This identification is called the “asymptotic character formula.” TheK0- invariant of the groupC-algebra is also determined

ROBERT BOYER

Received 31 August 2003 and in revised form 19 March 2005

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0- invariant of the groupC-algebra is also determined.

1. Preliminaries

The purpose of this paper is to develop the representation theory of infinite wreath prod- uct groups (defined inSection 2) by exploiting the relationship between their group alge- bras and conjugacy classes with those of the infinite symmetric group [16]. Furthermore, since these groups are not type I, that is, the unitary dual of these groups is not standard Borel space, their character theory and groupC-algebra play a special role in their rep- resentation theory. For instance, among the substitutes for the group dual are the space of all finite characters and the space of primitive ideals of groupC-algebra. Both these spaces are standard Borel spaces, so they can be effectively parameterized. Drawing at- tention away from irreducible representations to factor representations with characters allows the development of harmonic analysis for groups that are not type I.

The infinite wreath product groups are inductive limits of finite groups, so that their C-group algebra is, in fact, an AF-algebra; that is, an inductive limit of finite-dimen- sional C-algebras. Such algebras have been well studied by means of their K0-group invariant. For the wreath product groups, theK0-group has an order structure which is determined through the evaluation of both the finite and semifinite characters of theC- algebra as well as a natural multiplication which makes theK0-invariant into a special ordered ring, namely, a Riesz ring (seeSection 4). Another special feature for infinite wreath products is that their finite characters can be described as limits of normalized irreducible characters of the prelimit groups which is sometimes called the “asymptotic character formula.” The major principle of this paper is that all these important features of the representation theory of wreath products can be reduced to the known results for

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:9 (2005) 1365–1379 DOI:10.1155/IJMMS.2005.1365

the infinite symmetric group. An analogous reduction was found for the infinite unitary, symplectic, and orthogonal groups. See [5] for recent applications of character theory of inductive limits of classical groups to harmonic analysis and probability theory. See [10]

for an application of the asymptotic character formula to the study of multiplicities of specific representations.

We fix some notation concerning Young diagrams. AYoung diagramYis a finite ideal of (Z⁺)²relative to the usual product order structure. Let|Y|denote the number of points in the diagram; whiler_i(Y) denotes the length of itsith row andr#(Y) denotes the number of nonzero rows;ci(Y) denots itsith column. Finally, we denote the Frobenius coordinates ofYby

h1(Y),. . .,h_d#(Y)(Y)|v1(Y),. . .,v_d#(Y)(Y), (1.1) wherehj(Y), respectively,vj(Y), is the (horizontal, resp., vertical) coordinaterj(Y)−j, respectively,c_j(Y)−j, (length measured from the main diagonal).d#(Y) denotes the length of the main diagonal of the diagramY.

Letᐅndenote the set of all Young diagrams with exactlynnodes, and letᐅ=_∞

n=0ᐅn. Aninfinite Young diagramY is an infinite ideal of (Z⁺)². In addition to describing an infinite diagramY with its row and column lengths,Y can also be specified relative to a rectangular piece together with a finite diagram. Let

Ᏽk,=

(i,j) :i≤korj≤, (1.2)

withk+≥1. Then any infinite Young diagramY with finitely many rows and columns has the form

Y=Ᏽk,∪

Y0+ (k,), (1.3)

whereY0is a finite (possibly empty) Young diagram andY0+ (k,) means the usual trans- late of the Young diagram by the vector (k,).

As a convention, for a finite groupGwith dual spaceGof equivalence classes of irre- ducible representations, let p(π) denote the corresponding central projection inC(G) whilee(π) denotes a minimal subprojection ofp(π). WhenGis abelian, we may enumer- ate its dual as{ω1,ω2,. . .,ω_|G|}when convenient.

2. Algebraic structure ofK0(S(G))

For a finite groupG, letSn(G) denote its wreath product which is the canonical semidirect product ofGⁿ=G×G× ··· ×G(nfactors) whereS(n) acts onGⁿby permuting compo- nents. Write elements ofSn(G) as (σ;g1,. . .,gn) whereσ∈S(n) andg1,. . .,gn∈G. Given a setS, letFn(S,ᐅ) denote the set of all functionsf fromSintoᐅsuch that_s∈S|f(s)| =n.

