vol19_pp02-89. 3436KB Jun 04 2011 12:06:06 AM

MATRICES OF POSITIVE POLYNOMIALS* DAVID HANDELMAN**

Abstract. Associated to a square matrix all of whose entries are real Laurent polynomials in several variables with no negative coefficients is an ordered “dimension” module introduced by Tuncel, with additional structure, which acts as an invariant for topological Markov chains, and is also an invariant for actions of tori on AF C*-algebras. In describing this invariant, we are led naturally to eventually positivity questions, which in turn lead to descriptions of the Poisson boundaries (of random walks affiliated with these processes). There is an interplay between the algebraic, dynamical, and probabilistic aspects, for example, if the (suitably defined) endomorphism ring of the dimension module is noetherian, then the boundary is more easily described, the asymptotic behaviour of powers of the matrix is tractible, and the order-theoretic aspects of the dimension module are less difficult to deal with than in general. We also show that under relatively modest conditions, the largest eigenvalue function is a complete invariant for finite equivalence (early results of Marcus and Tuncel showed that it is not a complete invariant in general, but is so if the large eigenvalue is a polynomial).

Key words. positive polynomial, trace, point evaluation, Newton polyhedron, convex polytope, primitive matrix, random walk, order ideal, (topological) shift equivalence, finite equivalence, Choquet simplex, real analytic function.

AMS subject classifications. 37B10, 46B40, 46L80, 32A38, 60J05, 16W80, 19K14, 15A54. Introduction.

LetM be a matrix whose entries are polynomials indreal variables with no negative coefficients. Tuncel [Tu] initiated a study of “dimension modules” (certain ordered vector spaces) associated to such matrices, in connection with classification problems for (topological) Markov chains. (See [Tu, Tu2] for details.) In this article, we develop structure and classification theory in somewhat different directions.

LetA be eitherZ[x±_i 1]orR[x±_i1](Laurent polynomial rings indvariables), and letA+ denote the corresponding set of Laurent polynomials with no negative coefficients. ThenAis a partially ordered ring with positive coneA+_{, and the space of columns of sizen, denotedA}n_{, is an orderedA-module. IfMis in} MnA+(i.e.,M is ann×nmatrix with entries fromA+), thenM induces an order preservingA-module map, M : An → An. Iterating this map, we obtain the dimension module associated toM as the direct limit,

GM := limAn M−→An M−→An M−→. . . .

This is an orderedA-module, consisting of equivalence classes,[c, k]where(c, k)∈An×N, and[c, k] = [c′_{, k}′_{]if there existsN ≥ k, k}′ _{such thatM}N−k_{c = M}N−k′

c′_{;[c, k]is in the positive coneG}+

M if there existsN such thatMN_{c ∈ (A}n₎+_{. The shift map sending[c, k]7→ [c, k+ 1]is an order automorphism,}

with inverseMˆ :GM →GM defined viaMˆ([c, k]) = [M c, k]. The structural problems referred to earlier arise from attempts to say something about the positive cone ofGM. Explicitly, one of the crucial problems is

(S) Decide in terms ofM (in MnA+) andc (in An) whether there exists an integerN so that the columnMNchas all of its entries inA+.

(Of course,[c, k]belongs toG+_M if and only if such anN exists.) Whenn= 1,M is just a polynomialP inA+_{, andf is a Laurent polynomial with integer (or real) coefficients; in this case, (S) asks to decide}

* Received by the editors March 28, 2008. Accepted for publication August 31, 2009. Handling Editor: Robert Guralnick.

** Mathematics Department, University of Ottawa, Ottawa ON K1N 6N5, Canada (dehsg@uottawa.ca). Supported in part by a Discovery grant from NSERC.

if the Laurent polynomialPm_{f has no negative coefficients for some integerm. This was solved in [H3,} Theorems A and B]. Special cases had been solved by Meissner [Me]. In this situation, the solution leads to connections with a result of Ney and Spitzer [NS], determining the Martin boundary of the random walk onZdassociated withP. There were also connections to the weighted moment map of algebraic geometry [Od, p. 94], and to an affine AGL(d,Z)-invariant for compact lattice polytopes [H5], as well as a real affine (AGL(d,R)) invariant for general compact convex polytopes.

The problem (

S

) arises in other contexts than topological Markov chains. IfA=Z[x±_i1], the limiting dimension module is a direct limit of partially ordered abelian groups. LetR be an AF C*-algebra [El] whose ordered Grothendieck group is the limit obtained by evaluating all the variables appearing everywhere in (1) at 1—in other words, eachxi7→1:

(2) G_M(1) = limZn M_−−−→(1) Zn M_−−−→(1) Zn M_−−−→(1) . . . .

(Here M(1) an abbreviation of the real nonnegative matrixM(1,1, . . . ,1).) In other words, K0(R) =

GM(1), as ordered abelian groups. Then there is an actionαof the d-torus T = Td on R so that the equivariant Grothendieck group ofR, KT₀(R)isGM when viewed as a partially ordered module over the representation ring ofT, which of course isA. Alternatively,GM = K0(R×αT). This is a special case of more general results in [HR], which deals with classification results for locally finite actions of tori and other compact Lie groups on AF C*-algebras.

There is a probabilistic interpretation that is closely related. Begin with the matrixM = (Mij)with entriesM_ij inA+_{. Form the set of sites,S =Z}d_{× {1,2, . . . , n}. It is helpful to think ofS asndisjoint} copies of the latticeZdpiled on top of each other, with the second coordinate indexing the level. A particle at point(z, k)∈S(lattice pointz, levelk) is constrained to jump to leveljin the next unit of time; its motion in this level is governed by the entryMjk. Write Mjk = Pλwxw where we use multinomial notation xw _{= x}w(1)

1 ·. . .·x

w(d)

d , λw are all nonnegative real numbers, and we have suppressed the dependence on jk for expository convenience. Then on levelj, the particle moves fromz according to the rule that the probability of displacement by wisλw/Mjk(1). Then (

S

) amounts to asking which initial finitely supported signed distributions onSbecome nonnegative after a finite number of iterations of this process.

In the case thatn = 1, this is related to the Poisson boundary, although knowledge of the latter is insufficient to solve (

S

). In this case, letµbe the measure onZdobtained fromP: µ({w}) =λw/P(1), whereP =Pλ_wxw_{. Letν be a finitely supported signed measure, and setν}

0=|ν|(any positive measure

with the same support asνwill do just as well). The specialization of (S) asks if there existsmso that them -fold convolution ofµ, convolved (once) withνbe positive. The answer is affirmative if and only if for every extremal[0,∞]-valued harmonic functionh(onZd) finite with respect toν0,P_w∈suppνν(w)h(w) > 0. Ifν0is mutually absolutely continuous with respect toµor some convolution powerµ(k)=µ∗ · · · ∗µfor

somek, then the set of these extremal harmonic functions is compact (and in this situation is contractible). However, for generalν0, the set of such functions need not be either compact or connected (ifd= 1, it is

compact, but need not be connected).

Forn >1, there is a choice of initial (row of) measures so that a boundary, analogous to the Poisson boundary, occurs (it is the positive portion of the real maximal ideal space of a partially ordered ring naturally associated toM and is at least compact, although not generally connected). Other choices lead to very complicated situations that must be disentangled in order to attack (S).

One can also put the random walk interpretation (actually these admit the same formalism of the “matrix-valued random walks” occurring in [CW]—although the problems considered are almost entirely unrelated) in the context of subshifts of finite type. By symbol splitting, we may assume that all of the entries ofM are either monomials or zero (although the matrix size may change). FormS =Zd_{× {}1,2, . . . , n_} as we did before, and set

X=SN XM =

©

x= (w(x, i), k(x, i))∈SN¯¯Mk(x,i+1),k(x,i)=xw, w=w(x, i+ 1)−w(x, i)

ª .

That is, ifx(i) = (w(x, i), k(x, i))wherew(x, i)_∈Zdandk(x, i)_{∈ {}1,2, . . . , n_}, then the sequence x=x(i)belongs toXMprovided that for eachiinN, the transition fromw(x, i)tow(x, i+1)is permitted— that is, w = w(x, i+ 1)−w(x, i) is the exponent of the monomial appearing in thek(x, i+ 1), k(x, i)

entry ofM. One can think of this as a shift space on an enormous state space (S), but with relatively few transitions permitted. In general,XM is not compact, but we may find a lot of compact cylinder sets, by selecting a finite set of elements “z” inSand timesiinN,U ={(zj, i(j))}(jrunning over a finite index set), and set

KU ={x∈XM|there existsjsuch thatx(i(j)) =zj}.

The pathxbelongs toKU if it passes through at least one of thezj at timei(j). This corresponds to the selection of an initial measure.

We normally assumeM is primitive, i.e., some power ofMhas no zero entries; however, at times, we have to face up to the reducible case, which is somewhat more complicated.

The approach used whenn = 1 is elaborated in order to apply to the general case. Associated to the ordered vector spaceGM is the “bounded subring” (defined in section 1), denotedEb(GM). This is a partially ordered ring of endomorphisms of GM that are bounded in the appropriate sense. Much of the information concerning positivity of elements ofGM is reflected, although not terribly clearly, in the structure ofEb(GM). For example, the (densely defined) traces onGM in principal permit one to decide positivity of a specific element, and these traces factor through (in the appropriate sense)E_b(G_M). Recall that a trace on an ordered vector space is a nonzero positive linear functional. In this case, “densely defined” means the domain of the trace is an order ideal inGM. The (global, that is, everywhere defined) traces ofGM yield some information about positivity (for example, necessary for[c, k]to be positive is thatτ([c, k])>0

for every global trace onGM), but these are far from sufficient to determine the ordering, even ifd= 1(in contrast, whenn= d= 1, the global tracesdodetermine the ordering). The pure traces onEb(GM)that are not faithful determine ideals ofEb(GM), and part of the structure theory is to describe these order ideals, which turn out to correspond (not bijectively) with faces of the convex polytope associated toM by Marcus and Tuncel [MT]. (Whenn= 1, the association is a bijection and many very nice properties hold [H5].)

