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Gauss polynomials and the rotation algebra. Authors; Authors and affiliations. Man-Duen Choi; George A. Elliott; Noriko Yui. Man-Duen Choi. 1. George A. Elliott.
Inventiones mathematicae

Invent. math. 99, 225-246 0990)

9 Springer-Verlag 1990

Gauss polynomials and the rotation algebra Man-Duen Choi 1, George A. Elliott t'2, and Noriko Yui 3 1 Department of Mathematics, University of Toronto, Toronto, Canada M5S 1AI 2 Mathematics Institute, University of Copenhagen, DK-2100 Copenhagen 3 Department of Mathematics, Queen's University at Kingston, Kingston, Canada K7L 3N6

Summary. Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring, ba=~ab with ~ commuting with a and b, then the (generalized) binomial coefficient (n) k arisinginthe expansion

k= 0 k ~,,

(resulting from these relations) is equal to the value at 7 of the Gaussian polynomial

[:]

[ k ] [ n - k]

where [m] = ( 1 - x " ) ( 1 - x m- x)...(1 - x ) . (This is of course known in the case

~=1.) From this it is deduced that in the (universal) C*-algebra A o generated by unitaries u and v such that vu=e2':i~ the spectrum of the self-adjoint element (u+v)+(u+v)* has all the gaps that have been predicted to exist - provided that either 0 is rational, or 0 is a Liouville number. (In the latter case, the gaps are labelled in the natural way - via K-theory -- by the set of all non-zero integers, and the spectrum is a Cantor set.)

1. I n t r o d u c t i o n

The difference equation ~b(n+ 1) + ~b(n- 1)+ 2 c o s 2 n ( n 0 - fl) ~b(n)= E~b(n)

226

M.-D. Choi et al.

where 0, fl, and E are fixed real numbers, first introduced by Peierls in [16], has been studied by numerous authors. (A recent survey is given in [4].) It can be viewed as the eigenvalue equation for the bounded self-adjoint operator H0~ in l 2 (7~) defined by the equation

(HPo~)(n)=c~(n+ l ) + f ~ ( n - 1 ) + 2cos2rc(nO-fi)q)(n),

n~Z,

and it is natural to study the spectrum of this operator as a set. This set is independent of fl if 0 is irrational, and so if for each rational 0 one considers the union of the spectra for all fl in [0, 1], one obtains a compact set, U SpH0~ =

S(O),

//

depending on 0 alone. The set S(O) (or rather, any gap in this set) has been shown to depend continuously on 0 [14, 10, 3, 6]. Numerical calculations of the set S(O) for certain rational values of 0 (namely, for p/q with q < 50), reported by Hofstadter in [14], suggest strongly that for irrational 0 it is a Cantor set. One of the main results of the present paper is that S(O) is a Cantor set if 0 is a Liouville number (by which we mean, somewhat restrictively, that for every C > 0 there exists p/q with [0-P/ql < C-q) 9 Related results have been obtained by Bellissard and Simon in [6], and, more recently, by Helfer and Sj6strand in [12]. Bellissard and Simon showed that if the potential 2cos2rc(nO-fl) is multiplied by a coupling constant 2 > 0 , and the set which appears instead of S(O) is denoted by S(O, 2), then at least for a dense set of pairs (0, 2) in 1/2 (unspecified, but of course with 0 irrational), S(O, 2) is a Cantor set. Helffer and Sj6strand showed that for certain, specified values of 0 (not, however, dense in [0, 1]), S(O) has infinitely many gaps. (That S(O) must have at least one gap for some 0 - specifically, when [ 0 - 1/31 < 1/200 - w a s shown by one of us [1113 While Helffer and Sj6strand stopped short of proving that S(O) is a Cantor set 1, they obtained rather precise results about the structure of S(O), in terms of the continued fraction expansion of 0, which we are not able to duplicate. The main limitation 1 arising in their work is that part of the set S(O) escapes consideration, depending on the size of the integers a~, a2 . . . . in the continued fraction expansion 0 = [al, a2 .... ], which must be quite large (this means that 0 must be small, and furthermore belong to a Cantor set). Our restriction on 0 is quite different from this. (The Liouville numbers are dense.) There is, nevertheless, some overlap, and it is interesting to compare our results with those of [12] for values of 0 for which both are valid. Another result related to ours is that of Riedel in [18], which states, also for 0 a Liouville number, that the invertible elements of the irrational rotation C*-algebra Ao are dense. Recall that A o is defined as the C*-algebra generated (universally) by two unitary elements u and v with the relation v u = p u v, where p=exp2rciO. The relation of this to our result is not entirely clear, but in any i This l i m i t a t i o n has been o v e r c o m e in m o r e recent work by the same authors. (Their results still require t h a t the integers a l , az . . . . in the e x p a n s i o n 0 = [a~, a2, ...] be large.)