IfR(S_n(G)) denotes the standard additive group of representations ofS_n(G), then᏾= _∞

n=0R(Sn(G)) is a commutative graded ring with multiplication given by an induction product

π1◦π2=Ind^S_S^m+n_m(G)^(G)_×Sn(G)π1×π2

, (2.1)

whereπ1∈R(S_m(G)) andπ2∈R(S_n(G)). Let᏾0=_∞

n=0R(S(n)) denote the correspond- ing graded commutative ring for the symmetric groups. In [16], canonical isomorphisms Tω, for eachω∈G, are described from ᏾0onto a subring᏾(ω). Recall that a represen- tation π∈R(S_n(G)) lies in᏾(ω) if and only if the restrictionπ toGⁿ is a multiple of ω× ··· ×ω.

With the mapsTω, there is a natural parametrization of the irreducible representa- tions of the wreath productS_n(G) by f ∈F_n(G, ᐅ). This is done as follows. First, identify f(ω), forω∈G, with an irreducible representation of the symmetric group S(|f(ω)|).

As in [16],Tω(f(ω)) is an irreducible representation ofS_|f(ω)|(G). Finally, the induction product {T_ω(f(ω)) :ω∈G}will produce any irreducible of the wreath productS_n(G) wheren= |f|. To simplify notation, we will occasionally identify f(ω) with its image Tω(f(ω)) and treat f as the irreducible representation.

In this paper, we will only need the conjugacy classes ofS_n(G) whenG is abelian.

They are parametrized by the seth∈Fn(G,ᐅ). In outline, this is done as follows. Given a nonnegative integermandg∈G, letCm,gdenote the normalized characteristic function of the primitive conjugacy class ofS_m(G) which consists of all elements (σ_m;g1,g2,. . .,g_m) inSm(G) (σmis anm-cycle,g1,. . .,gm∈G) such that the cycle product ofg1,. . .,gmisg(see [16] for details). Next, for a Young diagramY∈ᐅandg∈G, letCY,g= ^r_j^#₌^(Y)1 Crj(Y),g. Finally, for a functionh∈Fn(G,ᐅ), letCh= {Ch(g),g:g∈G}. Every conjugacy class has this form.

The algebraic structure for᏾ extends to the infinite sum of complex-valued class functionsᏯ=_∞

n=0CF(Sn(G)), so it forms a graded commutative algebra. Further, each spaceCF(S_n(G)) admits an inner product from theL²-inner product onS_n(G) relative to normalized counting measure. Let Char(f) denote the character of the irreducible f ∈Fn(G,ᐅ) and let h∈Fn(G,ᐅ), thenChar(f),his the value of the character of the irreducible f at the conjugacy classh. Furthermore, the subrings᏾(ω) are mutually or- thogonal. LetᏯ0denote the corresponding algebra for the symmetric groups.

We record two basic formulas.

Branching law. Let f ∈Sn(G) and f∈Sn+1(G). Then f occurs in the restriction f to Sn(G) if and only if there existsω0∈Gsuch that f(ω)=f(ω) for allω=ω0and f(ω0) occurs in the restriction of f(ω0) toS(|f(ω0)|). Further, the representation f occurs in

fwith multiplicity dim(ω0).

Dimension theorem. Let f ∈Sn(G), then

dim(f)= n

n1,. . .,n_{p p}

j=1

dimfω_jdimω_j^|f(ωj)|, (2.2)

where G= {ω1,. . .,ωp},nj= |f(ωj)|, and dim(f(ωj)) is the dimension of the corre- sponding irreducible representation ofS(|f(ωj)|).

The following theorem follows from [4, Section 3].

Theorem2.1. As a ring,K0(S(G)) ᏾/(1−

{dim(ω)ω:ω∈G}).