In the case thatn= 1, the ordered ringEb(GM)(in the context ofn= 1, calledRP [H5]) is always a finitely generated commutative ring/algebra (depending on whether we useZorRin the definition ofA) with no zero divisors. It is easy to see thatEb(GM)may have zero divisors and need not be commutative, but generically the characteristic polynomial is irreducible overA, and that case, it is a commutative domain. Unfortunately, ifn_≥2, generically it is not finitely generated; it is not even noetherian; in fact, it does not even satisfy the ascending chain condition on order ideals. This means that the positivity problem is far more complicated than expected. We give necessary conditions in order that this ascending chain condition hold.

Among classification problems is the question of deciding finite equivalence of two such matrices; that is, given squareM andM′ with entries inA+, give criteria in order that there be a nonzero rectangular matrixX with entries in A+ _{such thatM X = XM}′_{. A necessary condition is given by equality of the} corresponding beta functions; for eachr = (ri)in(Rd)++(=

©

r_∈Rd¯¯ ri>0, i= 1,2, . . . , d ª

),M(r)

(all the entries ofMevaluated atr) is a real nonnegative matrix, and our initial primitivity assumption means the large eigenvalue, denotedβM(r)(or simplyβ(r)) has multiplicity one; it follows thatr 7→βM(r)is a real analytic function on(Rd)++_{. A necessary condition for finite equivalence is thatβ}

M =βM′ on(Rd)++. This was conjectured to be sufficient as well. However, Marcus and Tuncel [MT] gave two examples to show this was not the case; I showed (independently, but only in the form of a talk at the Adler conference) that the conjecture failed, for one of the pairs that Marcus and Tuncel considered, by completely different methods. However, it had been known already that ifd= 1, the conjecture is true (Parry), and Marcus and Tuncel [MT2] also showed that ifβ were a polynomial, the conjecture succeeds.

Here we give conditions onMandM′_{that guarantee thatβis a complete invariant for finite equivalence.} For any (primitive)M, we can find a left eigenvector forβwhose entries are polynomials inβwith coefficients

fromA, i.e., inR[x±_i1][β], and each entry is real analytic and strictly positive on(Rd)++_{. If we can arrange}

that each entry of the eigenvector is bounded above and below away from zero by a quotient of elements ofA+(this is a simplification of the real definition; the actual definition is more general), then we sayM satisfies (%) (a local criterion, using the facial structure ofM, can also be given, 8.7). For example, ifβis polynomial, or ifM is a companion matrix or the transpose thereof, or if the Newton polyhedra of all the entries of some power ofM are the same, then (%) holds. We sayM satisfies (#) if the ratio of the second largest absolute value of eigenvalues ofM(r)toβ_M(r)is bounded above by1₋ǫasrvaries over(Rd)++_.

This holds if for every faceFof the convex set associated toMF, there is a unique irreducible block which hits the spectral radius, and it is primitive (actually, in the statement of the result, the last “primitive” can be deleted), so it is relatively easy to test for.

Imprecisely, we show that among matrices satisfying (#) and (%), the beta function is a complete invariant for finite equivalence. It is conceivable that (#) implies (%), so the latter condition may be redundant. It is also possible that (%) alone is sufficientβto be a complete invariant for finite equivalence. In results leading up to this, we note that analytic and algebraic functions that are positive on(Rd)++can behave rather strangely, and this strangeness is reflected in boundary behaviour. Explicitly, ifρ : (Rd)++ _→

R++ is algebraic (i.e., satisfies a polynomial equation with coefficients fromA) and real analytic, and if X :R+→(Rd)++_{is the exponential of a ray inR}d_{with directional derivativev, then}

lim

t→∞

logρ(X(t)) logt

may depend on the starting point of the ray; ifρis bounded above and below by integer multiples of a rational function (whose numerator and denominator are inA+), then the limit does not depend on the starting point. This provides a notion of (local) “linkages” between the various blocks ofM_F (for a fixed faceF), which would be part of the data for shift equivalence. In the survey article [Tu2], Tuncel provides a global way of linking blocks.

Principal results. Theorem 4.5 describes completely the pure (global) traces onGM and its order ideals under the mildest of conditions, as well as the faithful pure traces on E_b(G_M). Roughly speaking, they factor through the left Perron eigenvector forβevaluated at points of(Rd)++_.

The unfaithful pure traces on order ideals ofEb(GM)and ofGM are described in several situations, completely if the necessary condition for noetherianness holds (sections 5 and 6). Roughly speaking, they factor through quotients corresponding to taking facial matrices. In fact, the unfaithful pure traces could be completely described in this form if an innocuous-looking conjecture were true.

The consequences, and subsequent non-genericity, of the ascending chain condition on order ideals of GM are discussed in section 7. Some of the relevant conditions can be simply expressed in terms of the weighted graph associated toM.

The finite equivalence result discussed above, Theorem 10.7, is proved in section 10.

In section 12, necessary and sufficient conditions on a size 2 matrixM are determined in order that it satisfy the condition (%) discussed previously. It turns out to be easy to express but difficult to prove: (%) holds if and only if the discriminant of the matrix divides a polynomial with no negative coefficients, Theorem 12.1. It follows that for size 2 matrices wherein the diagonal entries have no common monomials, (%) is equivalent toM +pI satisfying (#) for somepinA+(this property is called weak (#), and admits a much less awkward formulation).

Finally, in section 13, we have a weak realization theorem, which says that under suitable circumstances, given suitable functionsβ, there exists a primitive matrixM for whichβM =βN for some undetermined integerN; moreover,Mmay be selected to satisfy a strong condition, (**), which asserts that in some power ofM, the Newton polyhedra of all the entries are identical.

This paper is a considerable elaboration of results in the preprint, “Eventual positivity and finite equiv-alence for matrices of polynomials”, versions of which were distributed from 1987 on.

Table of contents 1 The bounded endomorphism ring

2 A brief look at the bounded endomorphism ring as an invariant 3 The many faces ofM

4 Faithful traces

5 Attack of the perfidious traces 6 Return of the perfidious traces

7 A F O Ggy example and generic nonnoetherianess 8 LinkingM&Ms

9 Local and global characterization of (#)

10 Finite equivalence for matrices satisfying (%) and weak (#) 11 Structural similarity

12 Two by two matrices

13 Eventual weak similarity to matrices satisfying (**) Appendix Nonnegative real matrices

References

1 The Bounded Endomorphism Ring

In this section, we define and develop the basic properties needed of the ring (algebra) of bounded endomor-phisms. Curiously, even in the case of zero variables, the notion is of some interest.

We recall some notation and definitions from the introduction. The ring of Laurent polynomials in the dvariablesx1,x2,. . . ,xdover either the integers or reals is denotedA, and the set of elements ofAhaving no negative coefficients is a positive coneA+ _{for A, making the latter into a partially ordered ring. The}

ring ofn×nmatrices with entries fromAwill be denoted MnA; of course, MnA+ will denote the set of matrices with entries fromA+_{. For an integern,A}n _{will denote the set of columns of sizen, viewed as} an orderedA-module (with the direct sum ordering). FixnandM in MnA+. The real matrix obtained by evaluating all the variables at1will be denotedM(1), an abbreviation forM(1,1, . . . ,1). Define the direct limit, as orderedA-modules,

(1) GM = limAn M−→An M−→An M−→. . . . This is the set of equivalence classes,

{(a, k)_|a_∈An, k_∈N_}/_∼,

where(a, k) ∼(a′_{, k}′_{)if there existsm > k, k}′_{such thatM}m−k_a=Mm−k′_a′_{. TheA-module structure} is obvious, andGM admits an ordering making it into a directed partially orderedA-module, by means of the positive cone

G+_M =©[a, k]∈GM ¯

¯there exists(a′, k′)∼(a, k)witha′∈(An)+ª.

We observe that[a, k] = [M a, k+ 1], andM induces an order isomorphism (that commutes with the action ofA) by means ofMˆ[a, k] = [M a, k]; its inverse is given by a “shift”,Mˆ−1_{[a, k] = [a, k+ 1]. More}

generally, ifN is in MnAand commutes withM, thenN induces anA-module endomorphism ofGM via

ˆ

N[a, k] = [N a, k]. This will be positive (i.e., order preserving) if and only if there exists an integermso thatN Mk _{belongs to M}

nA+. Thus bdescribes an assignment (actually a homomorphism ofA-modules and of rings), from the centralizer ofMin MnA,CA(M), to the ring ofA-module endomorphisms ofGM, EndAGM.

The matrixMis an element of the centre ofCA(M), and we may formally invertM; if the determinant ofM is nonzero, this amounts to adjoiningM−1_{from the matrices with entries in the field of fractions of}

A(the rational functions in thedvariables). In general, the construction is given as limit ring, CA(M)[M−1] = limCA(M)−−→×M CA(M)−−→×M CA(M)−−→×M . . . .

We have been a little sloppy in notation; if the determinant of M is zero, then the ideal of CA(M),

{N ∈CA(M)|MnN = 000}will be factored out. So, despite the notation,CA(M)will not be a subring of CA(M)[M−1]. Note that sinceM belongs to ann×nmatrix ring over a field (the rational functions in thedvariables), for any columnasuch thatMs_{ais zero, it follows thatM}n_{ais also zero. In any case, b} induces a map fromCA(M)[M−1]to EndAGM.

DefineE(GM) = n

e∈EndAGM ¯ ¯

¯ eMˆ = ˆM eo; then the range ofb :CA(M)[M−1]→ EndAGM is obviously insideE(GM). In fact the range is onto, and the map is an isomorphism.

Lemma 1.1

The map

b :CA(M)[M−1]→E(GM)

is an isomorphism of rings and

A

-modules.

Proof: It is routine to verify the map is a well-defined homomorphism of rings andA-modules. We first show the map is onto. SelecteinE(GM). LetEj (j = 1,2, . . . , n) denote the standard basis elements for An_{. Then we may find elementsa}

j inAn together with integersk(j)such thate[Ej,1] = [aj, k(j)]. We may assume thatk(j)are all equal, say tok, so thate[Ej,1] = [aj, k]for allj. DefineN0to be then×n

matrix with entries fromAwhosejth column isaj. Obviously,( ˆM)−(k−1)Nˆ0has the same effect aseon

the[Ej,1]’s. It is a routine verification thatPj[Ej,1]Ais essential (as anA-module) inGM, so that if two A-module endomorphisms agree on_{[Ej,1]}, they are equal. As the mapCA(M)→CA(M)[M−1]is not necessarily one to one, it is not immediate thatN0commutes withM. However, it easily follows that there

exists an integersso thatN0Mscommutes withM. SetN =N0Ms, and observe that( ˆM)−(k−1+s)Nˆ =e.

