ g, for all g ∈ G .It is clear that λ G is a supercharacter of G , and so in virtue of Theorem 2.5 it isuniquely determined by a unique Γ-invariant measure on A ◦ . In what follows, wecharacterise λ G as a supercharacter of G by understanding the measure on A ◦ whichis associated with λ G ; in particular, we provide a criterion for the regular characterto be an indecomposable supercharacter (or equivalently for the corresponding Γ-invariant measure to be ergodic), and determine its integral decomposition overSCh + ( G ).For the moment, we consider a slightly more general situation where A = G ◦ is the Pontryagin dual of an arbitrary compact abelian group G ; hence, A is adiscrete abelian group, and it is countable if and only if G is second countable (or,equivalently, metrisable). Since G is a compact group, there is a unique probabilityHaar measure on G which we denote by η (we recall that G is unimodular, andthus the left and right Haar measures coincide); on the other hand, we denote by κ the counting measure on G ◦ (which is a Haar measure on G ◦ ; notice that, since G ◦ abelian, it is trivially unimodular). For every ϕ ∈ L ( G , η ), we define the Fourier transform of ϕ to be the continuous map b ϕ : G ◦ → C given by b ϕ ( ϑ ) = Z G ϕ ( g ) ϑ ( g ) dη, ϑ ∈ G ◦ ;for details on the Fourier transform on arbitrary locally compact abelian groups,we refer to [21, Chapter 4]. In the case where b ϕ ∈ L ( G ◦ , κ ) for ϕ ∈ L ( G , η ),the Fourier inversion formula holds (see [21, Theorem 4.32]; see also [21, Proposi-tion 4.24]): for η -almost every g ∈ G , we have ϕ ( g ) = Z G ◦ b ϕ ( ϑ ) ϑ ( g ) dκ ;indeed, this formula holds for all g ∈ G whenever ϕ is continuous. Finally, wemention Plancherel theorem (see [21, Theorem 4.32]) which asserts that, whenrestricted to L ( G , η ) ∩ L ( G , η ), the Fourier transform extends uniquely to a unita-ry isomorphism F : L ( G , η ) → L ( G ◦ , κ ) (so that the Fourier transform is definedin the space L ( G , η ) + L ( G , η )).Now, let Γ be a group which acts on the left of G ◦ via continuous automorphisms;as before, we write γ · a = γ ( a ) for all γ ∈ Γ and all a ∈ G ◦ . Then, Γ actscontinuously on the left of G via the contragradient action, and thus every γ ∈ Γdefines a measure γ · η on G by the rule ( γ · η )( B ) = η ( γ − · B ) for all Borel subset B of G ; notice that ( γ · η )( G ) = η ( G ) = 1, and hence γ · η is a probability measure.It is well-known that, since every element γ ∈ Γ is a continuous automorphism of G with continuous inverse, the composition η ◦ γ is also a (left) Haar measure on G ,and thus the Haar measure η on G is Γ-invariant. As it turns out, the ergodicity ofthe Haar measure η with respect to the Γ-action on G depends on the existence offinite Γ-orbits on G ◦ . Indeed, the following result is an obvious generalisation of acertainly well-known theorem on classical ergodic theory on compact groups (see,for example, [40, Theorem 1.10]); a proof is included for convenience of the reader. Theorem 3.1.
Let G be a compact abelian group equipped with the normalisedHaar measure η , and let Γ be a group of continuous automorphisms of G . Then, η is ergodic (with respect to the Γ -action on G ) if and only if every nontrivial character ϑ ∈ G ◦ has an infinite Γ -orbit.Proof. Firstly, we assume that η is ergodic, and let ϑ ∈ G ◦ be such that the Γ-orbitΓ · ϑ ⊆ G ◦ is finite. Then, since the character ϕ Γ · ϑ = P τ ∈ Γ · ϑ τ of G is clearly Γ-invariant, the ergodicity of η implies that there is a constant c such that ϕ Γ · ϑ ( g ) = c for η -almost all g ∈ G , and thus c = Z G ϕ Γ · ϑ ( g ) dη = h ϕ Γ · ϑ , G i G = | Γ · ϑ | h ϑ, G i G where h· , ·i G denotes the usual inner product on L ( G , η ), and where G ∈ G ◦ is thetrivial character of G . By the orthonormality of the indecomposable characters of G , we conclude that c = 0 unless ϑ = G , in which case Γ · ϑ = { G } ; on the otherhand, if c = 0, then we obtain0 = Z G ϕ Γ · ϑ ( g ) ϑ ( g ) dη = h ϕ Γ · ϑ , ϑ i G = 1 , a contradiction. Consequently, { G } is the unique finite Γ-orbit on G ◦ , as required.Conversely, suppose that { G } is the unique finite Γ-orbit on G ◦ . In order toprove that η is an ergodic Γ-invariant measure, we show that every Γ-invariant function in L ( G , η ) is constant η -almost everywhere. Let ϕ ∈ L ( G , η ) be Γ-invariant, and consider the Fourier transform b ϕ = F ( ϕ ) ∈ L ( G ◦ , κ ). Since theHaar measure η is Γ-invariant, it is straightforward to check that b ϕ ( γ · ϑ ) = b ϕ ( ϑ )for all γ ∈ Γ and all ϑ ∈ G ◦ (which means that b ϕ is Γ-invariant), and thus Z Γ · ϑ | b ϕ ( ϑ ′ ) | dκ = κ (Γ · ϑ ) | b ϕ ( ϑ ) | . Since b ϕ ∈ L ( G ◦ , κ ), we conclude that b ϕ ( ϑ ) = 0 whenener ϑ ∈ G ◦ has an infiniteΓ-orbit (recall that G ◦ is discrete, and that κ is the counting measure on G ◦ ). Itfollows that b ϕ = c G where c = b ϕ ( G ), and so the Fourier inversion formula impliesthat ϕ is η -almost everywhere constantly equal to c , as required. (cid:3) We now return to the case where G = 1 + A is an arbitrary countable discretealgebra group (associated with a nil k -algebra A ) and Γ = G × G . Since thenormalised Haar measure η on A ◦ is Γ-invariant, it uniquely defines a supercharacter ξ η ∈ SCh( G ) by the rule ξ η ( g ) = Z A ◦ ϑ ( g − dη = δ g, , g ∈ G ;hence, ξ η is the regular character λ G of G . Let H η = L ( A ◦ , η ), and considerthe super-representation ( T η , H η ) of G . Since ( T η , H η ) affords the regular charac-ter of G , it follows that ( T η , H η ) is quasi-equivalent to the regular representation( π, L ( G, κ )); in fact, these representations are equivalent.
Proposition 3.2.