We now reduce the character theory of infinite wreath products to the abelian case by using the ergodic method of finding characters for locally finite groups. We briefly review this method from the theory of AFC-algebras. See [6, Chapter 1] for a general exposition. Because the groupC-algebra ofS(G) is an AF-algebra, it possesses an asso- ciated dynamical system (X,Γ), whereX is a certain path space andΓis a locally finite group of path permutations [11]. In particular,X consists of infinite paths through the

“multigraph” whose nodes are given byf ∈S_j(G), j≥0, such that the two vertices f and fare connected bykedges if f ∈S_j(G), f∈S_j+1(G), and f ≤ fwherekis the multi- plicity of f in f. The finite characters are determined by theΓ-invariant andΓ-ergodic probability measuresµonX. The measureµis determined by its values on the cylinder setsX_f, where f ∈S_n(G), for somen, andX_f consists of all paths that pass through the nodef. The ergodic method states that the measureµis uniquely determined by the limit

Nlim→∞

#Path(0,f,G)#Pathf,f_N,G

#Path0,f_N,G ⁼µX_f, (2.3) where 0 is the trivial representation ofS0(G)= {e}, fN∈SN(G), and Path(f,f,G) is the set of all finite paths whose initial node is f and whose final node is f.

We recognize that #Path(0,f,G) is the dimension of the irreducible representation f∈S_|f|(G) which we write as dimG(f) or dim(f), if the choice ofGis clear, while

#Path(f,f,G) equals the multiplicity of the representation f in the restriction f to S_|f|(G) and will be denoted by dimG(f\ f) or dim(f\ f), again if the choice of G is clear. For the symmetric group, this multiplicity dim(Y\Y), where Y,Y∈ᐅand Y⊂Y, is also known as the dimension of the skew-diagram formed by the set-difference of the Young diagramsYandY.

Lettbe the associated character of theC(S(G)). Then the value ofton the minimal projectione(f) is given by

te(f)= lim

N→∞

dimG fN\f dimG

fN . (2.4)

By the branching law and dimension theorem, we have the relationship dimG(f_\f)=dim_Zp(f_\f)

ω∈G

dim(ω)^{|f(ω)|−|f(ω)|}. (2.5)

Hence, the ratios of the orders of the path spaces are related as dimG

fN\f dimG

fN

=

ω∈G

1 (dimω)^|f(ω)|

dim_ZpfN\f dim_ZpfN

, (2.6)

wherep= |G|. Hence, the limit in (2.6) exists for the groupGif and only if it exists forZp. We sum up this discussion in the following.

Theorem2.2. LetGbe any finite group. Then the character theory of the infinite wreath product depends solely on the cardinality ofG. If |G| = |Zp|, identifyGwithZp, so both Sn(G)andSn(Zp)can be identified withFn(X,ᐅ)where|X| =p. If f ∈Fn(X,ᐅ), letcf =

{(dimω)^|f(ω)|:ω∈G}. Then for any finite charactert_GonS(G), cftG

eG(f)=t_Zpe_Zp(f). (2.7) It would be interesting to have a version of this result for nonfinite compact groups Gbut it is not clear what the replacement ofZp is. For the remainder of the paper, we assumeGis a finite abelian group.

3. Finite characters and primitive ideals

We give a brief review of the known classification of the finite characters ofS(∞) to- gether with their asymptotic character formula and the ergodic method. Letx denote two nonincreasing sequences{aj}and{bj}of nonnegative reals such thatc=1−

(aj+ bj)≥0. We set

F(x,t)=e^ct ∞ j=1

1 +ajt 1−bjt⁼

∞ n=0

pn(x)tⁿ, (3.1)

wherepn(x)=_∞

=1(a)ⁿ−(−b)ⁿare the generalized power sums. ForY∈ᐅ, letsY(x) denote the corresponding Schur function given by the power seriesF(x,t), sos_Y(x)= det[pri(Y)−i+j(x)]. Then the corresponding finite charactertx is multiplicative relative to disjoint cycles and is given bytx(n-cycle)=pn(x) andtx(e(Y))=sY(x), wheree(Y) is any minimal projection determined by the diagramY∈S(n). Every finite character has this form [12]. For further discussion of these results, see also [6,9,13,15]. We callx the Thoma parametersof the character.