Thus the map is onto.

It is immediate from the definitions that the map is one to one: NˆMˆ−t_{= 0implies[N E}

j,1] = [0,1], so thatMsN Ej = 0for somes(and allj), whenceMsN = 0, which meansN is in the kernel of the map

CA(M)→CA(M)[M−1]. •

Remark. 1: The portion of the preceding that does not involve the ordering applies if A is a general commutative unital ring andM is an arbitrary square matrix (see [BoH, p. 125ff] where_E(_·)is defined for square matrices with entries in a commutative ring).

Now we discuss the positive cone ofE(GM). The natural definition of a positive cone on a ring of endomorphisms is the following:

E(G_M)+ =©e_∈E(G_M)¯¯e(G+_M)_⊆G+_Mª. Lemma 1.2

Under the map

b :CA(M)[M−1]→E(GM),

an element of

C_A(M)[M−1_]

_{is sent to a positive element}

_Mˆ−k_Nˆ

_of

_E(G

M)

if and only if there

exists a positive integer

m

such that

Mm_N

_{belongs to M}

nA+

.

Proof: Letedenote the endomorphism described byMˆ−k_Nˆ_{. Ife≥ 0,e[E}

j,1]≥ [0,1]for allj. Hence there existsksuch that for allj,e[Ej,1] = [aj, k]whereajbelongs to(An)+. It follows from the definitions that multiplication by some power ofMwill render all the columns ofN positive, which is what we want.

The converse is trivial. •

Now we observe that every element ofE(GM)is a difference of elements of its positive cone (i.e., as a partially ordered abelian group,E(GM)isdirected).

Finally, we can define the bounded subring ofE(G_M). Let I :G_{M →}G_Mdenote the identity element ofE(GM). Set

Eb(GM) ={e∈E(GM)|there existsk∈Nsuch that −kI ≤e≤kI} Eb(GM)+=Eb(GM)∩E(GM)+

We observe that with the relative ordering,Eb(GM)is a partially ordered ring; it need not be commu-tative, although this is generic; it is an order ideal inE(GM), and moreover I is an order unit forEb(GM). ObviouslyGM is an ordered module over Eb(GM). However, thatAdoes not act onEb(GM), because (informally) all of the former’s elements (except the constants) are unbounded in any reasonable sense.

For now, we make the standing hypothesis that the real matrixM(1)is primitive. This is equivalent to saying that there exists a power ofM so that every entry is not zero. One of the reasons for doing this is to guarantee that ifHis any nonzero order ideal ofGM then its bounded subring is the same asEb(GM), so that all order ideals ofG_M may be viewed as ordered modules over a common partially ordered ring.

Now letHbe an order ideal ofG_M. Recall from the introduction that to describe the ordering onG_M we must analyze all the pure states of the order ideals (that have order units).

Specifically, suppose that G is a partially ordered unperforated abelian group anda is an element thereof. If there exists an order ideal with order unit ofG, I, containingasuch that for all (pure) traces τ of I, τ(a) > 0, then ais in the positive cone of G. If no such order ideal exists, then ais not in the positive cone; in particular, there need not be a minimal order ideal containinga, or there could be a trace on a minimal order ideal with order unit among those containinga, at whichτ(a)<0.

To analyze the traces on an order idealI, we examine the bounded subringsEb(I); these automatically containIas an order ideal, and their pure trace spaces are compact. Moreover, there is a natural map from the pure trace space of the latter to that ofI. The pure trace spaces are described in sections 4–6.

We may formE(H) = _{e_∈E(G_M)_|eH _⊆H_}, and put the relative order on it. It is an ordered subring. Unfortunately, even ifHadmits an order unit,E(H)need not admit the identity as an order unit, so the latter is not generally equal to its bounded subring (which we shall define shortly) even whenn=d= 1. Example 1.3

An order ideal

H

in

GM

which has an order unit but for which

E(H)

does not

admit the identity as an order unit.

Heren=d= 1, soM =P a polynomial in one variable. SetP = 1 +x+x3_{. As in [H5],G}

M is the ring R[x, x−1_{, P}−1_{], and the bounded subring,E}

b(GM)is justRP =R[x/P,1/P]. The positive cone of the latter is generated additively and multiplicatively by©1/P, x/P, x3_/Pª_{[H5, I.4]. LetH =Ibe the ideal of}

RP generated by{1/P, x/P}. As in [H5], this is also an order ideal ofRP (hence ofGM), and(1 +x)/P is an order unit forI. Since each ofx/P2_{, x}2_/P2_{, andx}3_/P2 _{belong toI, multiplication by x}2_{/P is a}

positive endomorphism ofI, which obviously extends to a positive endomorphism ofGM. However, it is not bounded by a multiple of the identity endomorphism (so that the identity is not an order unit forE(H)). To see this, we it is sufficient to show that there does not exist an integerksuch that(x2/P)·1 ≤ k·1, i.e.,x2_{/P ≤ kis impossible. Ifx}2_{/P ≤k, then according to the definition of the ordering, there would}

existKso thatx2_PK−1_≤kPK _{with respect to the usual ordering onA=R[x}±1_{]. In other words, every}

monomial appearing inx2(1 +x+x3)K−1would have to appear inPk. It is a simple exercise to show that x3K−1_{appears in the former but not in the latter.}

In the case thatn= 1, failure ofE(H)to admit an order unit ifHis an order ideal ofR_P forces the latter ring to be not integrally closed in its field of fractions (see [H5, Section III]). _•

IfHis an order ideal ofGM, we define its bounded subring

E_b(H) =_{e_∈E(H)_|there existsk_∈Nsuch that ₋kI _≤e_≤kI_} Eb(H)+=Eb(H)∩E(H)+.

We will show thatEb(H) =Eb(GM), at least whenM(1)is primitive. ThusEb(H)is independent of the choice of (nonzero)H. This requires a series of lemmas.

Lemma 1.4

Suppose that

M(1)

is primitive. If

e

belongs to

E(G_M)+

_and

_{eh= 0}

_{for some}

_h

_in

Proof. We may write h = [a, k] for some a in (An₎+ _{and some integer k, and we may also assume}

that e = ˆM−s_Nˆ _{for some matrix N commuting with M, all of whose coefficients lie in A}+_{. Then}

eh= [N a, s+k] = [0, s+k]. This forcesMtN a= 0for some positive integert. Evaluate all the variables at(1,1, . . . ,1). This yieldsN(1)Mt_{(1)a(1) = 0. SinceM(1)is primitive and the entries ofa(1)are all} nonnegative real numbers (and not all zero, unlesshwere zero to start with), there exists a larger value of tso that all the entries ofMt(1)a(1)are strictly positive. Since all the entries ofN(1)are nonnegative,

N(1)a(1)₆= 0, a contradiction. •

Corollary 1.5

Suppose that

M(1)

is primitive and

H

is an order ideal in

GM

. If

e

belongs to

E(G_M)+

_and

_{eH = 0}

_{, then}

_{e= 0}

_.

We think ofM as giving rise to a graph on thenpoints _{1,2, . . . , n_}(each of which will be called a state), with edges being labelled either by the exponents or the corresponding monomials. Thus ifxw appears in the Laurent polynomialM_i,j position, there is an edge joiningitojwith weightw. (We do not worry about multiplicities for now.) This will be called the (multiplicity-free) graph ofM.

Lemma 1.6

If

M(1)

is primitive, then there exist

w

in

Zd

and a positive integer

k

such that

xwM−k

belongs to

Eb(GM)

.

Proof. The conclusion amounts toxw_{I ≤KM}l _{for some positive integerK, the ordering computed with} respect to that on E(GM). Form the graph ofM. Select a loop from 1to1 that contains every state in

{1,2, . . . , n} at least once (this is of course possible, becauseM(1)is primitive). Letkbe the length of the path; so ifw is the sum of the weights along the loop,xw _{appears in the(1,1)entry ofM}k_{. For any} particular statei, we can obtain a loop of the same length by starting at the first occurrence ofiinstead of the first1. Since the resulting loop is a cyclic permutation of the original, it has the same sum of weights. This means that each diagonal entry ofMkcontainsxw; it follows immediately thatxwI ≤KMl, as desired.•

Since everything remains the same for our immediate purposes if we replaceM byx−w_Mk_{, we may} even assume that I ≤ KM (that is,0 ∈ LogMi,i for alli). We will have to be a bit more careful about this when we come consider the invariants∆andΓof Krieger (our assumption for now is that they are the same).

Lemma 1.7

Suppose that

M(1)

is primitive. If

H

is a nonzero order ideal of

GM

, then

A[ ˆM±1]_·H=G_M

and

A[ ˆM±1]+_·H+=G+_M.

Proof.The order ideal contains an element of the form[xw_E

j, k]for some integersjandk, and some lattice pointw. By multiplying by an element of the formax−w_(forainA+_{), we obtain[aE}

j, k]. By multiplying by a power (positive or negative) ofMˆ, we obtain anything of this form but with arbitraryk. Finally, by primitivity ofM(1)and the convexity ofH, we can always findw′andk′(depending onj) so that for any statej,[xw′

E_j, k′_{]belongs toH. Hence by applying suitable elements ofaand powers ofM}ˆ_{, an arbitrary}

element ofG+_M is attainable; this yields the second assertion and the first follows immediately. _• Proposition 1.8

Suppose that

M(1)

is primitive. Let

H

be a nonzero order ideal of

GM

, and let

e

be an element of

E(GM)

such that

eH = 000

. Then

e

is zero.

Proof. Select [xwEj, k]inH. ApplyM using the identity[M a, k+ 1] = [a, k]. From the convexity of H together with the primitivity ofM(1), we may findk′_{, together with lattice pointsw}

j such that for all j = 1,2, . . . , n,[xwjE

j, k′]belongs toH. Writee = ˆMsNˆ. AsMˆ is an automorphism, we must have

ˆ

N[xwj_E

j, k′] = 000for allj. ThusxwjMnN Ej are all zero, and this obviously forcesMnN = 000. Hencee

is zero. _•

Proposition 1.9

Suppose that

M(1)

is primitive. If

H

is an order ideal in

G_M

, then

E_b(G_M) =

Proof. We first show thatEb(H) is the order ideal ofE(GM) generated by the identity. By definition, Eb(H)is a subring ofE(GM). Suppose thatebelongs toE(GM)ande(H+)⊆H+; we wish to show that e(G+_M)⊆G+_M. This follows from Lemma 1.7. Thus the relative ordering onEb(H)agrees with the natural ordering.

