Brouwer and Weyl Revisited:Proposals for a Noncommutative Continuum

Vladimir Tasic

Abstract

The paper aims to capture some of the mathematical features of the in-tuitionist continuum, loosely based on the ideas of Brouwer and Weyl, in away that does not require any changes in the foundational framework. Theapproach presented here relies on the theory of semigroup C∗-algebras andsemigroup crossed products. It is intriguing that algebras arising in this con-text are of independent interest in mathematical physics and noncommutativegeometry.

1 Introduction

The intuitionist concept of the continuum was introduced by Brouwer in the 1910sand elaborated by Weyl during his “revolutionary” phase in the 1920s. A concisedescription of the guiding idea appears in Brouwer’s 1914 review of a then-standardtext on set theory, where he criticizes the mainstream concept of the reals: thecontinuum should be regarded not as a set of points, but as the collection of “sequencesof nested intervals whose measure converges to zero.”

By nested sequences of intervals Brouwer and Weyl meant dyadic rational inter-vals [m−1

2n, m+1

2n] ⊆ Q, m,n ∈ Z. The construction presented in this essay includes this

case, but allows different sets of intervals; indeed, the algebras depend on the choiceof discretization. It is also convenient to allow half-open intervals in some representa-tions. The key idea adopted from Brouwer and Weyl is that nested interval sequencesthemselves should be considered the real numbers (see [19]):

We call such an indefinitely proceedable sequence of nested intervals a point P

or a real number P . We must stress that the point P is the sequence [. . . ] itself;

not something like “the limiting point” to which, according to the classical view,

the intervals converge [. . . ]. Every one of these intervals [. . . ] is therefore part

of the point P .

In Brouwer’s view, shaped by his philosophy of mathematics, the interval se-quences should be viewed as developing or unfolding; at each stage we are givenonly an initial segment, which can unfold in many different ways. Rather than a setof static and absolute points—the real numbers—the continuum should be seen asdynamical in some sense.

1

The guiding principle of this inquiry is to look for an algebraic structure thatcaptures these views and simulates them within the framework of standard (that is,non-intuitionist) mathematics. The structure should (1) represent descending intervalsequences, (2) be a dynamical system (in some sense), and (3) have enough algebraicand analytic structure to allow for interesting connections with contemporary math-ematics.

The models proposed here are C∗-dynamical systems: crossed products of a com-mutative C∗-algebra by a semigroup associated in a natural way to descending se-quences of intervals. This approach in essence merely reorganizes available techniqueswith a different interpretation in mind. To motivate this interpretation, the article iswritten in a verbose mode and includes informal discussions of intermediate steps aswell as some curious asides.

Let me begin with an idea that relates to practical computation rather than tofoundational questions.

2 First attempt: a group algebra

A relevant (commutative) algebra of generalized intervals was studied in the 1950s byWarmus [20] in the context of numerical analysis and computer arithmetic. Warmus’salgebra requires admitting “imaginary” intervals such as [1, 0], which amounts todefining [a, b] = [mina, b,maxa, b]. With this convention, operations are definedso that ε2 = 1 for ε = [−1, 1] and every (generalized) interval can be written uniquelyin the form x + εy = [x − y, x + y]. The interval algebra is isomorphic to P =R[ε]/(ε2−1), the algebra of paracomplex numbers. Numerical analysts have developeda convenient graphic representation by plotting ε-values on the vertical axis, so thate.g. [1, 4] = 5

2+ 3

2ε:

versities. A web site at the University of Texas atEl Paso (www.cs.utep.edu/interval-comp) pro-vides links to these groups as well as a usefularchive of historical documents. The journal Reli-able Computing (formerly Interval Computations) isthe main publication for the field; there are alsomailing lists and annual conferences. Implemen-tations of interval arithmetic are available bothas specialized programming languages and as li-braries that can be linked to a program written ina standard language. There are even intervalspreadsheet programs and interval calculators.

One thing the interval community has been ar-dently seeking—so far without success—is sup-port for interval algorithms in standard comput-er hardware. Most modern processor chips comeequipped with circuitry for floating-point arith-metic, which reduces the process of manipulatingsignificands and exponents to a single machine-language instruction. In this way floating-pointcalculations become part of the infrastructure,available to everyone as a common resource.Analogous built-in facilities for interval compu-tations are technologically feasible, but manufac-turers have not chosen to provide them. A 1996article by G. William Walster of Sun Microsys-tems asks why. Uncertainty of demand is surelyone reason; chipmakers are wary of devoting re-sources to facilities no one might use. But Walstercites other factors as well. Hardware support forfloating-point arithmetic came only after theIEEE published a standard for the format. Therehave been drafts of standards for interval arith-metic (the latest written by Dmitri Chiriaev andWalster in 1998), but none of the drafts has beenadopted by any standards-setting body.

GotchasAlthough the principles of interval computingmay seem obvious or even trivial, getting the al-gorithms right is not easy. There are subtleties.There are gotchas. The pitfalls of division by aninterval that includes zero have already beenmentioned. Here are a few more trouble spots.

In doing arithmetic, we often rely on mathe-matical laws or truths such as x + –x = 0 and(a + b)x = ax + bx. With intervals, some of theserules fail to hold. In general, an interval has noadditive inverse; that is, given a nondegenerateinterval [u–, u–], there is no interval [v–, v–] for which[u–, u–]+[v–, v–] = [0,0]. There is no multiplicative in-verse either—no pair of nondegenerate intervalsfor which [u–, u–]![v–, v–] = [1,1]. The reason is clearand fundamental: No valid operation can ever di-minish the width of an interval, and [0,0] and [1,1]are intervals of zero width.

The distributive law also fails for intervals. Inan expression such as [1,2]! ([–3,–2] + [3,4]), itmakes a difference whether you do the additionfirst and then multiply, or do two multiplicationsand then add. One sequence of operations givesthe result [0,4], the other [–3,6]. Strictly speaking,either of these results is correct—both of them

bound any true value of the original expression—but the narrower interval is surely preferable.

Another example: squaring an interval. Theobvious definition [x–, x–]2 = [x–, x–]! [x–, x–] seems towork in some cases, such as [1,2]2 = [1,4]. Butwhat about [–2,2]2 = [–4,4]? Whoops! The squareof a real number cannot be negative. The sourceof the error is treating the two appearances of[x–, x–] in the right-hand side of the formula as ifthey were independent variables; in fact, what-ever value x assumes in one instance, it must bethe same in the other. The same phenomenon canarise in expressions such as 2x/x. Suppose x isthe interval [2,4]; then naïvely following the rulesof interval arithmetic yields the answer [1,4]. Butof course the correct value is 2 (or [2,2]) for anynonzero value of x.

Comparisons are yet another murky area.Computer programs rely heavily on conditionalexpressions such as “if (x < y) then....” When xand y are intervals, the comparison gets tricky. Is[1, 3] less than [2, 4], or not? Whereas there arejust three elementary comparisons for pointlikenumbers (<, = and >), there are as many as 18well-defined relations for intervals. It’s not al-ways obvious which one to choose, or even howto name them. (Chiriaev and Walster refer to“certainly relations” and “possibly relations.”)