Recall the asymptotics for theS(∞)-characters. LetY_N∈ᐅN correspond to an irre- ducible ofS(N) with normalized shifted Frobenius coordinatesaN,=(h(YN) + 1/2)/N andbN,=(v(YN) + 1/2)/N, with=1, 2,. . .,d#(YN). LetxN denote the sequence pair {a_N,,b_N,}. Following [16], letz_m denote the normalized characteristic function ofm- cycles inS(m). Identify as usualσ∈S(m) with its image inS(m+k) where its image fixes the complementary indices to{1,. . .,m}among{1,. . .,m+k}. Then the character value ofY_Nat anm-cycleσ_mis given byChar(Y_N),z_mz₁^N−mand

CharYN

,zmz^N₁^−m

dimYN =

d#(YN) =1

aN,m

−

−bN,m +O

1 N

, (3.2)

wherezmdenotes the normalized characteristic function of the conjugacy class of allm- cycles inS(m) andz_mz₁^N−mis the induction product inᏯ0. Then the limit of the normal- ized sequence of characters exists if and only ifaN,/N→a andbN,/N →b such that (a+b)<∞. This can be restated relative to the generalized power sums given by the generating functionF_N(x_N,t)= ^N₌1(1 +a_N,t)/(1−b_N,t).

This framework can be rephrased in terms of the ergodic method. By [6], we have the asymptotic estimate

dimYN\Y₀^∗

dimYN =s_Y0^∗xN +O

1 N

, (3.3)

wheres_Y0^∗(x_N) is the Schur function relative toF_N(x_N,t). Hence, for any finite character twith Thoma parametersx, there exists a sequence{YN},YN∈S(N), such that t(eY0^∗)= s_Y0^∗(x) if and only ifr(Y_N)/N→a,c(Y_N)→b, and(a+b)<∞.

By means of the asymptotic character formula and the ergodic method, we will be able to classify all the finite characters ofS(G).

Theorem3.1. The sequence{Char(fN)/dim(fN)}^∞_N=1of normalized irreducible characters fN∈SN(G)has a limit if and only if

Nlim→∞

fN(ω)

fN =qω,

ω∈G

qω=1, (3.4)

and wheneverqω>0, the following limits exist for allj≥1:

Nlim→∞

r_jf_N(ω)

f_N(ω) ⁼a_j(ω), lim

N→∞

c_jf_N(ω)

f_N(ω) ⁼b_j(ω), ∞ j=1

a_j(ω) +b_j(ω)<∞. (3.5) Proof. Because of the multiplicative nature of finite characters [7,13], it is enough to establish the limits of the normalized characters on the primitive conjugacy classes which play the role ofn-cycles in the infinite symmetric group.

Let f ∈F_N(G,ᐅ) =S_N(G). We wish to find the character value Char(f) at the conju- gacy classCh which is nothing more than the inner productChar(fN),Ch. We will be using the formalism of [16]. By [16, Section 7], we have for a primitive conjugacy class C_m,gthat

Cm,g= ω(g)Tω

zm

:ω∈G, (3.6)

whereg∈Gandmis a nonnegative integer.

With the usual embedding ofS_m(G) intoS_N(G), we analyze the asymptotic behavior of the inner product

CharfN

,Cm,g

C1,e

_N−m

dimfN . (3.7)

LetG= {ω1,. . .,ω_|G|}, so _|G|

j=1

ω(g)Tωj

zm

N−m

=

N−m n1,. . .,n_|G|

T_ωⁿ¹₁z1

···Tω^n||^GG^||

z1

, (3.8)

where the sum is over all nonnegative integersn1,. . .,n_|G|that sum toN−m.

By orthogonality, the only nonzero contributions ofC_m,g(C_1,e)^N−min the inner prod- uct

CharfN ,Cm,g

C1,eN−m

=

CharfN ω1

···CharfN ω_|G|

,Cm,g

C1,eN−m (3.9) occur when for some j0, we havem+nj0= |f(ωj0)|and for j= jo,nj= |f(ωj)|. Again we emphasize that the products here are induction products inᏯ.