Since the identity element ofEb(H)is automatically an order unit for it,Eb(H)is directed. Finally, suppose thate′_{≤ewhereeande}′_{are positive elements ofE(G}

M)andebelongs toEb(H). Ase(H+)⊆ H+andH is an order ideal, it easily follows thate′(H+) ⊆H+, so thate(H)⊆H. Hencee′belongs to E(H); bute′_≤KI

Hfor someK(since the same inequality holds fore), so thate′belongs toEb(H). HenceE_b(H)is an order ideal ofE(G_M)and has the identity as an order unit. However, by definition Eb(GM)is the order ideal generated by the identity. So the two must be equal. • Now we show thatEb(GM)is a shift invariant forM. Recall that ifRis a (unital) partially ordered commutative ring, andMandM′_{are square matrices with entries fromR}+_{, then we say that they are(lagl)}

shift equivalent(over_R+) if there exist rectangular matricesRandSwith entries from_R+such that M R=RM′ SM =M′S

RS=Ml SR= (M′)l.

If all the matrices involved have entries in_Rbut not necessarily in the positive cone, then we sayM and M′_{arealgebraically shift equivalent(overR).}

The reasonGM itself isnota shift invariant is that it depends on the choice ofA, that is, on the number of variables, and how they are labelled. Let B be a subgroup ofZd, and suppose that all the entries of M have all of their exponents lying inB. LetA0 denote the subring (ifA = Z[x±_i1]) or subalgebra (if

A =R[x±_i1]) generated byB. We say thatM isdefined overA0. We may defineGM,0by replacingAn

withAn0 throughout (1). Then it is immediate thatGM is naturally isomorphic (in all possible ways) with the tensor product of orderedA0-modules,GM,0⊗A0 A.

Lemma 1.10

Under these conditions,

E_b(G_M) =E_b(G_M,0).

Proof: SelectbinG+_M,0, and formH0, the order ideal generated bybtherein. We claim thatH is also an

order ideal inGM. WriteA = F[B1](giving the nameB1to the standard copy ofZd) andA0 = F[B],

whereF is either the integers or the reals. LetT be a transversal ofB inB1; that is,T is a subset ofB1

such thatB =_∪t∈T(t+B)and for allt=6 t′inT,t−t′does not belong toB. We may assume that zero belongs toT. Obviously,A=⊕t∈TxtA0. Since the coefficients of the entries ofM all lie inA0, we have

that for allt,M((xt_A

0)n)⊆(xtA0)n. It easily follows thatGMdecomposes as⊕t∈TxtGM,0, as ordered

A0[ ˆM±1]-modules. Moreover, each summand is an order ideal inGM, so thatH (being an order ideal in the summand corresponding tot = 0) is also an order ideal. (Order ideals in order ideals of a dimension group are themselves order ideals in the big dimension group.)

Next, we observe that ifG = ⊕Gi, a direct sum of partially ordered (directed) abelian groups with the coordinatewise ordering onG, ande : G → Gis a positive endomorphism such thate ≤ KIG (as endomorphisms), then for eachi,e(Gi) ⊆Gi. To see this, selectginG+_i . Thene(g) ≤Kg, so thate(g) belongs to the order ideal generated byg; this obviously is contained inGi. SinceGiis directed, it follows thate(Gi)⊆Gi.

Hence ifeis inEb(GM), for allt,e(xtGM,0) ⊆ xtGM,0. We have an order-preserving ring

homo-morphism,∆ : E_b(G_M,0) →Eb(GM), by permittingeto commute with eachxt. By Proposition 1.8, an element ofEb(GM)is determined by its effect on a nonzero order ideal. Hence the map is onto (since these endomorphisms must commute with elements ofA). _•

2. A brief look atEb(GM)as a shift invariant

Suppose thatM andM′_{are matrices with entries fromA}+_{(a common Laurent polynomial ring) and they}

are shift equivalent; by enlarging the ring we may assume that the entries in the implementing matricesRand Sare also inA(this is not really necessary—a result in [PT] asserts that the equivalence can be implemented by matrices with entries inA+_).

Proposition 2.1

If

M

and

M′

_{are defined over a common Laurent polynomial ring}

_A

_{, and are}

shift equivalent (with the shift equivalence implemented over

A

), then the shift equivalence

induces an order-isomorphism of ordered

A

-algebras,

E(GM)→E(GM′)

, and an isomorphism

of ordered rings

E_b(G_M)_→E_b(G_M′)

.

Proof.The equationM R=RM′_{induces an order-preservingA-module homomorphism,R}ˆ _:G

M′→G_M by means of [a, k]M′ 7→ [Ra, k]_M. It is a triviality to check that MˆRˆ = ˆRMˆ′. Similarly, S induces

ˆ

S :GM →GM′, that also intertwinesMˆ′andMˆ. In general, having two such intertwining homomorphisms is not sufficient to obtain a ring homomorphism, either E(GM) → E(GM′) or between their bounded subrings (examples exist in the case thatd= 0). However, we also see fromRS =Ml_{andSR = (M}′₎l_, thatRˆSˆ = ˆMl andSˆRˆ = (Mˆ′₎l_{. Defineφ}

R : E(GM′) → E(G_M)as follows. Picke′ inE(G_M′)and denoteφ_R(e′_{)bye. ForhinG}

M, define

e(h) = ( ˆM)−lReˆ ′( ˆSh).

The intertwining relations guarantee thate(h) = ( ˆR)e′³_Sˆ_{( ˆM)}−l_h´_{, as well. The so-definedeis clearly} linear. It commutes with the actions ofAandMˆ (and thus withMˆ−1_{), because of the following identities:}

e(aMˆth) = ˆM−lReˆ ′³Sˆ_·a_·Mˆth´

= ˆM−lReˆ ′³aMˆ′t_Shˆ ´

= ˆM−lRˆ_·a_·Mˆ′t_e′_{( ˆSh)}

=aMˆ′t_{e( ˆSh)} Ife′_{≥000, then it follows thatφ}

R(e′)is also positive. SoφR(e′)is at least well-defined. We check thatφR is a ring homomorphism.

φR(e′1◦e′2)(h) = ˆM−lRˆ(e′1◦e′2)( ˆSh) = ˆR(e′1◦e′2)( ˆSMˆ−lh) =φR(e′1)

h

ˆ

Re′₂( ˆSMˆ−lh)i = ˆRhSˆMˆ−lReˆ ′₂( ˆSMˆ−lh)i

= ˆRe′₁³e′₂( ˆSMˆ−lh)´=φR(e′1◦e′2)(h).

Hence under the assumptionsM R=RM′,SM =M′S, andSR = (M′)l, we deduce thatφRis an order-preservingA-module ring homomorphism that intertwines the action of the original pair of matrices. One definesφS in a similar fashion, and a five-line computation reveals that they are mutually inverse.

It is routine to verify that the bounded subrings are mapped isomorphically to each other under these

Beginning with shift equivalentM andM′_{, find a Laurent polynomial ring over which they and the} implementing matrices are all defined. We have just obtained a pair of natural inverse ring isomorphisms between the rings of endomorphisms (after enlarging the polynomial rings). These isomorphisms induce an isomorphism on the bounded subrings; however, the latter are independent of the choice of coefficient ring. So the bounded subring is a shift invariant. (In fact, a little more is true—if the first three equations hold, we obtain a map induced byφRfrom the bounded subring corresponding toM′to that corresponding toM.) Example 2.2

Zero variables.

Suppose thatd = 0, so thatM = M(1) = M0is a matrix of constants with entries from eitherZorR.

Assume the former, and thatM0is primitive. ThenGM0is simply Krieger’s dimension group, and it is simple

(as a dimension group) with a unique trace. LetvinR1×nbe the unique (up to positive scalar multiple) strictly positive left eigenvector ofM0, with corresponding eigenvalueρ, which is of course the spectral radius. The

unique traceV :GM →Ris determined viaV[c, k] =v·c/ρk. ThenG+_M\ {000}=V−1((0,∞)). IfNis a matrix commuting with(M0), thenVNˆ =λ(N)V, whereλ(N)is the eigenvalue ofN whose eigenvector isv. It follows easily thatE(GM0) =Eb(GM0). (This equality can only occur ifd= 0, or other equally

degenerate situations.)

ThusE_b(G_M0) =CZ(M0)[M

−1

0 ]. This is a finer shift invariant (overZ) thanZ[1/ρ]studied in [BMT],

but not so fine as the complete invariant (whend= 0) which isGM0as a module overEb(GM0).

For example, letaandbbe positive integers. Set M0=

· a 8b b a

¸

M₀′ =

· a 4b

2b a ¸

.

Both matrices have ρ = a+ 2b√2. The centralizer ofM0 is the centralizer of

h

0 8 1 0

i

(subtractatimes the identity and divide byb); this is the companion matrix for the polynomialx2−8. HenceCZ(M0) ∼=

Z[2√2], and thusEb(GM0)∼=Z[2

√

2][(a+ 2b√2)−1_{](the isomorphisms are given by the effect on the left}

eigenvector, or the functionλdefined earlier). On the other hand, the centralizer ofM′

0 is that of

h

0 2 1 0

i . ThusEb(GM′

0)∼=Z[

√

2][(a+ 2b√2)−1_{]. It is straightforward to check that the two bounded endomorphism}

rings are isomorphic if and only if

(_∗) Z[2√2][(a+ 2b√2)−1] =Z[√2][(a+ 2b√2)−1]

(note the equality). This will occur if and only if there exists a positive integerkso that(a+ 2b√2)k√_{2 =} c+ 2d√2for some integerscandd. On taking norms, this would imply2(a2_−8b2₎k _=c2_−8d2_{. Hence2}

dividesc, so4divides the right side. Thus2dividesa2. Conversely, ifais even,ρ2= 2(aρ−2a2−4b2), so that invertibility ofρentails invertibility of2and thus (_∗) holds.

We have deduced that (_∗) holds if and only ifais even. However,Z[√2]is a principal ideal domain, soZ[√2][(a+ 2b√2)−1_{]is as well. Hence, with very little effort, we obtain thatM}

0is shift equivalent to

M0′if and only ifais even.