Finally, look at what happens if a naïve im-plementation of the sine function is given an in-terval argument. Sometimes there is no prob-lem: sin([30°,60°]) yields the correct interval[0.5,0.866]. But sin([30°,150°]) returns [0.5,0.5],which is an error; the right answer is [0.5,1.0].What leads us astray is the assumption that in-terval calculations can be based on end pointsalone, which is true only for monotonic functions(those that never “change direction”). For otherfunctions it is necessary to examine the interior ofan interval for minima and maxima.

In fairness, it should be noted that many cher-ished mathematical truths fail even in ordinary

2003 November–December 487www.americanscientist.org

[–3,2]

0 1–1 2–2 3–3 4–4 5–5 6–6

1

2

3

4

5

6

[1,4]

Figure 4. Diagrammatic scheme introduced by Zenon Kulpa of thePolish Academy of Sciences represents an interval as a point on aplane, somewhat like the representation of a complex number. Positionof the point along the horizontal axis gives the value of the midpoint ofthe interval; height on the vertical axis encodes the radius (or half thewidth) of the interval. Diagonals extended to the horizontal axis revealthe interval itself. Any point within the shaded region represents aninterval that includes the value zero.

Warmus noted various properties of what he called “approximate numbers”, in-cluding the analog of the trigonometric form of complex numbers (with hyperboliccosine and sine). The same algebra had been studied by Clifford, among others, inthe context of mechanics in non-Euclidean spaces.1

1It would perhaps be interesting to review the applications of paracomplex differential geometryin mathematical physics (see [6]) in the light of Warmus’s interpretation of P as an algebra ofgeneralized intervals. This is not a direction I have pursued, but it is useful to keep in mind aconnection with special relativity discussed, for example, by Kauffmann [12]: P has the structure ofthe Minkowski space R1,1, with the metric given by the modulus function |a + bε| =

√a2 − b2.

2

From the perspective I am adopting here, the algebra P is not the right settingfor modelling the continuum as understood by Brouwer and Weyl. The notion of“generalized intervals” seems unnatural and the algebra does not reflect the mainfeature of the Brower-Weyl approach: the subset order of intervals, from which thecontinuum is constructed by considering descending chains. Tracking the order ofintervals involves, as it turns out, a passage to the noncommutative world.

Let us discard the “imaginary” intervals from the algebra P, those with negativeε-component; we also remove points, that is, intervals of zero length. We are left withP+ = a+εb | a, b ∈ R, b > 0, which can be identified with the connected componentof identity of the real affine group, Aff0(R) ∼= RoR×+. We can now impose the groupmultiplication on intervals. In effect, we think of intervals a + bε as matrices µ(a, b)in the lower-triangular matrix representation of the affine group:

µ(a, b)µ(c, d) =

(1 0a b

)(1 0c d

)=

(1 0

a+ bc bd

)= µ(a+ bc, bd).

The significance of this choice of multiplication of intervals is that the group operation,unlike paracomplex multiplication, reflects the subset order:

Lemma 1 Let a + bε, c + dε ∈ P+. Then a + εb ⊆ c + εd if and only if there is ax+ εy ∈ P+ such that |x|+ y ≤ 1 and µ(a, b) = µ(c, d)µ(x, y).

Proof: For c+ εd ∈ P+, let αc,d denote the affine map t→ c+ dt represented by thematrix µ(c, d). Then the interval c+εd = [c−d, c+d] is the image of ε = [−1, 1] underαc,d. For x + εy ∈ P+, |x| + y ≤ 1 iff αx,y[−1, 1] = [x − y, x + y] = x + yε ⊆ [−1, 1],hence

(c+ εd)(x+ εy) = c+ dx+ εdy

= αc+dx,dy[−1, 1]

= αc,dαx,y[−1, 1]

⊆ αc,d[−1, 1]

= c+ εd.

Conversely, if a+εb ⊆ c+εd, then αa,b[−1, 1] ⊆ αc,d[−1, 1], so α−1c,dαa,b[−1, 1] ⊆ [−1, 1].

Therefore x+εy = (c+εd)−1(a+εb) = α−1c,dαa,b[−1, 1] ⊆ ε, so (c+εd)(x+εy) = (a+εb)

and |x|+ y ≤ 1.

Thus the reverse inclusion of intervals corresponds to right multiplication by subin-tervals of ε = [−1, 1]. These elements form a submonoid MR of Aff0(R), which can beidentified with the monoid of affine maps αa,b(t) = a+ bt preserving the base interval:αa,b[−1, 1] = a + bε ⊆ [−1, 1]. Semigroup words represent descending sequences ofintervals: for s1, . . . , sn ∈MR

s1 ⊇ s1s2 ⊇ · · · ⊇ s1s2 · · · sn.

Note that by Lemma 1 the lattice of subintervals of [−1, 1] embeds into the latticeof right ideals of the monoid MR by sending an interval to the principal right ideal it

3

generates in MR:

a+ bε ⊆ c+ dε ⇐⇒ (a+ bε)MR ⊆ (c+ dε)MR.

It is also clear that the multiplication (a+ bε)(c+ dε) is defined so that the result isthe image of the interval c+ dε = [c− d, c+ d] under the affine map αa,b(t) = a+ bt.This action of affine contractions is the main idea behind the approach proposed here,but this idea will have to be refined. For the moment I only note that (a submonoidof) the affine group somehow reflects inclusion, and that I should be more preciseabout it. We still have to define addition.

Since matrix addition takes us outside of the group, this operation will have tobe introduced by artifice. Rather than making an arbitrary choice, take the mostgeneral algebra containing the affine group. The real group algebra, for example,has the requisite universal property. In this picture, the continuum has differentrepresentations, among which only the simplest one, induced by trivializing the entiregroup, gives the algebra of real numbers.

It seems desirable to refine the picture by introducing analytic tools. This couldbe done in a number of ways; choice is not limited by a philosophical commitment toBrouwer’s foundational framework. This allows us to use all the available instrumentsof modern analysis, including operator theory. For example, one may pass to thegroup C∗-algebra, which has a rich analytic structure and a universal property withrespect to unitary representations of the group.

Since Aff0(R) is amenable, the universal and reduced C∗ algebras are isomorphic.Hence a preliminary “C∗ model” of the continuum can be identified with the algebragenerated by unitary operators Ug that act by affine transformations on the space ofL2-functions on the set of approximate measurements (the affine group, consideredas the set of intervals): for f ∈ L2(Aff0(R)), Ug(f(h)) = f(g−1h) = f(α−1

g [h]). Hereα−1g [h] denotes the inverse image of the interval h under the affine map αg determined

by αg[−1, 1] = g.The algebra C∗(Aff0(R)) is not quite the model I have in mind. To capture

descending sequences in particular, we should work with a semigroup algebra and withrational intervals instead of real ones. That is indeed how models of the continuumwill be defined below. Nevertheless, at this preliminary stage it seems worth notingthat the interpretation of C∗(Aff0(R)) as an interval algebra may be of interest inmathematical physics.

The following section briefly discusses this interpretation and its relation to analgebra that has been studied in mathematical physics, but is not required for thesubsequent arguments. I include it to emphasize that the relation to algebraic struc-tures of interest in physics, such as paracomplex numbers, is preserved; indeed, forreasons that seem to me surprising and not entirely clear, a connection with math-ematical physics is maintained also in the construction of continua based on ideassuggested above and elaborated in later sections.