For convenience, setn_i= |f(ω_i)|, and letA(N−m,j) denote the multinomial coef- ficient _m1^N_,...,m^−m_|G|wheremi=ni, fori=j, and mj=nj−m. Further, setMj to be the monomial product inᏯ:

Mj=Tω1

z1

n1

···Tωj−1

z1

nj−1

Tωj

z1

nj−m

Tωj+1

z1

nj+1

···Tω|G|

z1

n|G|

. (3.10) We also need the elementary asymptotic estimate for multinomial coefficients where p,n0,k1,. . .,kpwill be treated as fixed:

N−n0

K1−k1,. . .,K_p−k_p

N K1,. . .,Kp

= K1

N k1

···

K_p N

kp

+O 1

N

. (3.11)

Hence, the ratio_K1−k^N_1,...,K⁻ⁿ⁰

p−kp

/_K1,...,K^N

p

has a limit if and only if for eachj=1,. . .,p, the sequence{Kj/N}converges with limitqj; hence the limit of the quotient is q^k_j^j.

We now have the simplifications Charf_N,C_m,gC_1,e^N−m=

|G|

j=1

ω_j(g)A(N−m,j)Charf_N,T_ωjz_mM_j

=

|G|

j=1

ωj(g)A(N−m,j)CharfN ωj

,Tωj

zm Tωj

z1

|fN(ωj)|−m

×

i=j

CharfN

ωi

,z^|1^fN(ωi)|

=

|G|

j=1

ω_j(g)A(N−m,j)Charf_Nω_j,z_mz^|₁^fN(ωj)|−m

i=j

dimf_Nω_i,

(3.12)

where we used the fact that Char(fN(ωi)),zm^|f(ω)| is the value of the character of the irreducible representation f_N(ω_i) ofS(|f_N(ω_i)|) at the identity which reduces to the di- mension of the representation.

We conclude that the normalized character value is CharfN

,Cm,g

C1,eN−m dimfN

=

|G|

j=1

ωj(g)A(N−m,j) dim(fN)

CharfN ωj

,zmz1^{|fN(ωj)|−m}

i=j

dimfN ωi

.

(3.13)

Since the character value onS(N),Char(f_N(ω_j)),z_mz₁^{|fN(ωj)|−m}can be handled with the asymptotic character formula forS(∞), we will give now a full asymptotic expansion for the normalized character value:

CharfN ,Cm,g

C1,eN−m

dimfN =

|G|

i=1

ωi(g) fN

ωi N

m

+O 1

N

×

_d#(fN(ωi))

=1

a fωim

−b fωim

+O 1

fN ωi

, (3.14) wherea(ω)₌(h(f_N(ω)) + 1/2)/_|f_N(ω)_|andb(ω)₌(v(f_N(ω)) + 1/2)/_|f_N(ω)_|are the normalized shifted Frobenius coordinates.

If|fN(ωi)|/N →0, then this term has no contribution to the sum. So, by passing to a subsequence, if necessary, assume|fN(ωi)|/N→qω>0. By [13, Lemma 2], (3.13) pos- sesses a limit for suchωif and only if botha(f(ω)) andb(f(ω)) possess limits, saya(ω) and b(ω), respectively, for all, together with the normalization a(ω) +b(ω)= 1−c(ω)≥0. In particular, this term has contributionqωpm(z(ω)) to the sum. Since the charactersωare a basis for functions overG, we find that distinct limits give distinct finite

characters.

Theorem3.2. The finite characterstofS(G)are parametrized by(qω,x(ω) :ω∈G) where qω≥0withqω=1andx(ω)=(a(ω),b(ω))are Thoma parameters for eachω. On the primitive conjugacy class withg∈Gandma nonnegative integer, the value oftis

tC_m,g= ω(g)

∞ =1

q_ωa(ω)^m−

q_ωb(ω)^m:ω∈G

, (3.15)

and, on the minimal projectione(f)with f ∈Fn(G,ᐅ), is q^|ω^f(ω)|sf(ω)

xω:ω∈G. (3.16)

In particular,qω=t(e(ω))wheree(ω)is the minimal projection corresponding toω∈S1(G) inC(S1(G)).

Proof. The evaluation of the finite characterton a primitive conjugacy class follows from the proof of the previous theorem.