For every countable discrete algebra group G = 1+ A (associatedwith a nil k -algebra A ), the linear operator L : H η → L ( G, κ ) defined by L ( β )( g ) = Z A ◦ β ( ϑ ) ϑ ( g − dη, β ∈ H η , g ∈ G, defines an invertible intertwining operator between the representations ( T η , H η ) and ( π, L ( G, κ )) of G whose inverse L − : L ( G, κ ) → H η is defined on the functionswith finite support ς ∈ C c ( G ) by the rule L − ( ς )( ϑ ) = X g ∈ G ς ( g ) ϑ ( g − , ϑ ∈ A ◦ . Proof.
Firstly, we note that the map ν : G → A (defined by ν ( g ) = g − g ∈ G ) induces a unitary isomorphism of Hilbert spaces ν ∗ : L ( G, κ ) → L ( A , κ ).Furthermore, we have L = F − ◦ ν ∗ , and hence it only remains to show that L isan intertwining operator. To see this, for every g, h ∈ G and every β ∈ H η , weevaluate L (cid:0) T η ( g ) β (cid:1) ( h ) = Z A ◦ (cid:0) T η ( g ) β (cid:1) ( ϑ ) ϑ ( h − dη = Z A ◦ β ( g − ϑ ) e g ( ϑ ) ϑ (1 − h ) dη = Z A ◦ β ( g − ϑ ) ϑ ( g − ϑ (1 − h ) dη = Z A ◦ β ( g − ϑ ) ϑ ( g − h ) dη = Z A ◦ β ( ϑ ) ( gϑ )( g − h ) dη = Z A ◦ β ( ϑ ) ϑ (1 − g − h ) dη = Z A ◦ β ( ϑ ) ϑ ( g − h − dη = (cid:0) π ( g ) L ( β ) (cid:1) ( h )where the fourth equality holds by the G -invariance of η . The result follows. (cid:3) Consequently, the regular representation of G = 1 + A is completely determinedby the normalised Haar measure η on the dual group A ◦ ; in particular, the regularcharacter λ G is an indecomposable supercharacter if and only if η is an ergodicΓ-invariant measure. Since supp( η ) = A ◦ , Proposition 2.6 imply the followingimmediate corollary to the previous proposition. Theorem 3.3.
Let G be a countable discrete algebra group associated with a nil k -algebra A . Then, the regular character λ G is an indecomposable supercharacterof G if and only if { } is the unique finite superclass. Therefore, we see that the nature of the regular character, as an indecomposablesupercharacter, is not intrinsic to the class of countable discrete algebra groups,but rather to the nature of the Γ-action on A ◦ (or, equivalently, on the Γ-action on A ). Corollary 3.4.
Let G = 1 + A be a countable discrete algebra group associatedwith a nil k -algebra A , and suppose that A is finite dimensional. Then, the regularcharacter λ G is not an indecomposable supercharacter of G .Proof. Every finite dimensional nil algebra is a nilpotent algebra, and thus there isa non-zero element a ∈ A satisfying ab = ba = 0 for all b ∈ A . It is obvious thatthe { a } is a finite superclass of G . (cid:3) By the way of example, if k be an arbitrary countable infinite discrete field, thenthe regular character of the infinite unitriangular group U n ( k ) is not an indecom-posable supercharacter; indeed, U n ( k ) = 1 + u n ( k ) and u n ( k ) is finite-dimensional(nilpotent) k -algebra. On the other extreme, every non-trivial superclass of thelocally finite unitriangular group U ∞ ( k ) is infinite, and hence the regular characterof U ∞ ( k ) is an indecomposable supercharacter.At this point, it is worth to mention that λ G is an indecomposable character of G if and only if the unique finite conjugacy class is the trivial conjugacy class { } (see [28, Lemma 5.3.4]); hence, Theorem 3.3 may be though as a supercharacteranalogue of this result.As shown by Thoma (see [33, 34]; see also [10]), the connection between theregular character of an arbitrary countable discrete group G and its finite conju-gacy classes is much deeper. Indeed, it is possible to define a Plancherel formuladetermined by the normal subgroup G fc consisting of all elements of G with finiteconjugacy class. The conjugation action of G on G fc induces a natural action onthe set of characters Ch( G fc ) of G fc ; let Ch G ( G fc ) denote the convex subset ofCh( G fc ) consisting of all G -invariant elements of Ch( G fc ), and let Ch + G ( G fc ) denotethe corresponding set of extreme (or indecomposable) elements. Given an arbitrary ϕ ∈ Ch + G ( G fc ) we denote by e ϕ its extension by zero of ϕ to G ; hence, for every g ∈ G , e ϕ ( g ) = ( ϕ ( g ) , if g ∈ G fc ,0 , if g ∈ G \ G fc .It is straightforward to check that e ϕ ∈ Ch( G ) for all ϕ ∈ Ch( G fc ); indeed, as provedin [11, Proposition 1.F.9] (see also [13], or [21, Theorem 6.13]), if ( π ϕ , H ϕ ) is theGNS representation of G fc associated with ϕ , then the GNS representation ( π e ϕ , H e ϕ )of G associated with e ϕ is equivalent to the induced representation Ind GG fc ( π ϕ ), and thus there is a cyclic vector υ ∈ H e ϕ such that e ϕ ( g ) = (cid:10) Ind GG fc ( π ϕ )( g ) υ, υ (cid:11) , g ∈ G. If we set f Ch + G ( G fc ) = { e ϕ : ϕ ∈ Ch + G ( G fc ) } , then [34, Satz 4] asserts that the intersection f Ch + G ( G fc ) ∩ Ch + ( G ) is non-empty;moreover, if ω denotes the Choquet measure on Ch + ( G ) which is associated withthe regular character λ G , then supp( ω ) = f Ch + G ( G fc ), and thus we have(3.1) λ G ( g ) = Z f Ch + G ( G fc ) χ ( g ) dω, g ∈ G. Since the regular character λ G is afforded by the regular representation of G andsince Ch + ( G ) can be understood as a dual space for G , it is somewhat customary torefer to ω as the Plancherel measure of G ; more details about the regular characterof nilpotent discrete groups can be found in [9]. (Recent developments on thePlancherel formula for countable groups can be found in [10].)As it turns out, there is also a counterpart for this result in terms of finitesuperclasses of a countable discrete algebra group. Let G = 1 + A be an arbitrarycountable discrete algebra group (associated with a nil k -algebra A ), and let ν bethe unique probability measure on SCh + ( G ) such that(3.2) λ G ( g ) = Z SCh + ( G ) ξ ( g ) dν, g ∈ G ;in direct analogy with the integral decomposition of λ G with respect to indecom-posable characters, we refer to the measure ν as the super-Plancherel measure . Ouraim is to describe this measure ν in terms of the supercharacters of the algebrasubgroup G fsc of G consisting of all elements having a finite superclass; we start byproving that G fsc is indeed an algebra subgroup of G . Lemma 3.5.