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3 Interval algebra and λ-Minkowski spacetime

Consider two self-adjoint operators x and y satisfying the λ-Minkowski relation foran unspecified real parameter λ > 0: [x, y] = iλy. The universal envelope U of thisLie algebra is interpreted as the coordinate algebra of a noncommutative deformationof the Minkowski spacetime. The corresponding Lie group has been called the λ-Minkowski group Mλ; it is the semidirect product R oα R, with the action of t ∈ R(the second component) given by the automorphism of ϕt ∈ Aut(R,+): ϕt(s) = e−λts.Explicitly, the group operation is (s, t)(u, v) = (s+ e−λtu, t+ v) [2]. By an analog ofWeyl quantization, or a noncommutative Fourier transform, this group is representedby compact operators on L2(R) (see [2, 8, 16] for details). The algebraMλ generatedby this unitary representation is then regarded as the “momentum algebra” of λ-Minkowski spacetime. As an immediate consequence, we have:

Proposition 1 The momentum algebraMλ of λ-Minkowski spacetime is a represen-tation of the interval algebra C∗(Aff0(R)). Up to equivalence, the only two noncom-mutative representations of C∗(Aff0(R)) are the algebras Mλ with λ > 0 or λ < 0.

Proof: Mλ is a reparametrization of the affine group. Recalling that the groupoperation in Mλ is given by (s, t)(u, v) = (s+e−λtu, t+v), an easy matrix computationverifies that the bijection

(s, t)→(

1 0s e−λt

)

is an isomorphism Aff0(R) ∼= Mλ. By definition,Mλ is a representation of C∗(Mλ) ∼=C∗(Aff0(R)). By a well known result of Zep [21], C∗(Aff0(R)) has only two nontrivialirreducible representations. One is Mλ with λ > 0; the other with λ < 0.

The λ-Minkowski spacetime (or “κ-Minkowski” if λ is replaced by κ = 1/λ) isa noncommutative deformation meant to reflect the impossibility of fully knowingthe short-distance structure of spacetime [2]. However, difficulties arise with physicalinterpretations of the λ-Minkowski relation. When the operators x and y are taken torepresent the time and space axes, noncommutative effects are directly proportionalto one of these observables. For example, Dabrowski and Piacitelli [8] identify ywith the space axis and note: “contrary to any physical expectations, [λ-Minkowski]exhibits very large noncommutative effects at large (e.g. cosmic) distances from aprivileged point of the space.”

The interpretation discussed here might offer a solution of this puzzle. The affinegroup, which I identified with intervals, can also be identified with the upper half-plane model of the hyperbolic plane. The λ-Minkowski group registers this in termsof canonical coordinates (x, e−λy), with e−λy representing interval half-length. Theclassical limit, from the interval perspective, should be the limit in which intervalscollapse to points: the line at infinity of the half-plane model. Thus it is naturalthat noncommutative effects are more pronounced for large values of y, which nowcorrespond not to “large distances” but to small interval lengths (“quantum scale”measurements).

5

In a sense, Mλ represents the algebraic and order structure of intervals ratherthan the structure of spacetime: noncommutativity reflects limits on the precision ofmeasurements, but in our setting this is a property of the mathematical framework,not of a physical system. It would therefore be all the more interesting to revisit theresults on λ-Minkowski spacetime, in particular its relation to quantum groups [15, 16]and noncommutative differential calculus, in the light of the interval interpretation.

The main objective here, however, is to exhibit an analogous algebra of rationalintervals, which are represented by the rational affine group in the same way as above.Hence a model of the continuum should be sought within the algebra C∗(Aff+(Q)).By standard results, the algebra is isomorphic to C∗(Q) o Q×+ ∼= C(Q) o Q×+. The

dual group of Q is Q ∼= AQ/Q, the quotient of the adeles by the rationals, which is

homeomorphic to [0, 1)× Z. Therefore the rational interval algebra is C([0, 1)× Z)oQ×+.

The underlying space is the product of a real interval and Cantor space Z =∏

p Zp,so that, loosely speaking, the rational interval algebra can be expected to exhibitboth smooth and “fractal” or p-adic features. The action of Q×+ is multiplication,

with generators 1/n acting as contractions of the real interval and expansions of Z,and in this informal sense the dynamics of the rational interval algebra resemble thehyperbolic structure of an Anosov map or a Smale space.

We are after a “C∗ model” of the Archimedian continuum, and hence the monoidI choose is biased to one side of the broader dynamics. Nevertheless, the ultrametricside sometimes reappears: for example, some representations contain the Bost-Connesalgebra C(Z) oN×, a C∗ dynamical system arising in number theory [4].

4 Continuum as a semigroup algebra

Lemma 1 has the obvious rational analog, and the remarks following it apply. Thusthe set of subintervals of [−1, 1] ∩ Q is a submonoid of Aff+(Q), and it reflects de-scending chains of intervals by right multiplication. Denote this submonoid by M ;it consists of intervals a + bε with a ∈ Q, b ∈ Q+, and |a| + b ≤ 1. I also view it,equivalently, as the monoid of the corresponding matrices µ(a, b) ∈ Aff+(Q); and asthe monoid of affine maps αa+bε(t) = a + bt with the conditions on a and b ensuringthat the maps are contractions of the base interval.

The monoid M lives in an amenable group, and the semigroup algebra can bedefined by restricting the left regular representation of the group. With this remarkand motivation discussed above, the following algebra seems adequate as a model ofthe continuum:

Definition 1 The basic model of the continuum is the semigroup C∗ algebra C(M)of the submonoid M of the rational affine group, defined as the restriction to `2(M)of the left regular representation of Aff+(Q).

The situation corresponds exactly to Nica’s construction of semigroup C∗-algebrasfor quasi-lattice ordered groups [18], which I briefly review. Let G be a group and P

6

a submonoid of G such that P ∩ P−1 = 1. Then P induces a left invariant partialorder on G by x ≤P y ⇐⇒ x−1y ∈ P ⇐⇒ yP ⊆ xP . The group G is said to bequasi-lattice ordered if in the poset (G,≤) all finite subsets that have a common upperbound in P , have the least common upper bound in P . The conditional supremum ofx and y, if it exists, is denoted by x∨ y. The (reduced) semigroup algebra C∗(G,P ),as defined by Nica [18], is the algebra generated by the isometries vx (x ∈ P ), actingon the space `2(P ) by left translation: vx(y) = xy.

In the special case of M and Aff+(Q), by Lemma 1, the partial order inducedon Aff+(Q) by M is the reverse inclusion of intervals: x ≤ y iff y ∈ xM iff y ⊆ x.Therefore x ∨ y is the largest interval in M contained in x ∩ y, if there is an intervalcontained in x∩ y. This does not happen only when x and y are disjoint or intersectin exactly one point. Thus, in our setting, x ∨ y = x ∩ y if x ∩ y is a proper interval;otherwise x∩ y is empty or a point, hence contains no interval and x∨ y is undefined(sometimes this is denoted by “x ∨ y =∞”, a notation I do not use).