To obtain the values of the finite characterton minimal projections inC(G), we use the ergodic method. For f₀^∗∈Sn(G), we find

dimf_N\f₀^∗=

f_N−f₀^∗ f₀^∗ω1,. . .,f₀^∗ω_|G|

_|G|

j=1

dimf_Nω_j\f₀^∗ω_j. (3.17) Hence, the value of finite charactertat the minimal projectione(f₀^∗) is given by the limit:

tef₀^∗= lim

N→∞

dimfN\f₀^∗ dimfN

= lim

N→∞

f_N−f₀^∗ f₀^∗ω1,. . .,f₀^∗ω_|G|

|G|

j=1dimf_Nω_j\f₀^∗ω_j f_N

fN

ω1,. . .,fN ω_|G|

|G|

j=1dimfN

ωj .

(3.18)

Their limits are

ω∈G

qω^|f(ω)|ef0^∗(x). (3.19)

In particular,

Nlim→∞

fN ωi

f_N ⁼tef_1,i=teω_i, (3.20) where f₀^∗= f1,i∈S1(G)=Gsuch that|f1,i(ωi)| =1 and|f1,i(ωj)| =0 for j=i. Hence,

qωhas the desired interpretation.

We will call the parameters (qω,x(ω) :ω∈G) from Theorem 3.2for a finite character tofS(G) itsinvariant parameters.

The method in [13] or [3] can be used to classify the primitive ideals ofC(S(G)) by means of the branching law. For eachω∈G, we let Y(ω) be an infinite Young diagram described inSection 1, soᏵk(ω),(ω)+Y0(ω), wherek(ω) is the number of infinite rows of Y(ω) and(ω) is the number of infinite columns ofY(ω). Note: we allow eitherk(ω) or (ω) to be∞. In either case, we setY0(ω)= ∅.

A primitive idealJis determined by theG-tuple:

k(ω),(ω),Y0(ω) :ω∈G= k,,Y0

. (3.21)

Hence, the minimal projectione(f)∈J(k,,Y0), where f ∈Sn(G), if and only if f(ω) is not a subset ofᏵk(ω),(ω)+Y0(ω), for someω∈G.

The multiplicative nature of finite characters together with the determination of the kernels of finite characters of S(∞) in [7] give the kernels of the finite characters of

infinite wreath products. Supposethas invariant parametersx= {a_j(ω),b_j(ω) :ω∈G} and{q_ω:ω∈G}. If q_ω=0, then k(ω)=(ω)=0, soᏵk(ω),(ω)=(Z²)⁺. Ifq_ω>0 and c(ω)>0, thenk(ω)=(ω)= ∞. Ifqω>0 andc(ω)=0, letk(ω), respectively,(ω) be the largest index such thataj(ω)>0, respectively,bj(ω)>0. Then the kernel (or the prim- itive ideal) determined bytis{(k(ω),(ω),∅) :ω∈G}. An immediate consequence of this calculation is the following.

Theorem3.3. A primitive idealJ=J(k,,Y0)is the kernel of a finite character if and only ifY0= ∅, for allω∈G.

We close the section with two observations.

(1) The unitary (1-dimensional) characters ofS(G) are given byΩ0,+(σ;g1,. . .,g_n)= ω0(g1···gn) orΩ0,−(σ;g1,. . .,gn)=sgn(σ)ω0(g1···gn) whereω0∈G. The invariant pa- rameters forΩ0,+, respectively,Ω0,₋, areqω=0 forω=ω0;qω0=1 anda1(ω0)=1, re- spectively,b1(ω0)=1.

(2) The parameters for the finite characters ofS(G) that come from the quotient group S(G)/G_∞ S(∞) whereG_∞is the normal subgroup^∞_n=1Gⁿare determined byqω1=1 whereω1is the constant character 1 onG.

4. Infinite characters

The aim of this section is to classify the infinite characters ofS(G) using Riesz ring tech- niques. We recall some concepts from Riesz group and ring theory. First, an ordered groupGis a Riesz group provided given any elementsai,bj∈G,i,j=1, 2, there exists an elementc∈Gwitha_i≤c≤b_j for any choicei,j=1, 2. For any unital AF-algebra, its K0-group is a Riesz group. A Riesz group is a Riesz ring provided it has a multiplication that is consistent with the order structure.

IfJ is the kernel of a charactert, whether finite or infinite, then tcorresponds to a faithful character of the primitive quotientA=C(S(G))/J. Further, with the induction product described inSection 1,K0(A) is a commutative ring. In particular,K0(A) for a primitive quotientAis a Riesz ring.