In the above notation, let A fsc be the subset of A consisting of allelements with finite Γ -orbit. Then, A fsc is a (nil) subalgebra of A and G fsc =1 + A fsc . In particular, G fsc is an algebra subgroup of G ; furthermore, G fsc is anormal subgroup of G .Proof. It is obvious that A fsc is closed under scalar multiplication. On the otherhand, if a, b ∈ A fsc , then Γ · ( a + b ) ⊆ Γ · a + Γ · b , and hence Γ · ( a + b ) is a finiteΓ-orbit, which means that a + b ∈ A fsc . Finally, if a, b ∈ A fsc , thenΓ · ( ab ) = { gag − gbh : g, h ∈ G } ⊆ { gag − : g ∈ G } (Γ · b ) ⊆ (Γ · a ) (Γ · b ) , and so the Γ-orbit Γ · ( ab ) is finite, which means that ab ∈ A fsc . The result is nowclear. (cid:3) The group Γ acts continuously on the left of A fsc , and thus we naturally obtaina continuous left action of Γ on the Pontryagin dual ( A fsc ) ◦ of the additive group( A fsc ) + . Following the terminology used in Proposition 2.1, let Ch Γ ( A fsc ) denotethe set of all Γ-invariant characters of ( A fsc ) + , and let Ch +Γ ( A fsc ) denote the setconsisting of all indecomposable Γ-invariant characters of ( A fsc ) + . By Proposition2.1, we know that Ch Γ ( A fsc ) is affinelly homeomorphic to the space M +Γ (( A fsc ) ◦ )of Γ-invariant probability measures on ( A fsc ) ◦ , so that the subset Ch +Γ ( A fsc ) of indecomposable Γ-supercharacters is in one-to-one correspondence with the set ofergodic measures in M +Γ (( A fsc ) ◦ ).On the other hand, if we set Γ fsc = G fsc × G fsc , then Theorem 2.5 asserts thatthe set SCh( G fsc ) of supercharacters of G fsc is affinelly homeomorphic to the space M +Γ fsc (( A fsc ) ◦ ) of Γ fsc -invariant probability measures on ( A fsc ) ◦ . In particular, sincewe clearly have M +Γ (( A fsc ) ◦ ) ⊆ M +Γ fsc (( A fsc ) ◦ ), we conclude that every Γ-invariantprobability measure µ ∈ M +Γ (( A fsc ) ◦ ) corresponds uniquely to the supercharacter ξ µ ∈ SCh( G fsc ) defined by the rule ξ µ ( g ) = Z ( A fsc ) ◦ ϑ ( g − dµ, g ∈ G fsc . Henceforth, we define SCh Γ ( G fsc ) = { ξ µ : µ ∈ M +Γ (( A fsc ) ◦ ) } , and denote by SCh +Γ ( G fsc ) the subset of SCh Γ ( G fsc ) consisting of all supercharac-ters of G fsc which correspond to the ergodic Γ-invariant measures in M +Γ (( A fsc ) ◦ );furthermore, for every ξ ∈ SCh Γ ( G fsc ), we denote by e ξ the extension by zero of ξ to G . Our next goal is to prove the following result. Theorem 3.6.
Let G = 1+ A be a discrete countable algebra group associated with anil k -algebra A , and let ξ ∈ SCh +Γ ( G fsc ) be arbitrary. Then, e ξ is an indecomposablesupercharacter of G . In order to proceed with the proof of this theorem, we consider a slightly moregeneral situation where B is an arbitrary Γ-invariant subalgebra of A ; we considerthe Pontryagin dual B ◦ of the additive group B + , and the natural contragradientΓ-action on B ◦ . By Proposition 2.1, we know that the space Ch Γ ( B ) consistingof all Γ-characters of B + is affinelly homeomorphic to the space M +Γ ( B ◦ ) of Γ-invariant probability measures on B ◦ , and so the subset Ch +Γ ( B ) consisting of allindecomposable Γ-characters of B + is in one-to-one correspondence with the setof ergodic measures in M +Γ ( B ◦ ). On the other hand, if we set Λ = H × H , thenTheorem 2.5 asserts that Theorem 2.5 the space SCh( H ) of supercharacters of thealgebra group H = 1+ B (which is countable and discrete) is affinelly homeomorphicto the space M +Λ ( B ◦ ) of Λ-invariant probability measures on B ◦ . In particular, sincewe clearly have M +Γ ( B ◦ ) ⊆ M +Λ ( B ◦ ), we conclude that every Γ-invariant probabilitymeasure µ ∈ M +Γ ( B ◦ ) corresponds uniquely to the supercharacter ξ µ ∈ SCh( H )defined by the rule ξ µ ( h ) = Z B ◦ ϑ ( h − dµ, h ∈ H. Henceforth, we set SCh Γ ( H ) = { ξ µ : µ ∈ M +Γ ( B ◦ ) } , and denote by SCh +Γ ( H ) the subset of SCh Γ ( H ) consisting of all supercharacters of H which correspond to the ergodic Γ-invariant measures in M +Γ ( B ◦ ); furthermore,for every ξ ∈ SCh Γ ( H ), we let e ξ denote the trivial extension by zero of ξ to G .Now, let B ′ be an additive subgroup of A such that A decomposes as the directsum A = B ⊕B ′ ; notice that, since B is a k -linear subspace of A , it admits a k -linearcomplement B ′ (which is obviously an additive subgroup of A ). Furthermore, let B ⊥ = { ϑ ∈ A ◦ : B ⊆ ker( ϑ ) } be the closed subgroup of A ◦ which is orthogonal to B , and note that B ⊥ is homeomorphic to the Pontryagin dual of B ′ ; indeed, B ′ is a canonically isomorphic to the quotient group A / B , and it is well-knownthat the Pontryagin dual ( A / B ) ◦ of A / B is naturally isomorphic to B ⊥ (see [21,Theorem 4.39]). It follows that A ◦ is homeomorphic to the product space B ◦ × B ⊥ ;since B ◦ is metrisable (because B is countable), [25, Corollary to Theorem 8.1]guarantees that every Borel subset of A ◦ is of the form B × B where B is aBorel subset of B ◦ and B is a Borel subset of B ⊥ . Therefore, every Γ-invariantprobability measure ν ∈ M +Γ ( A ◦ ) is uniquely factorised as a product measure ν = ν × ν where ν ∈ M +Γ ( B ◦ ) and ν ∈ M +Γ ( B ⊥ ), and hence the associatedsupercharacter ξ ν ∈ SCh( G ) factorises as follows: if a ∈ A and a = b + b ′ for b ∈ B and b ′ ∈ B ′ , then ξ ν (1 + a ) = Z A ◦ τ ( a ) dν = (cid:18) Z B ◦ τ ( b ) dν (cid:19)(cid:18) Z B ⊥ τ ′ ( b ′ ) dν (cid:19) . We are now able to prove the following auxiliary result.