The structure of the algebra C(M) = C∗(Aff+(Q),M) can be described as follows.For every interval x ∈ M , let Px denote the projection on the subspace `2(xM) ⊆`2(M). Since the principal right ideal xM C M consists of intervals contained inx, Px is the projection onto the subspace generated by subintervals of x. Theseprojections generate an abelian algebra BM since PxPy = Px∩y (with the conventionthat Px∩y = 0 if x∩y is empty or a point). The monoid M acts on the algebra BM byendomorphisms ϕx(Py) = Pxy. The endomorphisms are implemented by isometries vxsuch that vxPyv

∗x = Pxy (and hence vxv

∗x = Px). The algebra C(M) = C∗(Aff+(Q),M)

is generated by the isometries vx, and is the semigroup crossed product BM oϕM .It is useful to have in mind a dual description. Let ΩM be the space of filters

on M (considered as a poset). With the product topology induced from 2M , ΩM isa compact space now known as the Nica spectrum, and C(ΩM) is the dual of thecommutative algebra generated by projections Px = vxv

∗x, x ∈ M . The dual of Px

is the principal filter xM−1 ∩M , that is, the set of intervals in M that contain x;ultrafilters in ΩM correspond to points in the real interval [−1, 1]. Since M acts onthe projections by left translation, there is a dual action on C(ΩM): x ∈ M sendsthe filter aM−1 ∩M ∈ ΩM to βx(aM

−1 ∩M) = xaM−1 ∩M . The maps βx have leftinverses β−1

x (aM−1 ∩M) = x−1(x ∩ a)M−1 ∩M , and this induces an action of M onC(ΩM) by endomorphisms: vx(f) = f β−1

x . By Nica’s results, C(M) is isomorphicto the crossed product C(ΩM) oM .

In topos-theoretic foundations of physics the lattice of right ideals of a monoid—which has the structure a Heyting algebra—has been used to define a semantics ofpropositions, for example by Isham [11]. Our model of the continuum has a similarfeature, but does not require changes in the foundational framework.

Theorem 1 The maximal abelian subalgebra of the continuum C(M) provides a nat-ural Heyting algebra semantics, with propositions represented by projections associatedto right ideals of the monoid M .

Proof: The maximal abelian subalgebra is generated by projections Px, x ∈M , cor-responding to principal right ideals xM . Since PxPy = Px∩y represents the principal

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ideal xM ∩ yM = (x∩ y)M (with the convention that (x∩ y)M = ∅ if x∩ y is emptyor a point), and Px + Py − PxPy represents the ideal xM ∪ yM , finite intersectionsand unions of principal right ideals of M are represented by projections. Let I / Ma right ideal, and let I = xk | k ∈ N be an enumeration of its elements. ThenI = ∪k≥0Ik with Ik = x0M ∪ x1M ∪ . . . ∪ xkM . The projections representing theideals Ik form an increasing sequence, which therefore converges. The limit of thissequence is the projection representing the ideal I. The Heyting algebra structureis defined as in the lattice of right ideals of M . For example, PI ⇒ PJ is PK , theprojection corresponding to the largest ideal K such that I ∩K ⊆ J . Negation ¬PIis defined as PI ⇒ 0: the projection corresponding to ¬I, the largest ideal K suchthat I ∩K = ∅. This is in effect an embedding of the Heyting algebra of right idealsof M into the maximal abelian subalgebra of C(M).

Note that there is no nontrivial ideal I for which PI ∨ ¬PI = 1. In particular,when x1, . . . , xn is a partition of the base interval, the projections Pxi do not addup to 1 since M 6= x1M ∪ . . . ∪ xnM . Consequently:

Corollary 1 The continuum C(M) contains all Cuntz-Toeplitz algebras En.

Proof: Given any partition x1, . . . , xn of the base interval, let vi be the isometryof `2(M) corresponding to the left multiplication by xi. Since the intervals are dis-joint, these isometries have orthogonal ranges `2(xiM), and hence satisfy the relationsv∗i vj = δij. Therefore the subalgebra generated by v1, . . . vn is a homomorphic imageof the Cuntz-Toeplitz algebra En. Note that ε is not contained in any of the xi, soviv∗i (ε) = 0 for all i, and ε is mapped to zero by the projection

∑ni=1 viv

∗i . Hence the

kernel of the homomorphism En → C∗(v1, . . . vn) does not contain the unique propernonzero ideal of En, generated by 1−∑n

i=1 viv∗i ; the kernel is therefore zero, and the

homomorphism is an isomorphism.

Recent results of Li [14] give a description of semigroup C∗-algebras in termsof generators and relations, with the generators of the maximal abelian subalgebraindexed by certain families of right ideals of the semigroup. In the special case of M ,this family consists of principal right ideals, so generators can be indexed by elementsof M :

Theorem 2 The continuum C(M) is the universal C∗ algebra generated by projec-tions ex |x ∈M∪e∅ and isometries vx | x ∈M subject to the relations e∅ = 0,e1 = 1, and for all x, y ∈M :

1. vxy = vxvy

2. vxeyv∗x = exy

3. exey =

ex∩y if |x ∩ y| > 10 if |x ∩ y| ≤ 1

.

8

From this description it is clear that C(M) has a continuous one-parameter groupof automorphisms σt, t ∈ R, determined by its action on the generators vx: σt(vx) =|x|itvx, where |x| denotes the half-length of the interval x. Since half-length of x isjust the ε-component of x, namely the determinant of the matrix representing x, themapping x → |x| is the restriction to M of the group homomorphism Aff+(Q) =QoQ×+ → Q×+, so |xy| = |x||y| for all x, y ∈M .

Corollary 2 The one-parameter automorphism group σt is implemented on the space`2(M) by the Hamiltonian ∆ξx = ln(|x|)ξx: for all v ∈ C(M),

σt(v) = eit∆ve−it∆.

Proof: It suffices to check this on the generators vx, which act on `2(M) by leftmultiplication in M . For y ∈M we have eit∆ξy = |y|itξy and hence

eit∆vxe−it∆ξy = eit∆vx|y|−itξy

= |y|−iteit∆ξxy= |y|−it|xy|itvxξy= |x|itvxξy

Therefore σt(vx) = eit∆vxe−it∆.

4.1 Cone representation

For x = a + bε ∈ M , let Tx = Ta+bε denote the closure of the right ideal xM as asubset of R2. It is a triangle of height b with the interval [a− b, a+ b] as the baselinecentred at a. More precisely,

Ta+bε =

(s, t) ∈ R× R≥0 | |a− s| ≤ b− t, b− t ≥ 0.

Borrowing the terminology of special relativity, Tx is the restriction to R × R≥0 ofthe past cone of x. The closure of xM−1 in R2 is the future cone of x in the usualsense: with C+ = (s, t) ∈ R2 | |s| ≥ t ≥ 0 and C+(a, b) = (a, b) + C+, the closureof (a + bε)M−1 is C+(a, b). Since future cones are principal filters in the poset ofintervals, it is easy to verify that a + bε ⊆ c + dε iff C+(c, d) ⊆ C+(a, b). Thus themaximal abelian subalgebra C(ΩM) can be seen as the algebra of continuous functionson a compactification of the poset of future cones that are close enough to the origin(that is, have lower bounds in ε; these are precisely the future cones C+(a, b) witha+ bε ∈MM−1).