LetH denote a Riesz group with nonzero positive elementsaandb. We say thatais infinitely smallrelative tobifna≤bfor alln∈Z⁺. Ifais infinitely small, thenamust lie in the kernel of every state ofH. In particular, ifH is a unital Riesz ring with a nonzero infinitely small element, thenHcan have no faithful finite extremal states.

We need the following theorem found in Antony Wassermann’s unpublished Univer- sity of Pennsylvania Ph.D. thesis. Although the thesis was widely circulated, it was never published. A version of this result with a proof by Wassermann appears in an appendix to [3].

Theorem 4.1. IfR is a Riesz ring with no positive zero divisors, thenR has no faithful infinite characters.

It is routine to check that the primitive quotients by the idealsJ=J(k,,Y0) where Y0(ω)= ∅, for allω∈G, have no positive zero divisors. In another study [4], we found for several inductive limit groups that a primitive idealJ is integral (i.e., theK0-ring of

the primitive quotient is a commutative ring with no positive zero divisors) if and only if Jis the kernel of a finite factor representation.

Throughout this section, we letJ=J(k,,Y0) where for someω0∈G, Y0(ω0)= ∅ where we use the notation introduced in (1.3). Recall thatkdenotes the number of infi- nite rows andthe number of infinite columns, so the shape of such an infinite Young diagram is an infinite “L.” The finite diagramY0denotes the remaining contribution to the shape. We will write f ⊂Y+I(k,) if f(ω)⊂Y(ω) +I(k(ω),(ω)) for allω. We use the same convention for other standard set operations.

We call a primitive idealintegralif the condition thata,b∈K0(S(G)/I) are nonzero and positive implies that their productabis nonzero. We letJ_adenote the smallest integral primitive ideal that containsJ, so thatJa=J(k,) (where we omitY0here since they are all equal to∅).

Proposition4.2. The primitive quotientA=C(S(G))/Jcontains a nonzero idealBwhich is stably isomorphic to an idealB_aofA_a=C(S(G))/J_a.

Proof. Consider the projectionse(f), wheref ∈S_n(G), inAsatisfy the condition

f\I(k,)=Y0+ (k,). (4.1)

We associate tof(ω), the Young diagramf_a(ω)=f(ω)\(Y0(ω) + (k(ω),(ω))). Next con- sidere(fa) as a projection inAa. By the branching law, the idealBinAgenerated by all projectionse(f) satisfying (4.1) is stably isomorphic to the idealB_a inA_agenerated by the projectionse(f⁰), where f⁰⊃ {(i,j) :i≤k(ω)andj≤(ω)}, which are rectangular

diagrams.

We need to strengthenProposition 4.2in order to show thatBis the norm closure of the ideal of definition of any faithful character ofA. We study this question in terms of multiplication ofK0(A).

Proposition4.3. LetJ=J(k,,Y0)with|Y0| ≥1with primitive quotientA=C(S(G))/J.

If f1\I(k,)=Y0+ (k,), then the following identities hold inK0(A):

(1)if f2⊂I(k,), thene(f1)◦e(f2)=0;

(2)if f2∈Ᏽ(k,), thene(f1)◦e(f2)=0.

Proof. For the first identity, it suffices to show thate(f)◦e(f)=0, where f is chosen, so f(ω) is the smallest diagram that containsY0(ω) + (k(ω),(ω)). Without loss of general- ity, we may assumeG= {e}; that is, we check only theS(∞) case.

We set k=r1(f)> k and=c1(f)>1. Let p be a term in the decomposition of f ◦f, which is described by the Littlewood-Richardson rule [8], such thatp⊂J(k,,Y0).

According to the Littlewood-Richardson rule, the termspare determined by the proper insertion of integers into the Young diagram. Leta_ibe the number ofi’s placed in theith row ofp. By hypothesis,a1≥1; moreover, eachai≥1, 1≤i≤, sincerj(p)≤k, j > . Thus, (+ 1) entries must be placed in the firstkdistinct columns ofp. This is impossible, soe(f)◦e(f)=0 inK0(A). The second identity follows by the same method by applying

the Littlewood-Richardson rule.

We record the lemma from [4, Section 5].