On the other hand, if

M0=

·

4 3 5 4

¸

M₀′ =

·

4 15 1 4

¸ ,

we have that the bounded endomorphism rings of both matrices areZ[√15](as4 +√15 is a unit). The matrices correspond to the two ideal classes ofZ[√15], and so are not shift equivalent (this is easy to see directly—as the matrices are invertible, shift equivalence is the same as conjugacy via GL(2,Z)).

If we add the identity to both matrices, the bounded endomorphism ring is thenZ[√15][(5 +√15)−1_].

This is a principal ideal domain (as the ideal inZ[√15]that contains5and√15represents the non-trivial ideal class). Hence in this case the matrices are shift equivalent. _•

Example 2.3

FOG examples (two variables).

Now letd= 2; instead ofx1,x2, we writex,y. We consider a pair of matrices suggested by Jack Wagoner

and Mike Boyle in connection with FOG (“finite order generation”) and other murky conjectures, most of which have been resolved:

M =



x0 1y 01 1 0 1



 M′=



y0 x1 01 1 0 1

 .

The second matrix is obtained from the first by interchangingxandy. These matrices arise from the graphs:

• •

ր ց ր ց

•

°y ←− °x• °x• ←− °y•

(The unlabelled arrows have weight1 = x0_y0_{.) The automorphism of A = Z[x}±1_{, y}±1_{]given by}

inter-changingx andy induces an order isomorphismGM → GM′, and correspondingly between the rings of endomorphisms, and the rings of bounded endomorphisms. These do not implement module isomorphisms, because they do not (in the first two cases) commute with the action ofA. Although the matrices are conjugate via an elementary matrix of GL(3,A), they are not shift equivalent; in fact they are not even finitely equiva-lent, a fact that was discovered independently by Marcus and Tuncel [MT2] using quite general techniques and by me (using completely ad hoc methods).

By examining the left eigenvector for the large eigenvalue function ofM, it is routine to check that in factM is conjugate (i.e., using GL(3,A)) to the companion matrix for its characteristic polynomial. Thus CA(M) =A[M], the polynomial algebra (overA) inM. SoE(GM) =A[M±1]—but the bounded subring, Eb(GM)is more difficult to compute.

We shall have more to say about this example in section 7. _• Now we prove a result which is related to the converse of Proposition 2.1. This is already known. Proposition 2.4

If

M

and

M′

_{are defined over a common Laurent polynomial ring (algebra)}

A

, then any order isomorphism between

GM

and

GM′

as

A[ ˆM]

-modules

is induced by a shift

equivalence between

M

and

M′

_{, up to a power of}

_Mˆ

_.

Proof. Supposeφ : GM → GM′ is a positiveA-module homomorphism satisfyingφMˆ = ˆM′φ. Then there exists an integerktogether with positive columnsaj inAn

′

such that for allj= 1,2, . . . , n, φ([Ej,1]) = [aj, k].

Define then′_×nmatrixR

0by setting its jth column to beaj. All the entries of R0 lie inA+, and we

observe that as φintertwines the shift(s) and is an A-module homomorphism, φ has the same effect on GM as applyingR0to the equivalence classes and following this withMˆ(k−1). In particular,Rˆ0is

well-defined and satisfiesRˆ0Mˆ = ˆM′Rˆ0. It follows immediately that(R0M −M′R0)Mn = 000. If we replace

R0 by R1 = R0Mn, then we see thatR1has all its entries inA+, intertwinesM andM′, and satisfies ˆ

R=φMˆn−k+1_.

Similarly, ifψ :GM′ →G_M is positive and intertwinesMˆ andMˆ′, there exists a rectangular matrix S with entries from A+ _{that intertwines M andM}′_{and implements ψ, up to a power ofM}ˆ _orMˆ′_{. By}

multiplying on or the other ofR1orSby a power ofMwe may assume that the power ofMˆ that indicates

the perturbation fromφandψis the same; sayRˆ1=φMˆlandSˆ= ˆMlψ.

Now suppose thatφψ= ˆMt_{for some integert. ThenS}ˆ_Rˆ_{1= ˆM}t+2l_{. As before,(SR}

1−Mt+2l)Mn= 000. ReplacingR1byR=R1Mn, we obtain thatSR=Mt+2l+n, andRis positive and still intertwinesM

andM′_{. By incorporating another power ofM}′ _{intoS, we obtain that ifφψ= ( ˆM}′₎t_{, thenRS is also a} power ofM′_{. In particular, this yields a shift equivalence. •}

Remark. 2. If we let A be an arbitrary commutative unital ring andM an arbitrary square matrix (see Remark 1), then the unordered versions of Proposition 2.1 and Proposition 2.4 still hold, where we now use algebraic shift equivalence.

In view of Proposition 2.1 and Proposition 2.4, what is the point of discussingEb(GM) (especially since it generally is more difficult to compute thanGM)? For determining the pure traces on the order ideals, it is essential (the trace space ofE(GM)is inadequate for this purpose). Moreover, the preceding depends on the choice ofA. There is a close relationship betweenE_b(G_M)andE(G_M)ifM is defined over the minimal possibleA.

Krieger [Kr] has defined two invariants for shift equivalence, which amount to the following (where tr denotes trace). Iff = Pλwxw is a polynomial indvariables expressed in monomial notation, then Logf =©w∈Zd¯¯λw6= 0

ª .

Γ(M) =_h

∞ [ j=1

Log trMj _i

∆(M) =

∞ [ j=1

¡

Log trMj ₋Log trMj¢.

The angle bracketsh iindicate the group generated by the contents. If S andT are subsets ofZd, then S−T ={s−t|s∈S, t∈T}. Note that∆(M)is already a group.

Lemma 2.5

If

M(1)

is primitive, then

Γ(M) =_h©w_∈Zd¯¯ xwM−l_∈Eb(GM)

for some

l∈Nªi

∆(M) =h©w−w0

¯

¯xwM−l, xw0_M−l _∈E

b(GM)

for some

l∈N

ª

i.

Proof: Selectz in Log trMl_{. This means thatzis the total weight of a loop of lengthlwhose initial and} terminal state1 isj for somej. There exists a loop of lengthkthat passes through every state (asM(1)

is primitive). Let its total weight be denoted w. In this latter loop, insert the original loop immediately adjacent to an occurrence ofj. This creates a loop of lengthk+lcontaining every state, of weightz+w. As the loop of weight wcontains every state, wbelongs to Log ¡Mk¢

ii for alli = 1,2, . . . , n and thus xwI ≤ Mk for some positive integerK (as endomorphisms ofGM. ThusxwMˆ−k belongs toEb(GM). Similarly,xz+w_Mˆ−(k+l)_{belongs toE}

b(GM). Hencez=z+w−wbelongs to the right side.

Conversely, ifxwMˆ−l is a member ofEb(GM), thenxwI[Ej,1]≤KMˆl[Ej,1]for someK and all j. Hence there exists an integerN so thatMN ¡_KMl_−xw_I¢_E

j has all of its entries inA+. This simply means that all the entries of the matrixKMl+N _−xw_MN _{belong toA}+_{. Taking a diagonal entry, we see}

that ifubelongs to Log¡MN¢_iifor somei, thenu+wbelongs to Log ¡Ml+N¢_ii. Hencew=u+w−u expresseswas the difference of two elements ofΓ(M).

The argument for∆(M)is parallel. •

Parry and Schmidt [PS] (viz. [MT, 1.9]) had shown that ifM(1)is primitive, there exists a diagonal matrixDwith monomial entries so thatDM D−1_{has all of its entries inZ[Γ(M)]. Since this implements a}

very strong form of shift equivalence, we may assume thatM already has support inΓ(M), i.e. LogM ⊂ Γ(M). For many purposes (including (S) ), we can replaceM by xw_{M (for any lattice point w), and} so even assume that LogM _⊂ ∆(M). The result in [PS] can be deduced from the following, which is interesting in itself.

Proposition 2.6

Suppose that for some

i

, no entry in the

i

th row and column of

M(1)

zero. Then

Log tr

M_∪

Log tr

M2_∪

Log tr

M3

generates

Γ(M)

as an abelian group, and there exists a diagonal matrix

D

with monomial

entries such that Log

DM D−1_⊂Γ(M)

_.

Proof.LetWdenote the subgroup ofΓ(M)generated by the three sets. For each(k, l), select any exponent w_k,l appearing in LogM_k,l, unless M_k,l is zero, in which case we avoid that choice of (k, l) for now. Whenever it makes sense,wk,l+wl,k ∈W, and similarly, for any three indices,wj,k+wk,l+wl,j ∈W. We deduce that ifwk,l is defined, then it is congruent moduloW towi,l−wi,k(which is always defined). Moreover, ifw′_k,lis another choice replacingwk,l, then their difference belongs toW(since we may use the samew_i,l andw_i,k forw′

k,l. SetD =diag(1, xw1,2,· · ·, xw1,n). Then(DM D−1)kk = Mkk, while for k ₆=l,(DM D−1₎

kl is either zero orx−wk,lxvk,lMkl for somevk,l inW. Thus Log(DM D−1)k,l ⊂W

(where Log0is the empty set). _•

Lemma 2.7

Suppose that

M(1)

is primitive and Log

M ⊂∆(M)

. If

H

and

H′

are nonzero order

ideals of

GM

, then

H∩H′

is not empty. (In other words, all nonzero order ideals of

GM

are

essential.)

Proof. We may find (nonzero positive) elements of the form [xw_E

j, k] and [xw ′

Ej′, k] in H and H′ respectively. By replacing these by[xw_Mt_E

j, k+t]and[xw ′

Mt_E

j′, k+t]for suitablet, and using the convexity property of order ideals (and the fact thatM(1)is primitive), we may even assume that originally j=j′. There exists an integerlso thatxw0−w_Mˆ−l_{is an element ofE}

b(GM). In particular, if it is applied to any element of an order ideal, the outcome still belongs to that order ideal. Applying it to[xw_E

j, k], the outcome is not zero, positive and lies inH; on the other hand, the image is[xw0_E

j, k+l]. Under the current assumption that I ≤KM, this element is dominated by an integer multiple of[xw′_E

j, k], so belongs toH′.