In the representation of intervals by elements of the affine group, Tε is the closureof M in the topology of R2 The mapping x→ x is a monoid and poset embedding ofM into Tε. Let τx : R2 → 0, 1 be the characteristic functions of the cones Tx, x ∈M ,and denote by D the commutative algebra of functions generated by τx | x ∈ M.We identify D with its standard representation by multiplication operators on theHilbert space L2(Tε), with Lebesgue measure induced from R2. Elements x ∈ M

9

act on the algebra D by endomorphisms ϕx(τy) = τxy and we want to show that thecrossed product D oϕM is a homomorphic image of the continuum C(M).

For x ∈M , define the operators Vx on L2(Tε) by Vxf(t) = τε(x−1t)f(x−1t). Then

V ∗x f(t) = f(xt), so

V ∗x Vxf(t) = V ∗x τε(x−1t)f(x−1t) = τε(x

−1xt)f(x−1xt) = f(t).

Thus the operators Vx are isometries. Furthemore, conjugation by Vx implements theendomorphism ϕx. To see this, note that τε(x

−1t) = 1 iff x−1t ⊆ ε iff t ⊆ xε = x(because inclusion is left invariant), so τε(x

−1t) = τx(t). Since xy ⊆ x, we also haveτxτxy = τxy. Using these identities,

VxτyV∗x f(t) = Vxτy(t)f(xt)

= τε(x−1t)τε(y

−1x−1t)f(xx−1t)

= τx(t)τxy(t)f(t)

= τxyf(t).

In particular, since τε(t) = 1 for all t ∈ Tε, we have VxV∗x = τx. Therefore VxV

∗x is

a self-adjoint projection onto the subspace L2(Tx). It remains to verify the relations(3) of Theorem 2.

Clearly, τxτy = τx∩y. Therefore the relations (3) hold when x ∩ y ∈ M , and alsowhen x ∩ y = ∅; the only remaining possibility is that x ∩ y is a point (hence not inM), in which case the relations (3) require that τxτy = 0. But if x ∩ y is a point,τx∩y(t) = 0 is true for all t with one possible exception, when t = x ∩ y = p forsome p ∈ Q; then t = x ∩ y ∈ Tε, so τx∩y(t)f(t) = f(p), which is not necessarilyzero. Nevertheless, τxτy = τx∩y = 0 holds almost everywhere in the L2 sense. Thusall the relations of Theorem 3 are fulfilled with vx = Vx and ex = τx. The algebraD oφM is a homomorphic image of C(M): the cone representation.

The argument of Corollary 1 applies:

Corollary 3 The cone representation of the continuum C(M) contains Cuntz-Toeplitzalgebras En for all n.

4.2 Real line representations

For x = [p, q] ∈M , denote by [x) the real interval [p, q), and let C#[−1, 1) denote thealgebra of functions [−1, 1)→ C generated by the characteristic functions Ex = χ[x)

of real intervals [p, q) with p, q ∈ Q ∩ [−1, 1]. With this notation, C#[−1, 1) isthe algebra generated by the commuting projections Ex | x ∈ M and consists offunctions on [−1, 1) that are right continuous at every rational point, have a finiteleft limit at every rational point, and are continuous elsewhere. The monoid M actson the algebra C#[−1, 1) by endomorphisms ϕx(Ey) = Exy.

Recalling that each x = a + bε ∈ M is the image of [−1, 1] under the affine mapαx(t) = a+ bt, define for x ∈M the operator Vx on C#[−1, 1) by

Vx(f(t)) = Eε(α−1x (t))f(α−1

x (t)).

10

Then Vxy = VxVy, so x→ Vx is a representation of M by endomorphisms of C#[−1, 1),and we have V ∗x (f(t)) = f(αx(t)). The operators Vx are isometries:

V ∗x Vx(f(t)) = Eε

(α−1x αx(t)

)f(α−1x αx(t)

)= f(t).

To see that conjugation by these isometries implements the action ϕ, first note thatEε α−1

y α−1x = Eε α−1

xy = Exy. Since xy ⊆ x, we have ExExy = Exy, and therefore

VxEyV∗x (f(t)) = Eε(α

−1x (t))Eε(α

−1y α−1

x (t))f(αx α−1x (t))

= ExExyf(t)

= Exyf(t)

Thus VxEyV∗x = Exy; in particular, VxV

∗x = Ex. Nica’s covariance condition is fulfilled:

VxV∗x VyV

∗y = ExEy = Ex∩y = Vx∩yV

∗x∩y.

It remains to verify the relations (3) of Theorem 2. These follow from the covarianceformula, since x ∩ y /∈ M iff [x) ∩ [y) = [x ∩ y) = ∅. Therefore C#[−1, 1) oϕ M is ahomomorphic image of C(M). Let us call it the local representation on the real line:C[−1,1)(M).

It is also possible to define a global representation on the real line, by letting Mact on the algebra C#

0 (R) generated by the characteristic functions of all real intervals[p, q) with p, q ∈ Q. Since M ⊆ Aff+(Q), the action Ux(f(t)) = f(α−1

x (t)), x ∈ M ,is just the left regular action of the affine group. The operators Ux are unitary, soC#

0 (R)oM ∼= C#0 (R)oAff+(Q). Denote this algebra by CR(M). In a sense, the local

representation C[−1,1)(M) is a “neighbourhood of zero”, and the global representationis built up from translates of the local ones:

Theorem 3 Let e = χ[−1,1) and let K be the algebra of compact operators on `2(Z).Then C[−1,1)(M) = eCR(M)e and CR(M) ∼= C[−1,1)(M)⊗K.

Proof: Let g ∈ Aff+(Q) and consider the unitary Ug ∈ CM(R). Then

eUgef(t) = χ[−1,1)Ugχ[−1,1)(t)f(t)

= χ[−1,1)(t)χ[−1,1)(α−1g (t))f(α−1

g (t))

= χ[−1,1)(t)χ[g)(t)f(α−1g (t))

= χ[−1,1)∩[g)(t)f(α−1g (t))

= χ[ε∩g)f(α−1g (t))

= Vxf(t)

with x = [−1, 1]∩ g ∈M . Thus, for intervals g ∈ Aff+(Q) that are too far from zero,in the sense of being disjoint from [−1, 1] (or intersecting it in one point), we haveeUge = 0: these intervals have no “causal interaction” with elements of C[−1,1)(M).

Let gn = 2n + ε = [2n − 1, 2n + 1] ∈ Aff+(Q), n ∈ Z, be the translates of thebase interval. Then en = Ugne = χ[2n−1,2n+1) is a family of orthogonal projections.

11

These projections are mutually equivalent: setting wn,m = ene∗n−m, one calculates

that w∗n,mwn,m = en and wn,mw∗n,m = em. We have

∑nk=0 Ugne = χ[−2n−1,2n+1), and

this family of projections is an approximate identity of CR(M). Therefore CR(M) ∼=C[−1,1)(M)⊗K.

Given a partition x1, . . . , xn of the base interval, it is easy to verify that therelation

∑ni=1 viv

∗i = 1 holds in the algebra C[−1,1)(M) for isometries vi = Vxi . Thus

the Cuntz-Toeptlitz subalgebras of the continuum C(M) are mapped in this image toCuntz algebras.

Corollary 4 The local representation C[−1,1)(M) contains all Cuntz algebras On.