Proposition 3.7.
Let G = 1 + A be a discrete countable algebra group associatedwith a nil k -algebra A , let B be a Γ -invariant subalgebra of A , and let H = 1 + B .Moreover, let ξ ∈ SCh Γ ( H ) , let e ξ ∈ SCh( G ) be the trivial extension by zero of ξ to G , and let e µ ∈ M +Γ ( A ◦ ) be the Γ -invariant probability measure associated with e ξ .Then, e µ factorises uniquely as the product e µ = µ × η B ⊥ where µ ∈ M +Γ ( B ◦ ) is the Γ -invariant measure associated with ξ and η B ⊥ is the normalised Haar measure of B ⊥ . In particular, e ξ is an indecomposable supercharacter of G if and only if bothmeasures µ and η B ⊥ are ergodic with respect to the Γ -action on B ◦ and on B ⊥ ,respectively.Proof. The proof is a matter of straightforward calculations; notice that, since B is Γ-invariant, B ⊥ is a Γ-invariant compact subgroup of A ◦ , and hence the Haarmeasure η B ⊥ is Γ-invariant. Indeed, let a ∈ A be arbitrary, and decompose a = b + b ′ where b ∈ B and b ′ ∈ B ′ are unique. Then, Z A ◦ ϑ ( a ) d ( µ × η B ⊥ ) = (cid:18) Z B ◦ τ ( b ) dµ (cid:19)(cid:18) Z B ⊥ τ ′ ( b ′ ) dη B ⊥ (cid:19) ;since B ⊥ is compact, the orthogonality relations imply that R B ⊥ τ ′ ( b ′ ) dη B ⊥ = δ b ′ , for all b ′ ∈ B ′ , and thus Z A ◦ ϑ ( a ) d ( µ × η B ⊥ ) = ( ξ (1 + a ) , if a ∈ B ,0 , otherwise,and hence e ξ ( g ) = Z A ◦ ϑ ( g − d ( µ × η B ⊥ ) . By the unicity of the measure e µ ∈ M +Γ ( A ◦ ) (see Theorem 2.5), we conclude that e µ = µ × η B ⊥ , and this completes the proof (the remaining assertions are clear). (cid:3) Finally, we consider the regular character λ G of the countable discrete algebragroup G = 1 + A . On the one hand, we recall from Eq. (3.1) that λ G ( g ) = Z f Ch + G ( G fc ) χ ( g ) dω, g ∈ G, where ω is the Choquet measure on Ch + ( G ) associated with λ G ; on the otherhand, since λ G ∈ SCh( G ), we know that there is a unique probability measure ν on SCh + ( G ) such that λ G ( g ) = Z SCh + ( G ) ξ ( g ) dν, g ∈ G (see Eq. (3.1)). By comparing these two integral decompositions, we conclude that ξ ( g ) = 0 for all g ∈ G \ G fc and ν -almost all ξ ∈ SCh + ( G ). We consider the adjoint(left) action of G on A given by g · a = gag − for all g ∈ G and all a ∈ A , anddenote by A fc the subset of A consisting of all elements with finite G -orbit (hence, G fc = 1 + A fc ). Moreover, we define A fc (Γ) = { a ∈ A fc : Γ · a ⊆ A fc } ;we note that, if a ∈ A \ A fc (Γ), then there is an element γ ∈ Γ such that γ · a / ∈ A fc and ξ (1 + a ) = ξ (1 + γ · a ) for all ξ ∈ SCh + ( G ), and thus ξ (1 + a ) = 0 for all a / ∈ A fc (Γ) and ν -almost all ξ ∈ SCh + ( G ). Using an argument similar to the proofof Lemma 3.5, it is easy to see that both A fc and A fc (Γ) are subalgebras of A ; it isalso obvious that A fsc ⊆ A fc (Γ) ⊆ A fc .Let ξ ∈ SCh + ( G ) be such that ξ (1 + a ) = 0 for all a ∈ A \ A fc (Γ), and ξ denotethe restriction to G fc (Γ) = 1 + A fc (Γ); notice that ξ ∈ SCh +Γ ( G fc (Γ)) and thatthe trivial extension by zero of ξ to G equals ξ , and thus Lemma 3.7 implies thatthe normalised Haar measure η A fc (Γ) ⊥ of A fc (Γ) ⊥ is ergodic with respect to theΓ-action.We are now able to conclude the proof of Theorem 3.6. Proof of Theorem 3.6.
It is clear that e ξ is a supercharacter of G , and thus Theorem2.5 guarantees there is a unique Γ-invariant probability measure e µ ∈ M +Γ ( A ◦ ) suchthat e ξ ( g ) = Z A ◦ ϑ ( g − d e µ, g ∈ G ;moreover, we know that e ξ ∈ SCh + ( G ) if and only if e µ is ergodic. Accordingto Proposition 3.7, e µ factorises uniquely as the product e µ = µ × η ( A fsc ) ⊥ where µ ∈ M +Γ (( A fsc ) ◦ ) is the Γ-invariant measure associated with ξ and η ( A fsc ) ⊥ is thenormalised Haar measure of ( A fsc ) ⊥ ; since ξ ∈ SCh +Γ ( G fsc ), we know that themeasure µ is ergodic, and thus e µ is ergodic if and only if η ( A fsc ) ⊥ is ergodic (withrespect to the Γ-action). Finally, since A fc (Γ) ⊥ is homeomorphic to the quotient A ◦ / A fc (Γ) ◦ and since there is a natural homeomorphism A ◦ / A fc (Γ) ◦ ∼ = A ◦ / ( A fsc ) ◦ × ( A fsc ) ◦ / ( A fc (Γ) ◦ , we conclude that the measure η A fc (Γ) ⊥ factorises as the product of the normalisedHaar measure of A ◦ / ( A fsc ) ◦ and the normalised Haar measure of ( A fsc ) ◦ / A fc (Γ) ◦ .Since η A fc (Γ) ⊥ is ergodic (as shown above), it follows that the normalised Haarmeasure of A ◦ / ( A fsc ) ◦ is ergodic (with respect to the Γ-action), and this completesthe proof (because A ◦ / ( A fsc ) ◦ is homeomorphic to ( A fsc ) ⊥ . (cid:3) We next describe the super-Plancherel measure . For simplicity, we denote by λ fsc the regular character of G fsc ; it is clear that λ fsc ∈ SCh Γ ( G fsc ) is associatedwith the normalised Haar measure η fsc on ( A fsc ) ◦ . Since SCh Γ ( G fsc ) is a Choquet simplex and SCh +Γ ( G fsc ) is its set of extreme elements, Choquet’s theorem assertsthat there is a unique probability measure ν fsc on SCh +Γ ( G fsc ) such that λ fsc ( g ) = Z SCh +Γ (Γ fsc ) ξ ( g ) dν fsc , g ∈ G. Theorem 3.8.