•

LetAbe one of the ordered ringsZ[x±_i 1]orR[x±_i1](Laurent polynomial rings indvariables), equipped with positive cone consisting of the Laurent polynomials with no negative coefficients. For two rectangular matrices of the same dimensions with entries fromA, we writeS_≺S′_ifS′_{has all of its entries inA}+_and

LogSij ⊆LogSij′ for all coordinates.

Theorem 2.8

If

M(1)

is primitive and

∆(M) =Zd

, then there exists

e0 =xv0Mˆ−l

in the centre

of

Eb(GM)

such that

E(GM) =Eb(GM)[e0−1]

. Additionally, for

h

in

GM

, if

e0h ≥0

then

h≥0

.

Proof.Iff is an element ofE(GM), we may replace it byfmultiplied by any power ofMˆ. Hence we may assume thatf = ˆN whereN is inCA(M).

First assume that trM ₆= 0and0belongs to Log trM. AsM(1)is irreducible, there existsk(k=n!

will do) together with a weightwso that there is a looplof lengthkcontaining every state from1tonwith total weightw. SetWt=

©

y ∈Zd¯¯xyM−(t+k) ∈Eb(GM) ª

.

Claim: Wt⊃S0≤s≤t(w+Log trMs).

Selectvappearing in thei, iposition ofMs_{. This corresponds to a pathl}′_{fromitoiwith total weight} v. Insertl′_{intolso that the initial stateiappears adjacent to an occurrence ofiinl. This creates a looplof}

lengthk+swith total weightw+v. Setm=t₋s. As0belongs to Log trM, there exists a statej′_{so that}

0lies in LogMj′_j′—yielding a loop of length 1 fromj′toj′. Insert them-fold concatenation of this loop inl∪l′, adjacent to an occurrence ofj′. The weight is unchanged, but of course the length is increased to k+t. As every state appears in the resulting loop, we have thatxw+v_Mˆ−(k+t)_{belongs toE}

b(GM). This proves the claimed statement.

Lets(1), s(2), . . . be positive integers, and lett≥Ps(j).

Claim: w+PLog trMs(j) _⊂W

t.

Selectv(j)in Log trMs(j)_{; letl(j)be a corresponding loop with initial state(i(j) and total weight}

v(j). Ifm=t₋Ps(j), setl0to be the loop consisting of the constant path atj′of lengthm. Insert each

l(j)adjacent to an occurrence ofi(j)inlandl0adjacent to an occurrence ofj′. The resulting loop has total

weightw+PLog trMs(j) _{and is of lengtht. This establishes the claim.}

The cone inZdwith vertexwgenerated by all of the Log trMs_generatesZd_{as an abelian group, by} hypothesis (∆(M) = Zd). Hence it contain an elementv0such that all ofw+v0±ei (i = 1,2, . . . , n) belong to it as well (hereeirun over the standard basis ofZd). By the last claim, there existst0such that {w+v0±ei} ⊆Wt0—in fact,x

w+v0±ei_{I ≺M}t0_.

ChoosezinZd, and writez′_{=z−w. Now decompose} z′ =X

I

aiei+ X

I′ bi′e_i′

where I ∪I′ is a partition of a subset of {1,2, . . . n}, ai are positive integers andbi are negative. Set m=Pai+P|bi|. We note that for allt≥mt0,

w+Xai(v0+ei) + X

(−bi′)(v₀−e_i′) ∈W_t.

Thusw+mv0+z lies inWt, whencexw(xv0)mxzI ≺ Mmt0+k. By the preceding, xw(xv0)mxzI ≺ Mm(t0+k)_{, and sox}z_{I ≺}¡_x−v0_Mt0+k¢m_{. Moreover, this holds withmreplaced by any integer exceeding}

it.

LetEijdenote the matrix with a 1 in thei, jposition and zeroes elsewhere. We shall show that for any zinZdand choice ofi, j,

xzEij ≺¡x−v0Mt0+k¢m (in other words,z_∈Log¡(x−v0_Mt0+k₎m¢

ij) for all sufficiently largem. Choosevin Log¡(x−v0_Mt0+k₎m¢

ij(note thatt0+k > n!, so every entry of the matrix is nonzero).This corresponds to a path of lengtht0+kfromitojwith total weightv. In particular,xvEij ≺x−v0Mt0+k. By the preceding, for all sufficiently largem,xz−vI ≺¡x−v0_Mt0+k¢m_{, so the result obtains on multiplication.}

Now letN be an element ofC_A(M). Since LogN is finite, for all sufficiently largem, for alliandj, LogN_{ij ⊂}Log ¡(x−v0_Mt0+k₎m¢

ij.

Hence there exists an integer K so that ₋Kx−v0_Mt0+k _{≤ N ≤ Kx}−v0_Mt0+k_{. In particular, N}ˆ _·

(xv0_Mˆ−(t0+k₎m1 _{belongs toE}

b(GM)for all sufficiently largem1. Applying the argument withz′=−w,

we deduce that for somem0, the element defined ase0 = (xv0Mˆ−(t0+k)m0 belongs to Eb(GM), and is actually in the centre. Thus ifm1is chosen to be a multiple ofm0,

ˆ

N = ( ˆN _·e0)e−₀1

whereN eˆ 0belongs toEb(GM)ande0is in the centre ofEb(GM). Moreover,e0is not a zero divisor in

If trM ₆= 0but0is not in Log trM, choosew0in Log trM, and replaceM byx−w0M.

Finally, if trM = 0 we observe that trMs _{6= 0for alls ≥ n!(and in facts ≥ n}2_{is sufficient). If}

g. c. d. (s, n!) = 1and N Ms _{= M}s_{N, then (M}n_{N)M = M(M}n_{N) (by Lemma A1.1). ThusM}n_N belongs toCA(M), soMdnNMˆ−n is inE(GM) and implementsN. In other words, CA(Ms)[ ˆM−s] = CA(M)[ ˆM]. So we can replaceM byMsin the preceding. • The methods used here suggest the following shift invariant. ForM in MnA+, define the subset of Zd_×N,

T(M) =©(w, t)_∈Zd_×N¯¯there existsssuch thatxwMs _≺Ms+tª.

Our first assertion is thatT(M)is a shift invariant forM. This follows immediately from the observation (practically a tautology) that(w, t)belongs toT(M)if and only ifxw_Mˆ−t _{is an element of the bounded} endomorphism ringEb(GM). There is a natural map fromT(M)to WPS(M) :=∪(Log trMk)/k, given byτ : (w, t)_7→ w/t; again verification is obvious. The image is in general a proper subset, and of course its image is also a shift invariant, but there is some information lost in going fromT(M)to its image. This will be seen as a consequence of the result to be proved below, that ifd= 1, cvxτ T(M) =cvx WPS(M). (The formal definition of WPS , introduced by Marcus & Tuncel [MT], will be given in section 3.)

First, we present an example of what can happen whend= 2. Suppose

M =



x1 y0 10 0 1 1

 ,

our familiar FOG example (2.3). Then it is fairly easy to see that

T(M) =©((a, b), t)∈Z2_×N¯¯ a≥0, b≥0, andt≥a+b+ 3ª.

Thus although(0,0)(one of the vertices of cvx WPS(M), the standard triangle inR2) belongs toτ T(M)

(via the element((0,0),3)—3 can be replaced by any larger integer, but no smaller one), neither(1,0)nor

(0,1)does. We do obtain that both of these points as limits of elements ofτ T(M), e.g.,_{τ((k,0), k+ 3)_}, but this is a general phenomenon. In this example,τ causes a loss of information in the sense that although

(0,0)is in its image, we cannot tell that((0,0),1)is not inT(M). This exampleEb(GM)will be investigated

more deeply in the next section. _•

Valerio de Angelis asked whether the following is true, in the form given in the third sentence. Proposition 2.9

Let

M

be a primitive square matrix of size

n

with entries from

A+

, where

d= 1

.

Then cvx

τ T(M) =

cvx WPS

(M)

. In particular, if

0

is the left endpoint of cvx WPS

(M)

, there

exist positive integers

s

and

t

such that

Ms_≺Ms+t

_.

Proof. The conclusion of the second sentence is that(0, t)belongs toT(M). It is clearly enough to prove this, since we may replacex(the lone variable) by its inverse and multiply by a suitable power ofxto obtain the other endpoint. The following proof yields extravagantly large choices for bothsandt, but it may be that this is necessary (particularly whennis large).

We assume that0is the left endpoint of cvx WPS(M); in particular,0itself belongs to WPS(M).The first step is to find a states in{1,2, . . . , n} and a positive integer ksuch that the set S = Log(Mk)ss satisfies:

(i) 0∈S

(ii) gcdS = gcd (_∪eLog trMe).

(iii) There exists an integerm >2ninSsuch that for all integersl_≥mlying in the right side of (ii),lcan be expressed as a nonnegative integer combination of elements ofS.

¯

¯¡trχ(N)¢_(r)/βN_(r)−1¯_{¯ < 1/3, and so} ¯_¯βN_(r)/¡_trχ(N)¢_(r)−1¯_{¯ < 1/2. IfN is increased, this} in-equality will be preserved.

ForNsufficiently large, selectQinA+such thatQ·trχ(N)also belongs toA+. SetPN =trχ(N)QN. We may replaceβN_byβN_QN_,RbyQR,χ=Xn₊P_p

iXibyXn+PpiQiXi, etc. With this renaming, °

°βN_/P N −1

°

°<1/2. Define

P =Xxw the sum overw_∈Zd_∩cvx LogP_Nd.

By e.g., [HM, V.4], P_Nd/P is bounded below away from zero on (Rd)++. Clearly for all a and b, τ_B,v(βN_/P

N)>0. Thus the same is true ofβN d/P. Consider the matrixR′ _{= R}N d_{/P in M}

nA[P−1]. We shall show that every entry ofR′ lies in the algebraRP. Since cvx LogP ∩Zd=LogP and cvx LogPisdtimes a lattice polytope (this is the reason for raisingPN to thedth power), it is sufficient to show that every entry ofR′is bounded as a function on

(Rd)++[HSLN, III.2 & III.4].

ReselectN so that in addition it is a multiple ofl. Then it follows from ³

trβN PN

´±1

being bounded that all of the entries ofR′_{are bounded below. The equationΛR=βΛyieldsΛR}′₌ βN d

P Λ. Fixj; we deduce X

λi(R′)ij = βN d

P λj.