The real line representation of the continuum contains other interesting subalge-bras, including some related to the ultrametric aspect of interval dynamics. Recallthat the Cuntz alebra QN is defined in [7] as the universal algebra generated by theunitary u and isometries sn (n ∈ N) acting on `2(Z) by uξk = ξk+1 and snξk = ξnk.

Corollary 5 The global representation CR(M) contains the Cuntz algebra QN andthe Bost-Connes algebra.

Proof: CR(M) ∼= C#0 (R)oAff+(Q). By Theorem 6.5 of Cuntz [7], C0(R)oAff+(Q) ∼=

C0(Af )oAff+(Q), where Af denotes the finite adeles. This algebra contains QN and

hence the Bost-Connes algebra. Since C0(R) ⊆ C#0 (R), the result follows.

5 Discretization by subsemigroups

The submonoids Fn associated with partitions of the unit interval are free submonoidsof M ⊆ Aff+(Q). They induce a partial order on Aff+(Q) in the same way as M .However, the order is coarser than that induced by M : if x ≤Fn y, then y ∈ xFn ⊆xM , so y ⊆ x as before; the converse does not hold in general. Therefore M gives thefinest “graining” of the continuum, with all descending sequences of rational intervalsallowed; whereas submonoids of M can be viewed as inducing coarser orders andhence coarser discretizations of the continuum. Thus the algebra of complex numbers(and its subalgebra R of self-adjoint elements) corresponds to the trivial submonoid.

This motivates the following definition.

Definition 2 Let S be any submonoid of M ⊆ Aff+(Q). The continuum correspond-ing to the discretization by intervals in S is the subalgebra C(S) ⊆ C(M) generatedby isometries vx | x ∈ S.

In the examples above, Fn is the monoid corresponding to a partition of the baseinterval into n subintervals, and C(Fn) is the Cuntz-Toeplitz algebra En. By an abuseof terminology we call any epimorphism of C∗ algebras C(S)→ A a “representation”of the continuum. With this convention, any representation of C(M) restricts to oneof C(S).

12

The free monoids associated with partitions of the base interval are only one classof discretizations by subsemigroups. A more general class of examples appeared inthe recent work of Carey, Phillips, Putnam and Rennie [5]. For λ ∈ (0, 1) ∩ Q, letZ[λ, λ−1] = mλn | m,n ∈ Z, and let Gλ be the subgroup of Aff+(Q) defined byZ[λ] o 〈λ〉. Further, let Cλ

0 (R) denote the algebra generated by the characteristicfunctions of intervals [a, b) with endpoints in Z[λ, λ−1]. The algebras Aλ are definedas Aλ = Cλ

0 (R) o Gλ where Gλ acts by affine transformations αg as earlier: forf ∈ Cλ

0 (R) and g ∈ Gλ, vg(f) = f α−1g . Finally, the subalgebras Qλ are defined as

eAλe, where e is the characteristic function of the interval [−1, 1). (Actually, the baseinterval [0, 1) is used in [5]; I am adapting the construction to the present setting byusing the interval ε = [−1, 1).)

Let Sλ = Gλ ∩M . Then Aλ and Qλ are isomorphic to the representations of thecontinuum C(Sλ), respectively obtained by restricting the representations CR(M) andC[−1,1)(M) to the subalgebra C(Sλ). Namely, CR(Sλ) = Cλ

0 (R) o Sλ, and elements ofSλ are represented in CR(Sλ) by unitary operators on Cλ

0 (R). Since Sλ generates Gλ,it follows that CR(Sλ) = Cλ

0 (R)oSλ ∼= Cλ0 (R)oGλ = Aλ. A calculation analogous to

one carried out in the proof of Theorem 3 shows that C[−1,1)(Sλ) = eCR(Sλ)e, whichis isomorphic to eAλe = Qλ. By Proposition 2.7 and Corollary 2.18 of [5], we have

Corollary 6 For λ = p/q in lowest terms, C[−1,1)(Sλ) ∼= Qλ ∼= Oq−p+1, and CR(Sλ) ∼=Aλ ∼= K ⊗Oq−p+1.

The Brouwer-Weyl monoid B consists of intervals m2n

+ 12nε with |m| < 2n, m ∈ Z,

n ∈ N. It is not difficult to see that B is generated by three elements −12

+ 12ε, 1

2+ 1

2ε,

and 12ε, and that it is not free on this set of generators. The semigroup algebra C(B)

is a model of the Brouwer-Weyl continuum. Its local representation on the real line,in the sense of Section 4.2, is the Cuntz algebra O2.

Theorem 4 The local representation C[−1,1)(B) on the real line is isomorphic to theCuntz algebra O2, and CR(B) ∼= O2 ⊗K.

Proof: CR(B) = CR(S1/2) ∼= C1/20 (R) o G1/2 since B and S1/2 are represented by

unitaries and generate the group G1/2. Hence CR(B) = CR(S1/2) ∼= A1/2 ∼= O2 ⊗K byCorollary 7. Similarly, C[−1,1)(B) = eCR(B)e = eCR(S1/2)e ∼= eA1/2e = Q1/2 ∼= O2.

The group G1/2 = Z[1/2] o 〈2〉 is one of the Baumslag-Solitar groups, and is alsoknown as “the wavelet group”. The structure of the dynamical system C∗(G1/2) =

C(Z[1/2]) oZ is well understood (see [3, 10]): it is the dyadic solenoid, the bilateralBernoulli shift. Topologically it is a Smale space, locally homeomorphic to the productof a real interval and the Cantor set of dyadic integers Z2.

J. Keesling

Figure 6. The solenoid as an inverse limit.

Figure 7. The local structure of the solenoid.

solenoids can be calculated using techniques determined by Abramov [1]. The topological entropy ofautomorphisms of general compact connected Abelian topological groups has been determined by Juzvin-sky [25]. The topological entropy of an autohomeomorphism of these may not be the same as the automor-phism to which it is associated by homotopy. So, solenoids are a special case.

4 The Knaster continuumSolenoids were the main motivation for the study of expanding hyperbolic attractors. The work of Williamswas in response to questions posed by Smale and his program of analyzing the structure of structurally stabledynamical systems [37]. His work determined the structure of expanding hyperbolic attractors. However,there are many attractors that do not fit into the category he studied. The simplest of these is the Knastercontinuum. Let n ≥ 2 be an integer. Define Kn = (I, fn) where fn : I → I is defined by

fn(t) =

n · t 0 ≤ t ≤ 1/n2 − n · t 1/n ≤ t ≤ 2/nn · t − 2 2/n ≤ t ≤ 3/n4 − n · t 3/n ≤ t ≤ 4/n

· · · · · ·

26

13

In geometric group theory, G1/2 is sometimes pictured in terms of a foliation ofthe upper half-plane model of the hyperbolic plane (see [9]), which gives an intuitivesense of the dynamics of the algebra C∗(G1/2):

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The upper and lower boundaries correspond to the 2-adic and real completions ofQ. In general, semigroup algebras should correspond to one-sided shifts, which canbe expected to include the algebras of functions on one of the boundaries. However,both boundaries appear as subalgebras of the real line representation CR(B) of theBrouwer-Weyl continuum. Denoting by Q2 the algebra C(Z2) o (Z o [2〉), where(m, 2n) ∈ Zo[2〉 acts on C(Z2) by affine endomorphisms ϕ(m,2n)f(z) = f(2−n(z−m)),we have:

Corollary 7 The real line representation CR(B) of the Brouwer-Weyl continuum con-tains the algebra C0(R) oG1/2

∼= C0(Q2) oG1/2∼= Q2 ⊗K.