Let G = 1 + A be a countable discrete algebra group associated witha nil k -algebra A , and let G fsc = 1 + A fsc be the algebra subgroup consisting of allelements having a finite superclass. If ν fsc is the Choquet measure on SCh +Γ ( G fsc ) associated with the regular character λ fsc , then the super-Plancherel measure ν on SCh + ( G ) is given by the pushforward of ν fsc by the map SCh +Γ ( G fsc ) → SCh + ( G ) defined by the mapping ξ e ξ . In particular, we have λ G ( g ) = Z SCh +Γ ( G fsc ) e ξ ( g ) dν fsc = Z g SCh +Γ ( G fsc ) ξ ( g ) dν, g ∈ G, where we set g SCh +Γ ( G fsc ) = { e ξ : ξ ∈ SCh +Γ ( G fsc ) } (which, by the previous proposi-tion, is a subset of SCh + ( G ) , easily seen to be closed).Proof. Since e λ fsc = λ G , we have λ G ( g ) = Z SCh +Γ ( G fsc ) e ξ ( g ) dν fsc , g ∈ G. On the other hand, Theorem 3.6 implies that e ξ ∈ SCh + ( G ) for all ξ ∈ SCh +Γ ( G fsc ),and the result follows by the uniqueness of the Choquet measure on SCh + ( G )associated with the regular character λ G . (cid:3) We end this section by exemplifying the super-Plancherel decomposition of theregular character in the rather simple case of the unitriangular group G = U n ( k )over an arbitrary infinite countable discrete field k . In this situation, the subgroup G fsc = U n ( k ) fsc is easy to describe: U n ( k ) fsc = 1+ k e ,n where, for all 1 ≤ i < j ≤ n , e i,j ∈ u n ( k ) denotes the usual elementary matrix having ( i, j )-th coefficient equalto one and zeroes elsewhere. Notice that U n ( k ) fsc is precisely the center of U n ( k ),and that it equals the subgroup U n ( k ) fc consisting of all elements having finiteconjugacy class; consequently, the Plancherel measure coincides with the super-Plancherel measure. Since U n ( k ) fsc is clearly isomorphic to the additive group k + of k , the Pontryagin dual of U n ( k ) fsc may be identified with the Pontryagin dual k ◦ of k + : with every ϑ ∈ k ◦ we associate the character ξ ϑ : U n ( k ) fsc → C definedby ξ ϑ (1 + αe ,n ) = ϑ ( α ) for all α ∈ k . Therefore, SCh +Γ ( U n ( k ) fsc ) = { ξ ϑ : ϑ ∈ k ◦ } where Γ = U n ( k ) × U n ( k ), and thus λ U n ( k ) ( g ) = Z k ◦ e ξ ϑ ( g ) dν where ν is the normalised Haar measure on k ◦ and where, for every ϑ ∈ k ◦ , e ξ ϑ denotes the extension by zero of ξ ϑ from U n ( k ) fsc to U n ( k ).The behaviour of the regular character of U n ( k ) provides a good example ofhow different phenomena arises in the representation of big groups : although theregular character of any finite algebra group decomposes as a convex sum of all itsindecomposable supercharacters, this is no longer true in the case of the infiniteunitriangular group U n ( k ), where the regular character decomposes as a “convexsum” involving only a “very small” set of indecomposable supercharacters. The infinite unitriangular groups U n ( k )This section is mainly devoted to description of the supercharacters of the infiniteunitriangular group U n ( k ) where k is the algebraic closure of a finite field of charac-teristic p . The group U n ( k ) is the prototype example of an approximately finite alge-bra group : an algebra group G = 1+ A over a field k (associated with a nil k -algebra A ) is said to be approximately finite if there are a chain k ⊆ k ⊆ · · · ⊆ k m ⊆ · · · of subfields of k and a chain G ⊆ G ⊆ · · · ⊆ G m ⊆ · · · of finite subgroups of G such that, for every m ∈ N , the subgroup G m = 1 + A m is an algebra group over k m associated with some k m -subalgebra A m of A , and G = [ m ∈ N G m ;for simplicity, we refer to such an algebra group G as an AF-algebra group . We notethat, since G m is a finite group, the subfield k m must be finite, and hence k musthave non-zero characteristic (as required); moreover, the k m -algebra A m must befinite-dimensional over k m , and hence it is a nilpotent k m -subalgebra of A . Onthe other hand, the inclusion G m ⊆ G m +1 clearly implies that A m ⊆ A m +1 , andthus A m is a k m -subalgebra of A m +1 . Finally, we observe that, with respect to thenatural inclusion maps, the AF-algebra group G is isomorphic to the direct limitlim −→ m ∈ N G m , and hence it is an amenable group (by [30, Proposition 13.6] becausefinite groups are trivially amenable); furthermore, it is also straightforward to checkthat k = S m ∈ N k m is also isomorphic to the direct limit lim −→ m ∈ N k m .Concerning our standard example, for every m ∈ N , let k m denote the finite field F p m ! with p m ! elements where p is the characteristic of k . Since k m is a subfield of k m +1 , we may consider the (finite) unitriangular group U n ( k m ) as a subgroup of U n ( k m +1 ). In this situation, k = S m ∈ N k m is the algebraic closure of the prime field F p with p elements, and the (infinite) unitriangular group U n ( k ) may be naturallyrealised as the union U n ( k ) = [ m ∈ N U n ( k m );in particular, we see that U n ( k ) is an AF-algebra group.In the general situation, let G = S m ∈ N G m be an arbitrary (fixed) AF-algebragroup, and consider the set SCl( G ) of superclasses and the set SCh( G )) of super-characters of G (notice that G is countable and discrete); moreover, for every m ∈ N ,we consider the set SCl( G m ) of superclasses and the set SCh( G m ) of supercharac-ters