Evaluating at r in(Rd)++, the boundedness below hypothesis yields that the entries ofR′ are bounded above, and thus bounded. So,R′_{belongs to M}

nRP. Setβ′to be the large eigenvalue function ofR′, which we regard as a continuous functions on the pure state space ofRP.

Letγbe a pure state ofRP. Restricted toRP, it is of the formτB,vfor some(B, v)inRd×Rd[HSLN, Section 3]. It is routine to verify thatτB,v(β′)is the large eigenvalue of the non-negative real matrixγ(R′), and the corresponding eigenvectors areτB,v(Λ)andτB,v(Θ). Sinceγis a limit of point evaluation states, (#) implies thatτB,v(β′)is larger than the modulus of all the other eigenvalues ofγ(R′). By [H1, 2.1], there exists an integerDso that for allD′_≥D,γ³_(R′₎D′´

is strictly positive. (A real nonnegative matrix with a largest eigenvalue of multiplicity one whose corresponding right and left eigenvectors are strictly positive, is primitive–consider the limit of powers of the matrix normalized so that spectral radius is 1.) Hence there exists a neighbourhoodUγ ofγ in the pure state space ofRP such that for allρinUγ,ρ

¡

(R′₎D¢_{and the} same expression withDreplaced byD+ 1are strictly positive—it follows that all sufficiently high powers are strictly positive; denote the minimum power,Eγ.

Compactness of the pure state space yields a finite open covering_{U_i}with correspondingE_i. Set E = max_{E_i}; thenγ¡(R′₎E+k¢_{is strictly positive for all nonnegative integersk. Ifais an element of} RP,γ(a)>0for all pure statesγ, then not only isapositive inRP, but it is actually an order unit. Thus all the entries of(R′)E+kare order units. There thus exists an integerF such that all the entries ofPFRE+k belong toA+_{, fork= 0,1. This means that all sufficiently large powers ofP}F_{Rhave entries fromA}+_{. To} check that¡PF_R¢E _{satisfies (**), we simply note that this follows from the entries of(R}′₎E _{being order}

units. _•

The intricate hypotheses make the lemma quite easy to prove. However, they really are necessary. Concerning (c) for example, start with any reasonable choice forR0with strictly positive eigenvectorsΛ andΘand eigenvalueβ etc., and find a matrixS whose entries are bounded rational functions such that S2 = 0,ΛS = 000, andSΘ = 000(‘reasonable’ means in particular that the irreducible polynomial satisfied byβ is of degree (overA) less than the size ofR0, or else no suchS can be found). Pick a lattice point wsuch that the growth ofxw_{is not dominated by that ofβ. SetR =R}

0+xwS. ThenΛandΘare also eigenvectors forR with eigenvalueβ. If we assume (for example) that the determinant ofR0 is not zero

(almost everywhere on(Rd₎++_{, for which it is enough that it not vanish at one point), then the argument of} the proof will fail, and it is easy to construct an example where the conclusion of the Lemma fails.

SetΛ = (3,1,2),Θ = (1,1,1)T (no variables!), andR0= ΘΛ +I∈M3Z+. Now put

S =



00 −11 −11 0 ₋1 1

 ,

soS2_{= 000, andΛSandSΘare both zero. AlsoR}

0S =SR0=S; finally,R=R0+x1SadmitsΛandΘas strictly positive eigenvectors, but no power can be multiplied by a polynomial without negative coefficients to a nonnegative matrix.

As we pointed out earlier, the hypothesis concerning dividing a polynomial with no negative coefficients is necessary. Moreover, necessary and sufficient conditions are known and easy to deal with [H2, V.6]; in fact, it is sufficient that theQof Lemma 13.1(d) belong toAand be nonzero.

To use these results, especially when R is the companion matrix of a polynomial (or a power of a companion matrix), we require some elementary algebraic results about manipulating pairs of vectors. Lemma 13.2

Let

A

be a commutative ring such that all finitely generated projective modulues

of rank

n₋1

are free. Let

v

and

v′

_{be rows of size}

_n

_{with entries from}

_A

_{, and let}

_w

_and

_w′

_be

columns of size

n

with entries from

A

. Suppose that

v·w=v′·w′

, and the latter is invertible

in

A

. Then there exists

P

in GL(

n

,

A

) such that

vP =v′

_and

_P−1_w=w′

_.

Proof. Obviously, we may assume v ·w = 1; it is enough to prove the result in the case that v′ = (1,0, . . . ,0) = (w′₎T_{. In particular, each ofv andv}′ _{is unimodular; as projectives are free, there exists}

P0 in GL(n,A) such thatvP0 = (1,0, . . . ,0). RelabelvP0 andP0−1w as v andw respectively, so that w = (1,_∗, . . . ,_∗)T_{. If we add anyA-multiple of the first entry of the currentw to any other, the inverse} elementary operation subtracts a zero from the first entry ofv—sovis left unchanged. We may thus assume thatw= (1,₋1,_∗, . . . ,_∗)T_{. LetWdenote the column of sizen−1obtained by removing the first entry of} w.

There existsQin GL(n−1,A) such thatQ−1_{W = (−1,0, . . . ,0)}T_{(sizen−1). ApplyingP}

1= 1⊕Q to our currentvandw, we may assume

v= (1,0, . . . ,0) and w= (1,₋1,0, . . . ,0).

Now subtract the second entry of the currentvfrom the first entry; the inverse operation adds the first entry

ofwto the second, and we are done. _•

LetP be an element ofA+_{, letK =cvx LogP, and formR}

K as in [HSLN]. ThenRK embeds as a dense ring of continuous real-valued functions on the pure state space ofKwith the additional property that all elements ofRK that are invertible as continuous functions are already invertible inRK. By [Sw], any invertible matrix in the (real-valued) continuous function ring can be uniformly approximated by invertible matrices with entries inR_K(and via determinants, we see that the inverses have entries inR_Kas well). By [HSLN, Section 4], the pure state space ofRis homeomorphic to the compact convex setK, so is contractible. Thus all projectives over the continuous function ring are free—we take the real-valued continuous function ring as our choice forAin Lemma 13.2.

Now letM be a square matrix with entries fromRP (orRK). Suppose thatM admits a largest real eigenfunction β (with values in the pure state space ofRP) and there exist right and left eigenvectorsv andwforβwhose entries are integral overRP (orRK) such thatγ(v·w)>0for all pure statesγ. As a continuous function on the pure state space,v_·wis invertible. By the lemma above, there exists a strictly positive pair—that is,v′ _andw′_{, whose entries are strictly positive continuous functions on the pure state}

matrix can be approximated (uniformly) by matrices with entries fromRK, the usual compactness argument yields a matrixBin GL(RK) such that bothvBandB−1ware strictly positive. Applying the earlier lemma, we deduce the following:

Proposition 13.3

Let

R

be a matrix with large eigenvalue

β

, such that for some integer

k

, there

exists

P

in

A+

_satisfying:

(i) Log

Rk _⊆

_Log

_P

_;

(ii)

β

satisfies (#) with respect to

χ

.

(iii) For all sufficiently large integers

J

, tr

χ(J)

_{has no negative coefficients.}

Then there exists

Z

in GL(

n

,

RK

) and an integer

N

so that for all

M ≥N

, the matrix

¡

P′_ZRZ−1¢M

has all of its entries in

A+

_{, and satisfies (**); here}

_P′

_{is an element of}

_A+

_{with Log}

_P′ ₌

_{Log tr}

_βk

for some integer

k

.

This litany of conditions is actually satisfied in a number of situations. For example, ifχsatisfies (iii) above andβis a root ofχsatisfying (#) (with respect toχ), then we chooseRto be the companion matrix forχ. To start, suppose that(β/P)±1is bounded, whereP belongs toA+_{. For example, relabelβ}n!_asβ, so that the trace ofχwill do. We may also assume thatRP is integrally closed (by alteringP somewhat, but not enough to affect the convex hull of its Log set). The companion matrix ofβ/P has entries inRP, so each will belong toRP. Then there exists a power ofP so thatPNR′ satisfies the conclusion in the proposition above.

If instead we begin withRin MnA+, then the characteristic polynomialχwill satisfy (iii) automatically; if we assume that the large eigenvalue satisfies (#), then the preceding paragraph applies to the companion matrix of the characteristic polynomial. There is a much more difficult argument to show that a power ofR can be replaced by a matrix in MnA+for which condition (i) holds. So some power ofRis conjugate to a matrix which after multiplication by a suitable polynomial, satisfies (**).

Appendix

Lemma A1.1

Let

D

be a commutative domain, and let

M

and

N

be

n_×n

matrices over

D

.

Suppose that

s

is an integer relatively prime to

n!

, and that

MsN = N Ms

. If the field of

fractions of

D

contains no non-trivial

s

th roots of unity, then

Mn_N

_{commutes with}

_M

_.

Proof: We may assumeDis already a field. LetK be the splitting field of the characteristic polynomial ofM. The relative primeness condition guarantees thatK contains no newsth roots of unity, so contains no non-trivial ones at all. We may thus assume thatM is in Jordan normal form. The centralizer ofM in MnKis obtained by fixing an eigenvalueλ, noting how the number of blocks of a given size occur in the generalized eigenspace attached toλ, and then taking the corresponding matrix ring over the polynomials in the block. ForMs_{, andλ 6= 0, the block sizes are the same: Ifλ}s _{= λ}′s_{, then the ratioλ/λ}′ _{is ansth}

root of unity. Since the0block ofMnis just the zero matrix, we see immediately that ifMsN =N Ms,

then(Mn_{N)M =M(M}n_{N). •}

IfMis a nonnegative real matrix with only one eigenvalue of absolute value equaling its spectral radius, then that eigenvalue belongs to a unique block ofM (whenM is written in block triangular form); this will be called the principal block. The following is probably well-known; certainly, portions of it are special cases of results in [Ga]. Compare this with [Tu2, Theorem 2, p. 293], which characterizes the existence of a positive left eigenvector.

Lemma A1.2

Let

M

be a nonnegative real matrix of size

n

. Suppose

M

has its spectral radius

as an unique eigenvalue of multiplicity one, and all other eigenvalues have strictly smaller

absolute value. The following are equivalent:

(b) in some power of

M

, all rows whose diagonal entry lies in the principal block of

M

are

strictly positive;

(c)

M

admits only one nonnegative right eigenvector (up to scalar multiple).