Proof: Since B generates G1/2, CR(B) = C1/20 (R) o B ∼= C

1/20 (R) o G1/2. This

algebra contains C0(R) o G1/2, which by results of Larsen and Li [13] is isomorphic

to C0(Q2) oG1/2∼= Q2 ⊗K.

5.1 Ultrametric continua

The continua defined from submonoids of M are calibrated to modelling descendingchains of rational intervals. Recent literature, perhaps inspired by the Bost-Connessystem and its relation to number theory, has been emphasizing dynamical systemswhich in our setting correspond to the inverses S−1 of submonoids S ⊆ M . Thesemigroup algebras can be defined in the same manner; however, the order inducedby M−1 is subset order of intervals (rather than reverse inclusion), and coarser partialorders arise from S−1 ⊆M−1.

In the algebras C(S−1), right multiplication corresponds to expanding intervals.In the Archimedean metric of Q, this is just the opposite of “sequences of intervalswhose measure converges to zero”. But in the ultrametric setting, these operatorsproduce sequences of subsets with vanishing measure. As in the Archimedean case,monoids S ⊆M can be seen as providing different discretizations. This motivates

Definition 3 Let S be a submonoid of M . The algebra C(S−1) is a model of ultra-metric continuum discretized by S−1.

14

We discuss two examples already encountered above: the Cuntz algebra QN de-fined in [7] and the algebra Q2 of Larsen and Li [13].

Let S ⊆ M be the submonoid of matrices µ(mn, 1n) with m ∈ Z, n ∈ N+ and

|m| < n. Then S−1 consists of matrices µ(m,n) with |m| < n, and is a subsemigroupof P = ZoNo

+ ⊆ Aff+(Q), on which S−1 acts by left multiplication, and in particularacts on the copy of Z in P : µ(m,n)µ(k, 1) = µ(nk+m, 1). Thus S−1 acts on `2(Z) byisometries w(m,n)ξk = ξnk+m, with |m| < n. Let us call the algebra CZ(S−1) generatedby these isometries the `2(Z)-representation of the continuum C(S−1). Clearly thisis a subalgebra of CZ(P ), but the adjoint operation in CZ(S−1) supplies additionalisometries and these algebras are in fact equal. To see this, note that CZ(P ) is preciselyQN, defined by Cuntz [7] as the universal algebra generated by the unitary uξk = ξk+1

and isometries snξk = ξnk on `2(Z). Therefore, to show that CZ(S−1) = CZ(P ) = QN,we only have to show that CZ(S−1) contains the unitary shift u. Now,

w∗(m,n)ξ(k) =

ξ(k−m

n) k ∈ m+ nZ

0 k /∈ m+ nZ .

Hence, for k ∈ 2Z,

w(1,2)w∗(0,2)ξk = w(1,2)ξk/2 = ξ2(k/2)+1 = ξk+1;

and for k ∈ 1 + 2Z,

w(0,2)w∗(−1,2)ξk = w(0,2)ξ(k+1)/2 = ξ2((k+1)/2) = ξk+1.

Therefore u = w(1,2)w∗(0,2) + w(0,2)w

∗(−1,2) ∈ CZ(S−1).

Corollary 8 The `2(Z) representation of the ultrametric continuum C(S−1) is iso-morphic to the Cuntz algebra QN ∼= C(Z)oS−1 and contains the Bost-Connes algebra.

The Cuntz algebra QN appears to be in some sense dual to the local representationof C(S) on the real line; by a choice of submonoid of the affine group, it singles outthe ultrametric part of the C∗-dynamical system C∗(Aff+(Q)) = C([0, 1)× Z) oQ×+.

Similarly, the inverse of the Brouwer-Weyl monoid consists of matrices µ(m,n)with |m| < 2n, which is a subsemigroup of Z o [2〉 ⊆ Aff+(Q). Therefore B−1 actson `2(Z) by isometries w(m,2n)ξk = ξ2nk+m. Denote this algebra by CZ(B−1). Now,the algebra Q2 is defined in [13] as the universal algebra generated by the unitaryuξk = ξk+1 and the isometry s2ξk = ξ2k. Therefore CZ(B−1) ⊆ Q2. But the algebrasare in fact equal because the unitary u can be expressed in terms of generators ofCZ(B−1): u = w(1,2)w

∗(0,2) + w(0,2)w

∗(−1,2) ∈ CZ(B−1). The results of Larsen and Li [13]

yield

Corollary 9 The `2(Z) representation of the ultrametric continuum CZ(B−1) is iso-morphic to Q2

∼= C(Z2) oB−1.

15

6 Higher dimensions; possible generalizations

The same basic idea can be used to construct models of higher dimensional continua.The group Γn = QnoQ×+ plays the role of the affine group in the case n = 1. ClearlyΓn can be identified with a subgroup of GLn+1(Q) by

(a, b)→ µ(a, b) =

(1n 0a b

).

The multiplication is the same as the composition of affine maps αa,b : Qn → Qn

given by x → a + bx. By identifying the closed ball B(a, r) of radius r centred at awith the matrix µ(a, r), we can impose the group multiplication on the set of balls asin the case of intervals; the unit ball ε = B(0, 1) is the identity element of the group.The submonoid Mn = B(x, y) | ||x||+ y ≤ 1, y > 0 plays the role of the monoid Min the case n = 1. Namely, matrices µ(x, y) with ||x||+y ≤ 1 correspond to the affinemaps preserving the unit ball and hence act as affine contractions. Furthermore, itis easy to see that B(a, b) = αa,b[B(0, 1)]. Thus we have a higher-dimensional analogof Lemma 1, and principal right ideals of Mn determine the inclusion order of balls:

Lemma 2 B(a, b) ⊆ B(a, d) iff B(a, b) ∈ B(c, d)Mn.

I omit the proof, which is essentially the same as the proof of Lemma 1. Withthese elementary observations, one could try defining the n-dimensional continuumC(Mn) as the semigroup algebra C∗(Mn). However, Nica’s construction cannot beapplied because the group Γn is not quasi-lattice ordered by the monoid Mn: unlikein the case of intervals, in n > 1 dimensions there is no largest ball contained in theintersection of two balls.

The construction has been generalized in the recent work of Li [14], where C∗

semigroup algebras are defined for all cancellative semigroups. Each of several def-initions discussed by Li applies to Mn, a submonoid of an amenable group. Forexample, the continuum C(Mn) could be defined as the semigroup algebra obtainedby restricting to Mn the left regular representation of Γn, or as the algebra that isuniversal for covariant representations of Mn (see [14] for various definitions). We aremostly interested in representations, so it is enough to know that a general definitionis possible.