of the finite algebra group G m = 1 + A m . Our aim is to exploit the relationshipbetween the pair (SCl( G ) , SCh + ( G )) and the pairs (SCl( G m ) , SCh + ( G m )) for every m ∈ N , in order to show that every indecomposable supercharacter ξ ∈ SCh + ( G ) is finitely approximated by a sequence ( ξ m ) m ∈ N , where ξ m ∈ SCh + ( G m ) for all m ∈ N ,in the sense that ξ is the pointwise limit of ( ξ m ) m ∈ N . This will be accomplishedusing Lindenstrauss’ pointwise ergodic theorem for tempered Følner sequences ; werecall that a Følner sequence for a group G is a family { F m : m ∈ N } of finitesubsets of G such that, for every g ∈ G ,lim m →∞ | F m △ ( gF m ) || F m | = 0where △ denotes the symmetric difference of sets, and that a Følner sequence { F m : m ∈ N } for an amenable group G is said to be tempered if there is a positive real number C > (cid:12)(cid:12)(cid:12)(cid:12) [ ≤ k ≤ m F − k F m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12) F m +1 (cid:12)(cid:12) , m ∈ N . Lindenstrauss’ pointwise ergodic theorem ([27, Theorem 1.3]) . Let G be anamenable discrete group acting on a probability space ( X, µ ) . Assume that themeasure µ is ergodic with respect to the G -action on X , and that there is a temperedFølner sequence { F m } m ∈ N for G . Then, for µ -almost every point x ∈ X and every f ∈ L ( X, µ ) , we have lim m →∞ | F m | X g ∈ F m f ( g − · x ) = Z X f ( x ) dµ. For every m ∈ N , let Γ m = G m × G m , and note that Γ = S m ∈ N Γ m ; as before,we set Γ = G × G . For every γ ∈ Γ, there is m ∈ N such that γ ∈ Γ m , and thusthe symmetric difference Γ m ′ △ ( γ Γ m ′ ) is empty for all m ′ ∈ N such that m ′ ≥ m .Therefore, { Γ m : m ∈ N } is obviously a Følner sequence for Γ which is clearly tem-pered. As a consequence of Lindenstrauss’ pointwise ergodic theorem, we deducethe following result; we recall that, for every m ∈ N , the indecomposable superchar-acters of the finite algebra group G m = 1 + A m are in one-to-one correspondencewith the Γ m -orbits on the dual group ( A m ) ◦ (by means of the formula (1.1)). Theorem 4.1 (Finite approximation property) . In the notation as above, let G = S m ∈ N G m be an arbitrary AF-algebra group, let µ ∈ M +Γ ( A ◦ ) be an ergodic Γ -invariant measure on A ◦ , and consider the indecomposable supercharacter ξ µ ∈ SCh + ( G ) which (uniquely) corresponds to µ ; hence, ξ µ ( g ) = Z A ◦ ϑ ( g − dµ, g ∈ G. For every ϑ ∈ A ◦ , let ϑ m ∈ ( A m ) ◦ denote the restriction of ϑ to A m , and let ξ m ∈ SCh + ( G m ) be the indecomposable supercharacter of G m which corresponds tothe Γ m -orbit Γ m · ϑ m . Then, for µ -almost every ϑ ∈ A ◦ , we have ξ µ ( g ) = lim m →∞ ξ m ( g ) , g ∈ G. Furthermore, every indecomposable supercharacter of G is of this form.Proof. For every m ∈ N , every ϑ ∈ A ◦ and every g ∈ G , we set A m,ϑ ( g ) = 1 | Γ m | X γ ∈ Γ m e g ( γ · ϑ ) = 1 | Γ m | X γ ∈ Γ m ( γ · ϑ )( g − γ · ϑ ) m = γ · ϑ m for all γ ∈ Γ m , we have A m,ϑ ( g ) = ξ m ( g )whenever g ∈ G m (see Eq. (1.1)).On the one hand, let ϑ ∈ A ◦ be fixed, and assume that the sequence ( A m,ϑ ( g )) m ∈ N converges for all g ∈ G . Then, as in the proof of Proposition 2.7, we may define anergodic Γ-invariant measure µ on A ◦ by the rule Z A ◦ f dµ = lim m →∞ | Γ m | X γ ∈ Γ m f ( γ · ϑ ) , f ∈ C ( A ◦ );in particular, we see that ξ µ ( g ) = Z A ◦ e g dµ = lim m →∞ A m,ϑ ( g ) , g ∈ G. Now, let g ∈ G be arbitrary, and choose the smallest m ∈ N such that g ∈ G m .Then, A m,ϑ ( g ) = ξ m ( g ) for all m ∈ N such that m ≥ m (because g ∈ G m ), andthus ξ µ ( g ) = lim m →∞ ξ m ( g ), as required.Conversely, if µ is an arbitrary ergodic Γ-invariant measure on A ◦ , then Lin-denstrauss’ pointwise ergodic theorem implies that, for µ -almost every ϑ ∈ A ◦ , wehave ξ µ ( g ) = Z A ◦ e g dµ = lim m →∞ | Γ m | X γ ∈ Γ m e g ( γ · ϑ ) = lim m →∞ A m,ϑ ( g ) , g ∈ G (recall that { Γ m : m ∈ N } is a tempered Følner sequence). Since A m,ϑ ( g ) = ξ m ( g )for all sufficiently large m ∈ N , the result is now clear. (cid:3) The previous theorem exhibits the asymptotic nature of the indecomposable su-percharacters of AF-algebra groups; in the case of indecomposable characters ofan arbitrary AF-group (that is, a direct limit of finite groups) an analogous re-sult was established by Kerov and Vershik (see [39]). In the particular example ofthe unitriangular group U n ( k ) over the algebraic closure of a finite field of non-zerocharacteristic p , since there are explicit formulas for the indecomposable superchar-acters of the finite unitriangular groups, it is possible to derive a formula for itsindecomposable supercharacters; in [8], a similar formula was accomplished (albeitby different methods, based on [39]) in the case of the locally finite unitriangulargroup U ∞ ( F q ) = S n ∈ N U n ( F q ) over an arbitrary finite field F q .We shall describe the supercharacters of U n ( k ) using the finite approximationproperty (Theorem 4.1); for simplicity, we set K = SCl( U n ( k )) and E = SCh + ( U n ( k )).We start by providing a brief characterisation of the superclasses and indecompos-able supercharacters of a finite unitriangular group; the details can be found in[2, 6] (see also [7]). For every m ∈ N , we set K m = SCl( U n ( k m )) and E