When (b) occurs, it will occur for all powers exceeding

n2_{−n+ 2}

_{, and no other rows will be}

strictly positive.

Proof. Without loss of generality, we may assume the spectral radius is1. Letvandwbe the left and right nonnegative eigenvectors for1 respectively, normalized so that the scalar productvw is1. Then©Mkª converges (any reasonable norm) to the matrixwv.

Hence, ifv is strictly positive, then for all sufficiently largek, the rows ofMk _{corresponding to the} positions ofwthat are not zero are strictly positive. Of course, these can only belong to the principal block, since the support ofwmust include the principal block.

Suppose (b) holds. Eventually, the rows involving the principal block are all strictly positive, which entails that no other rows are ever strictly positive (otherwise they would belong to the principal block). Moreover, it also follows that all the columns corresponding to the principal block must be zero below that block. WriteM =hA X

0 C i

, whereAis the principal block, and on replacing by a sufficiently high power, we may assumeAandXare strictly positive, andCis merely nonnegative. Letv= (vA, s)be the partitioned left eigenvector for1. Thenv_Ais the left eigenvector forAso is strictly positive; asv_AX+sC =s,sis strictly positive, so that so isv. Hence (b) implies (a).

Assume (a). Ifuis a right eigenvector for any eigenvalue other than1, we must havevu= 0; asvis strictly positive,ucannot be nonnegative. So (a) implies (c).

Suppose (c) holds; writeM in block upper triangular form. The right nonnegative eigenvector for the upper left block extends to an eigenvector for M by simply adjoining zeros; sinceM always has a nonnegative eigenvector whose eigenvalue is the spectral radius, uniqueness forces the upper left block,A, to be the principal block, and the unique nonnegative right eigenvector is simplyw = (w_A,0)T_{. Now a}

routine induction on the number blocks can proceed. _•

Theorem A1.3

Let

M

be a square nonnegative real matrix of size

n

and let

f

be a real column of

size

n

. Set

S =_{i_|(Mn_f)

j 6= 0

for some

j

equivalent to

i}

.

(a) There exists an integer

m

such that the column

Mm_f

_{is strictly positive if and only if}

_M

has no zero rows and

v_·f >0

for all nonnegative left eigenvectors

v

of

M

corresponding

to nonzero eigenvalues.

(b) There exists an integer

m

such that the column

Mmf

is nonnegative if and only if

vS·fS >0

for all nonnegative left eigenvectors

vS

of

MS

that correspond to nonzero eigenvalues.

Proof: (a). DefineGM = limM :Rn→Rnequipped with the limit ordering. Being a finite dimensional vector space, it admits an order unit. By [HCM, 1.3], the pure states are all obtained from left nonnegative eigenvectors ofM, in the sense that ifvis one such, we define a stateV onG_Mvia[f, k]_7→(v_·f)/sk−1_where sis the corresponding positive eigenvalue ofM. Notice that no left nonnegative eigenvector can correspond to a zero eigenvalue, asM has no rows consisting entirely of zeroes. All pure states are of this form. The stated conditions onfandMare clearly necessary. Suppose they hold. ObviouslyV([f,1])>0for all pure statesV follows from the hypotheses. AsG_M admits an order unit, and is obviously unperforated (even a dimension group), [HM, I.1] guarantees that[f,1]is an order unit ofGM. Sety = (1,1,· · · ,1)T ∈ Rn. Asf is an order unit, there exists a positive integerJ such thatJ[f,1]−[y,1]≥000inGM. From the way the limit ordering is defined, there exists an integerNsuch thatMN(Jf−y)has only nonnegative entries; thus every entry ofMN_{f is at least as large as1/J times that ofM}N_{y, which is of course strictly positive.} (b). By (a), that there will exist an integerN (which we may take to be greater thann) such that(M_S)N_f

S is strictly positive if and only if for all (extreme) left nonnegative eigenvectorsvSofMS,vS·fS >0, and thusMN_{f ≥000.}

Conversely, assume thatMN_{f ≥000for someN, andM}N_{f 6= 000we may assumeN > n. Suppose}

[i]≤[j], and(MN_{f)[j]6= 000. Applying a sufficiently large further power ofM (in fact we only need to use} the primitive matrixM[j]), we can even obtain that each entry of(MNf)[j]is strictly positive. It follows immediately that(MN_{f)[i]is not zero, and so applying more powers ofM (actuallyM[i]) if necessary,} we deduce that(MN_{f)[i]is also strictly positive. We easily conclude that for some sufficiently largeN,}

(MS)NfS is strictly positive. Hence, there will be affirmative answer to the question if and only if either MN_{f = 000, or for all nonnegative left eigenvectorsv}

Sof the cut-down matrixMS,vS·fS >0.

Suppose at the outset, we replacef byf0=Mbf, wherebismax{#([i])|i∈ {1,2, . . . , n}}. Then we eliminate the indices belonging to a minimal[i]iff0[i]is the zero column, and a simplification occurs.•

References

[A] L.V. Ahlfors, Complex Analysis, 2nd edition, McGraw-Hill (1966) New York.

[AE] L. Asimow and A. J. Ellis, Convexity theory and its applications in functional analysis, Academic Press (1980) New York.

[Alf] E. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Math. und ihre Grenzgebiete 57 (1971) Springer-Verlag.

[BH] B.M. Baker and D.E. Handelman, SU(2)-equivariant K-theory of a class of C*-algebras and nonho-mogeneous random walks, K-Theory Journal 8 (1994) 499–540.

[BoH] M. M. Boyle & D. E. Handelman, Algebraic shift equivalence and primitive matrices, Trans. Amer. Math. Soc. 336 (1993) 121–149.

[BMT] M. Boyle, B. Marcus, & P. Trow, Resolving maps and the dimension group for shifts of finite type,

Mem. Amer. Math. Soc. 377 (1987) American Math. Soc.

[EHS] E. G. Effros, D. E. Handelman, and C.L. Shen, Dimension Groups and Their Affine Representations,

Amer. J. Math. 102 (1980) 385–407.

[El] G. A. Elliott, On the classification of inductive limits of semisimple finite dimensional C*-algebras, J. of Algebra 38 (1976) 29–44.

[E1] D.E. Evans, The C*-Algebras of Topological Markov Chains, Tokyo Metropolitan University Lecture Notes (1984).

[E2] D.E. Evans, Quasi-Product States on C*- Algebras and their Connections with Topology and Ergodic Theory, Proceedings Busteni Conference on Operator Algebras, edited by H. Araki, C.C. Moore, S.S. Stratila, and D. Voiculescu, Springer Lecture Notes in Mathematics1132(1985) 129–151.

[Ga] F. Gantmacher,Applications of the theory of matrices,Interscience Publishers (1959) .

[G] K.R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys and Mono. 20 (1986) American Math. Soc.

[GH] K.R. Goodearl and D.E. Handelman, Rank Functions and Ko of Regular Rings, J. Pure App. Algebra 7 (1976) 195–216.

[GH2] K.R. Goodearl & D. E. Handelman, Metric completions of partially ordered abelian groups, Indiana J. of Math. 29 (1980) 861–895.

[H1] David Handelman, Positive matrices and dimension groups affiliated to C*-algebras and Topological Markov Chains, J. Operator Theory 6 (1981) 55–74.

[H2] David Handelman, Positive polynomials and Product Type Actions of Compact Groups, Mem. Amer. Math. Soc. 320 (1985) Amer. Math. Soc., Providence.

[H3] David Handelman, Deciding Eventual Positivity of Polynomials, Ergodic Theory and Dynamical Systems 6 (1986) 57–79.

[H5] David Handelman, Positive polynomials, convex integral polytopes, and a random walk problem,

Lecture Notes in Mathematics 1282 (1987) Springer-Verlag, New York and Heidelberg.

[H6] David Handelman, Reducible topological Markov chains via K0-theory and Ext1, Contemporary Mathematics 10 (1982) 41–76.

[H7] David Handelman, Effectiveness of an affine invariant for convex polytopes, J. Pure and App. Alg. 66 (1990) 165–184.

[H8] David Handelman, Iterated Multiplication of characters of compact connected Lie groups, J. Alg. 173 (1995) 67–96.

[H9] David Handelman, Representation rings as invariants for compact groups and limit ratio theorems for them, Int. J. Math. 4 (1993) 59–88.

[HR] David Handelman & Wulf Rossmann, Actions of compact groups on C*-algebras, Illinois J. of Math. 29 (1985) 51–95.

[INN] I. Iscoe, P. Ney, and E. Nummelin, Large deviations of uniformly recurrent Markov additive processes,

Adv. App. Math. 6 (1985) 373–412.

[Kr] W. Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980) 239–250. [MT1] B. Marcus and S. Tuncel, The weight per symbol polytope and scaffolds of invariants associated with

Markov chains, Ergodic Theory and Dynamical Systems 9 (1991) 129–180.

[MT2] B. Marcus and S. Tuncel, Matrices of polynomials, positivity, and finite equivalence of Markov chains,

J. Amer. Math. Soc. 6 (1993) 131–147.

[Me] E. Meissner, Uber positive Darstellung von positiven Polynomen,¨ Math. Ann. 70 (1911) 223–235. [Od] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Math. (1985) Springer-Verlag.

[P] W. Parry, Problems and Perspective in the Theory of Markov Shifts, Lecture Notes in Mathematics 1342 (1988) 626–637.

[Po] H. Poincar´e, Sur les ´equations alg´ebriques, C. R. Acad. Sci. Paris 93 (1883) 1418–1414.

[PS] W. Parry and K. Schmidt, Natural coefficients and something for Markov chains, Inventiones Math. 76 (1984) 15–32.

[PT] W. Parry and S. Tuncel, On the Classification of Markov Chains by Finite Equivalence, Ergodic Theory and Dynamical Systems 1 (1981) 303–335.

[Sw] R. Swan, Topological examples of projective modules, Trans. Amer. Math. Soc. 230 (1977) 201–234. [Tu2] S. Tuncel, A dimension, dimension modules, and Markov chains, Proc. LMS 46 (1983) 100–116. [Tu2] S. Tuncel, Faces of Markov chains and matrices of polynomials, Contemporary Mathematics 135

(1992) 391–422.

[VK] A. M. Vershik & S. V. Kerov, The K-Functor (Grothendieck Group) of the Infinite Symmetric Group,