Let Ωn be the set of filters on Γn that have lower bounds in Mn. This is acompact subspace of 2Γn , and x ∈ Mn acts on it by ω → xω = xg | g ∈ ω. As inthe one-dimensional case, the monoid Mn acts on the commutative algebra C(Ωn) byoperators Vx(f(ω)) = χΩ(x−1ω)f(x−1ω), and it is easy to see that VxVy = Vxy. Thealgebra generated by these isometries is isomorphic to the semigroup crossed productC(Ωn)oϕMn, and is a a representation of the continuum C(Mn). This is the analog ofwhat we called the cone representation in Section 4.1. Namely, future cones C+(a, b)correspond to principal filters in the poset Γn, in the sense that C+(a, b) is the closurein Rn+1 of the filter B(a, b)M−1

n .As in the one-dimensional case, submonoids of Mn correspond to different dis-

cretizations of the continuum. Algebras of this type have appeared in the work of

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Manin and Marcolli on fractals and error-correcting codes [17], where Cuntz-Toeplitzalgebras are associated to codes, and codes to fractals by an iterated function system(that is, a semigroup). Manin and Marcolli use n-dimensional cubes rather than balls,and it is interesting to note that some perturbations of the metric on Qn are possiblewhile maintaining the same basic scheme.

For example, if the Euclidean norm || · ||2 is replaced by the norm ||x||1 =∑ |xi|,

the cubes can be encoded by the affine group and the inclusion relation among cubesis captured by the structure of right ideals of the monoid Mn(|| · ||1) consisting ofcubes with ||x||1 + y ≤ 1. Hence the appropriate continuum could be defined as asemigroup algebra. It would be closer to the interval case in the sense that there isno partition of the unit ball into finitely many balls, but this can be done with cubes,so the continuum would contain all Cuntz-Toeplitz algebras. The question arises ofwhether this would work for a more general metric, with an appropriate monoid, andunder what conditions.

At a minimum, it appears that continua are operator algebras far more complexthan Rn, and that this approach to modelling Brouwer’s and Weyl’s idea providesintriguing links to different areas of mathematics and physics. For example, it isknown that endomorphisms of Cuntz algebras are related to solutions of the quantumYang-Baxter equation. One wonders whether there is a deeper reason why thesepurely mathematical considerations of an algebra of intervals lead to structures ofapparent significance in physics.

7 Odds and Ends

To describe the algebra, note that for each g ∈ Γn and each right ideal I of Mn,gI ∩Mn is either empty or a right ideal of Mn. Let J be the set of finite intersectionsof right ideals J(g) = gMn ∩ Mn, g ∈ Γn, together with the empty set. This setincludes all principal ideals, since J(x) = xMn for x ∈ Mn. Furthermore, for eachx ∈ Mn, xJ(g) = xgMn ∩ xMn = J(xg) ∩ J(x) and the inverse image of J(g) underthe left multiplication by x is x−1J(g) ∩Mn = J(x−1g) ∩ J(x−1) = J(x−1g) becauseJ(x−1) = x−1Mn ∩ Mn = Mn. The family of right ideals J is closed under leftmultiplication and taking inverse images under left multiplication by elements of Mn,and hence fulfills the conditions required by Li [14] (Definition 3.2).

Definition 4 The continuum C(Mn) is the universal algebra generated by commutingprojections eI | I ∈ J ∪ e∅ and isometries vx | x ∈Mn subject to the relationse1 = 1, e∅ = 0, and

1. vxvy = vxy for all x, y ∈Mn

2. eIeK = eI∩K for all I,K ∈ J

3. for all xi, yi in Mn with x−11 y1 · · ·x−1

m ym = 1 in Γn,

v∗x1vy1 · · · v∗xmvym =m∏

k=1

eJ(y−1m xm···y−1

k xk).

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Denote by PJ(g) the projection onto the subspace `2(J(g)) ⊆ `2(Mn), and let Ddenote the commutative algebra generated by these projections. The monoid Mn

acts on the right ideals in J by left multiplication: xJ(g) = J(xg) ∩ J(x). Hence itacts on D by endomorphisms ϕx(PJ(g)) = PxJ(g). This is a representation of Mn byendomorphisms of D. Namely, ϕx(PJ(g)) = PxJ(g) = PJ(xg)∩J(x) = PJ(g)PJ(x), and

ϕxϕyPJ(g) = ϕxPJ(yg)PJ(y)

= PJ(xyg)PJ(x)PJ(xy)PJ(x)

= PJ(xyg)PJ(xy)

= ϕxyPJ(g)

since J(xy) ⊆ J(x), so that PJ(x)PJ(xy) = PJ(xy) for x, y ∈Mn. Thus we can form thesemigroup crossed product D oϕMn.

Theorem 5 D oϕ Mn is isomorphic to the C∗ algebra generated by the isometriesVx(y) = xy on `2(Mn).

Proof: The endomorphisms ϕx are implemented on `2(Mn) by the isometries Vx.To see this, suppose y ∈ xJ(g) = J(xg) ∩ J(x). Then y ∈ J(x) and x−1y ∈ J(g),so VxPJ(g)V

∗x y = VxPJ(g)x

−1y = Vxx−1y = y. If y /∈ xJ(g) = J(xg) ∩ J(x), we have

to show that VxPJ(g)V∗x y = 0. Now, since y /∈ J(xg) ∩ J(x), either y /∈ J(x), in

which case V ∗x y = 0; or y /∈ J(xg), so x−1y /∈ J(g) and PJ(g)Vxy = 0. ThereforeVxPJ(g)V

∗x = PJ(g)PJ(x) = ϕxPJ(g). In particular, VxV

∗x = PJ(x) for x ∈Mn.

Similar calculations show that V ∗x PJ(g)Vx = Px−1J(g) = PJ(x−1g). Hence pro-jections corresponding to right ideals J(x−1y) with x, y ∈ Mn can be written asV ∗x VyV

∗y Vx = V ∗x PJ(y)Vx = PJ(x−1y). But each matrix µ(a, b) ∈ Γn can be expressed as

µ(0, 1m

)−1µ( 1m

a, 1mb), which is in M−1

n Mn for m large enough. Therefore every elementof Γn has the form x−1y for some x, y ∈ Mn, so the algebra generated by isometriesVx contains all the projections PJ(g), g ∈ Γn.

By Lemma 3.1 of [14], or by direct calculation, it follows that for each grouprelation x−1

1 y1 · · ·x−1m ym = 1 in Γn there is a corresponding relation in C(Mn):

v∗x1vy1 · · · v∗xmvym =m∏

k=1

PJ(y−1m xm···y−1

k xk).

For example, take x ∈Mn and the group relation 1−1xx−11 = 1. Then

VxV∗x = V ∗1 VxV

∗x V1 = PJ(1−1x)PJ(1−1xx−11) = PJ(x).

Similarly, the group relation x−1yy−1x = 1 yields

V ∗x PJ(y)Vx = V ∗x VyV∗y Vx = PJ(x1y)PJ(x−1yy−1x) = PJ(x−1y).

B(a, b) ⊆ B(c, d) ⇐⇒ αa,b[B(0, 1)] ⊆ αc,d[B(0, 1)]

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⇐⇒ α−1c,dαa,b[B(0, 1)] ⊆ B(0, 1)

⇐⇒ B(− 1

d(a− c),

b

d

)⊆ B(0, 1)

⇐⇒ 1

d||a− c||+ b

d≤ 1

⇐⇒ ||a− c|| ≤ d− b⇐⇒ (c, d)− (a, b) ∈ C+

⇐⇒ C+(c, d) ⊆ C+(a, b)

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