m =SCh + ( U n ( k m )); recall that k m denotes the finite field F p m ! with p m ! elements (whichwe consider as a subfield of k ).We denote by Sp ( n ) the set consisting of all set partitions of [ n ] = { , . . . , n } ,and write π ∈ Sp ( n ) as a sequence π = B /B / . . . /B k where B , . . . , B k aredisjoint subsets of [ n ] such that [ n ] = B ∪ B ∪ · · · ∪ B k ; we refer to B , . . . , B k asthe blocks of π . A pair ( i, j ) with 1 ≤ i < j ≤ n is said to be an arc of π ∈ Sp ( n )if i and j lie in the same block B of π and there is no k in B such that i < k < j ;we denote by D ( π ) the set consisting of all arcs of π . If π ∈ Sp ( n ), then a map α : D ( π ) → k m \ { } is called a k m - colouration of π . We denote by Col k m ( π ) theset consisting of all k m -colourations of π ∈ Sp ( n ), and defineΦ n ( k m ) = { ( π, α ) : π ∈ Sp ( n ) , α ∈ Col k m ( π ) } ;an element of Φ n ( k m ) is referred to as a k m -coloured set partition of [ n ].For every ( π, α ) ∈ Φ n ( k m ), we define e π,α ∈ u n ( k m ) to be the element e π,α = X ( i,j ) ∈D ( σ ) α ( i, j ) e i,j where e i,j ∈ u n ( k m ) stands for the elementary matrix having ( i, j )-th coefficientequal to 1 and zeroes elsewhere. It is straightforward to check that, for everysuperclass K ∈ K m , there is a unique ( π, α ) ∈ Φ n ( k m ) such that 1 + e π,α ∈ K , andthus the superclasses in K m are in one-to-one correspondence with the k m -coloured set partitions of [ n ]; we denote by K π,α the superclass in K m which is associatedwith ( π, α ) ∈ Φ n ( k m ).We next proceed with the description of the indecomposable supercharacters of U n ( k m ). Let k ◦ m denote the dual group of the additive group k + m ; notice that thereis a group isomorphism k + m ∼ = k ◦ m given by the mapping α ατ where τ ∈ k ◦ m isan arbitrarily fixed non-trivial character of k m , and where the character ατ ∈ k ◦ m ,for α ∈ k m , is defined ( ατ )( β ) = τ ( αβ ) for all β ∈ k m . Since the additive group u n ( k m ) + is clearly isomorphic to a direct product of n ( n − / k + m , everycharacter ϑ ∈ u n ( k m ) ◦ can be uniquely represented by a strictly uppertriangular n × n matrix ϑ = ( ϑ i,j ) where ϑ i,j ∈ k ◦ m for all 1 ≤ i < j ≤ n ; in fact, if τ ∈ k ◦ m isan arbitrarily fixed non-trivial character of k m and if we define τ a ( b ) = τ (Tr( a T b ))for all a, b ∈ u n ( k m ), where Tr denotes the usual trace map and a T denotes thetranspose matrix of a , then it is not hard to check that the mapping a τ a defines a group isomorphism between u n ( k m ) + and u n ( k m ) ◦ (we observe that theexistence of this isomorphism depends strongly on the self-duality of the finitefield k m , and that a similar isomorphism cannot exist when we replace k m by itsalgebraic closure k ). For our purposes, it is useful to extend this notation anddefine the character τ a ∈ u n ( k m ) ◦ for all τ ∈ k ◦ m and all a ∈ u n ( k m ); for simplicityof writing, we set τ ⋆ a ∗ = τ a . By the way of example, for every τ ∈ k ◦ m and every1 ≤ i < j ≤ n , the character τ ⋆ e ∗ i,j ∈ u n ( k m ) ◦ is given by ( τ ⋆ e ∗ i,j )( b ) = τ ( b i,j )for all b = ( b i,j ) ∈ u n ( k m ), and thus we see that every character ϑ ∈ u n ( k m ) ◦ decomposes uniquely as the sum ϑ = X ≤ i Proposition 4.2. Let ( π, τ ) ∈ Φ n ( k ◦ m ) and ( π ′ , α ) ∈ Φ n ( k m ) be arbitrary. Then,the (constant) value ξ π,τ ( K π ′ ,α ) of the (normalised) indecomposable supercharacter ξ π,τ on the superclass K π ′ ,α is equal to unless D ( π ) ⊆ R ( π ′ ) , in which case it isgiven by ξ π,τ ( K π ′ ,α ) = 1 p m ! nest π ( π ′ ) ϑ π,τ ( e π ′ ,α ) . We are now able to deal with the infinite unitriangular group U n ( k ), and todescribe its superclasses and its indecomposable supercharacters. As above, wedenote by Φ n ( k ) the set consisting of all k -coloured set partitions of [ n ]; by a k -colouration of a set partition π ∈ Sp ( n ) we mean a map α : D ( π ) → k \ { } . Noticethat there is an obvious inclusion Φ n ( k m ) ⊆ Φ n ( k m +1 ) for all m ∈ N , and thatΦ n ( k ) = [ m ∈ N Φ n ( k m ) . On the other hand, for every m ∈ N , let K ( m ) π,α ∈ K m denote the superclass of U n ( k m )which is parametrised by the set partition ( π, α ) ∈ Φ n ( k m ); hence, K ( m ) π,α = 1 + Γ m · e π,α . Since Γ m ⊆ Γ m +1 , it is clear that K ( m ) π,α ⊆ K ( m +1) π,α for all ( π, α ) ∈ Φ n ( k m )and all m ∈ N ; consequently, since u n ( k ) = S m ∈ N u n ( k m ), for every superclass K of U n ( k ), there is a unique ( π, α ) ∈ Φ n ( k ) such that 1 + e π,α ∈ K . Therefore, as in thecase of the finite unitriangular groups, we conclude that the superclasses of U n ( k )are in one-to-one correspondence with the k -coloured set partitions; as before, wedenote by K π,α the superclass in K which is associated with ( π, α ) ∈ Φ n ( k ).For the characterisation of E = SCh + ( U n ( k )), we first note that (as in thefinite case) every character ϑ ∈ u n ( k ) ◦ can be uniquely represented by a strictlyuppertriangular n × n matrix ϑ = ( ϑ i,j ) where ϑ i,j ∈ k ◦ for all 1 ≤ i < j ≤ n ;indeed, ϑ decomposes uniquely as the sum ϑ = X ≤ i The indecomposable supercharacter ξ π,τ of U n ( k ) is defined forall ( π, τ ) ∈ Φ n ( k ◦ ) . Moreover, for every ( π, τ ) ∈ Φ n ( k ◦ ) and every ( π ′ , α ) ∈ Φ n ( k ) , the constant value ξ π ′ ,τ ( K π ′ ,α ) of ξ π,τ on the superclass K π ′ ,α is equal to unless D ( π ) ⊆ R ( π ′ ) and nest π ( π ′ ) = 0 , in which case it is given by ξ π,τ ( K π ′ ,α ) = ϑ π,τ ( e π ′ ,α ) .Proof. Firstly, we note that, if σ ∈ k ◦ is non-trivial, then there is m ′ ∈ N suchthat σ m = for all m ∈ N such that m ≥ m ′ , and consequently, for an arbitrary( π, τ ) ∈ Φ n ( k ◦ ), there is m ∈ N such that τ ( i, j ) m = for all m ∈ N such that m ≥ m and all ( i, j ) ∈ D ( π ). Therefore, for every m ∈ N such that m ≥ m , themapping ( i, j ) τ ( i, j ) m defines the k ◦ m -colouration τ m : D ( π ) → k ◦ m \ { } , andhence ( π, τ m ) ∈ Φ n ( k ◦ m ); moreover, we havelim m →∞ τ ( i, j ) m ( β ) = τ ( i, j )( β ) , β ∈ k . On the other hand, Proposition 4.2 asserts that, for every m ∈ N such that m ≥ m ,the value ξ π,τ m ( K π ′ ,α ) is non-zero only if D ( π ) ⊆ R ( π ′ ), in which case we have ξ π,τ m ( K π ′ ,α ) = 1 p m ! nest π ( π ′ ) ϑ π,τ m ( e π ′ ,α ) . Therefore, ξ π,τ ( g ) = lim m →∞ ξ π,τ m exists for all g ∈ G , and in the case where D ( π ) ⊆ R ( π ′ ), we conclude that ξ π,τ ( K π ′ ,α ) = ( , if nest π ( π ′ ) = 0, ϑ π,τ ( e π ′ ,α ) , if nest π ( π ′ ) = 0,and this concludes the proof. (cid:3) As a consequence of Theorem 4.1, we obtain the following result. Theorem 4.4. The mapping ( π, τ ) ξ π,τ defines a one-to-one correspondencebetween Φ n ( k ◦ ) and SCh + ( U n ( k )) . Furthermore, this correspondence defines ahomeomorphism when Φ n ( k ◦ ) = lim ←− m ∈ N Φ n ( k ◦ m ) is equipped with the inverse limittopology (where, for every m ∈ N , the space Φ n ( k ◦ m ) is equipped with the discretetopology). The supercharacters of U n ( k ) enjoy (very) particular features which generaliseimportant properties of the supercharacters of the finite unitriangular groups (and,more generally, of finite algebra groups). Namely, the closure of every Γ-orbit on u n ( k ) supports a single ergodic Γ-invariant measure; if O is the closure of someΓ-orbit, then every ergodic Γ-invariant measure supported on O must determinean indecomposable supercharacter, and thus O contains one element of the form ϑ π,τ for some ( π, τ ) ∈ Φ n ( k ◦ ). However, one can check that, if ( π, τ ) and ( π ′ , τ ′ )are two distinct elements of Φ n ( k ◦ ), then the closures of the Γ-orbits Γ · ϑ π,τ andΓ · ϑ π ′ ,τ ′ are also distinct, and consequently the closure of every Γ-orbit supports aunique ergodic Γ-invariant measure.In more detail, let 1 ≤ i < j ≤ n be arbitrary, and let π ∈ Sp ( n ) be the uniqueset partition such that D ( π ) = { ( i, j ) } . For every τ ∈ k ◦ \ { } , let ξ i,j ( τ ) denotethe supercharacter ξ π,τ where τ ∈ Col k ◦ ( π ) is given by τ ( i, j ) = τ ; following [2], werefer to ξ i,j ( τ ) as the ( i, j )-th elementary character of U n ( k ) associated with τ . Onthe other hand, let ϑ i,j ( τ ) = τ ⋆ e ∗ i,j ∈ u n ( k ) ◦ , and let O i,j ( τ ) denote the closure ofthe Γ-orbit Γ · ϑ i,j ( τ ). It is straightforward to check that Γ · ϑ i,j ( τ ) consists of allcharacters τ ⋆ a ∗ ∈ u n ( k ) ◦ where a = ( a r,s ) ∈ u n ( k ) is any element which satisfies a r,s = , if 1 ≤ r < i or j < s ≤ n ,1 , if r = i and s = j , a i,s a r,j , if i < r < s < j ,and thus the Γ-orbit Γ · ϑ i,j ( τ ) is homeomorphic to the product space k j − i − (hence, it is countable and discrete). On the other hand, for every τ ∈ k ◦ , theset S = { ατ : α ∈ k } is clearly a subgroup of k ◦ with S ⊥ = { } where S ⊥ is theorthogonal subgroup of S in k ; since S ⊥ = S ⊥ where S denotes the closure of S in k ◦ , we conclude that S = S ⊥⊥ = k ◦ (hence, S is dense in k ◦ ), and this implies thatthe closure O i,j ( τ ) of Γ · ϑ i,j ( τ ) is homeomorphic to the compact space ( k ◦ ) j − i − .In general, it can be proved that, for every ( π, τ ) ∈ Φ n ( k ◦ ), the Γ-orbit Γ · ϑ π,τ decomposes as the (algebraic) sumΓ · ϑ π,τ = X ( i,j ) ∈D ( π ) Γ · ϑ i,j ( τ ( i, j )) , and consequently its closure O π,τ decomposes as the sum O π,τ = X ( i,j ) ∈D ( π ) O i,j ( τ ( i, j )) . From this, it is not hard to conclude that O π,τ is homeomorphic to the compactspace ( k ◦ ) r ( π ) where r ( π ) = (cid:12)(cid:12) S ( i,j ) ∈D ( π ) { ( i, k ) , ( k, j ) : i < k < j } (cid:12)(cid:12) .Since there is a one-to-one correspondence between SCh + ( U n ( k )) and the setconsisting of the closures of all Γ-orbit on u n ( k ) ◦ , it follows that, if O is the closure ofsome Γ-orbit on u n ( k ) ◦ and µ O is the unique ergodic Γ-invariant measure supportedon O , then the corresponding indecomposable supercharacter ξ O ∈ SCh + ( U n ( k ))admits the integral formula ξ O ( g ) = Z O ϑ ( g ) dµ O , g ∈ G, which is, not only a generalisation of the supercharacter formula for finite algebragroups, but also an analogue for the Kirillov character formula for nilpotent realLie groups (see [26] and also [14] for the case of nilpotent discrete groups over therational numbers). 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Lochon) Centro de An´alise Funcional, Estruturas Lineares e Aplicac¸˜oes (Grupode Estruturas Lineares e Combinat´orias), Departamento de Matem´atica, Faculdade deCiˆencias da Universidade de Lisboa, Campo Grande, Edif´ıcio C6, Piso 2, 1749-016 Lisboa,Portugal Email address ::