Gauss polynomials and the rotation algebra

227

case A o can certainly be represented on 12(7/), by mapping u into the bilateral shift (taking q~ into (~b(n-1)),~z), and v into the operation of multiplication by p" (taking ~b into (p"~b(n)),~z), and then the polynomial (u+v)+(u+v)* is mapped into the bounded self-adjoint operator H0~ Since Ao is simple (when 0 is irrational), the spectrum of the operator H0~ is the same as the spectrum of the element (u + v) + (u + v)* in A o. Our strategy is to work within the algebra A o, using the commutation relation v u = p u v . We shall employ the method of rational approximation of 0 used in E6], [-11], and [18], which is natural in view of the continuity of the set S(O). This method is based on the fact that if 0 is rational then the irreducible representations of Ao are finite-dimensional. Roughly speaking, we shall use this to establish sufficiently good estimates on the gaps in S(O) when 0 is rational (in Sect. 3) that, after a sufficiently good quantitative estimate of the continuity of S(O) (in Sect. 4), the desired estimates on the gaps of S(O) when 0 is very well approximible by rational numbers follow in the limit (Sect. 5). It turns out (and was used in [63) that proving that a gap exists in S(O) when 0 is rational is equivalent to showing that a certain pair of adjacent eigenvalues of (u + v) + (u + v)* are distinct in a certain irreducible representation of A o. These are roots of the characteristic polynomial, and in considering this and other polynomials in ( u + v ) + ( u + v)*, we have been led to what might be called the non-commutative binomial theorem - with respect to the commutation relation v u = p u v. (This is derived in Sect. 2.) Thus, we solve our original question about the spectrum S(O) by means of an algebraic analysis of polynomials in the rotation algebra, closely akin to number theory. The coefficients in our binomial theorem turn out to be the familiar Gaussian polynomials. We need to evaluate these polynomials at roots of unity. This invites comparison with the recent use of Gaussian polynomials in quantum statistical mechanics - where they are evaluated at positive numbers ([2, 15]). The Gaussian polynomials do not appear to have been evaluated at special numbers before - except of course at the number 1 ! Let us recall some elementary properties of the rotation C*-algebra Ao. Let ~ denote the unit circle. There exists a canonical action (Zl, Z2) I---~ . . . . . of ]1"2 on Ao, such that :~..... (u)=zlu and c~..... (v)=z2v. Any element of Ao fixed by ct is a scalar multiple of 1. There is a unique tracial state v of Ao invariant under 5. If 0 is rational, 0 = p/q with (p, q) = 1, then there is an irreducible representation ~r of A o on t12~ taking u into the diagonal operator diag(1, p . . . . . p0-1) and v into the operator which permutes the canonical basis (e~) cyclically, sending e2 into el, e3 into e2, and so on. It is not difficult to see that every representation of Ao is unitarily equivalent to ~z..... = ~ ..... for some (zl, Zz)~T 2, and that two such representations g ..... and ~z;, ~ are unitarily equivalent if, and only if, z ' l = p " z l and Z'2=pnz2 for some m, n = 0 . . . . . q - - l . (Since u q and vq are central in A o if pq= 1, the image of Ao in any irreducible representation of Ao is spanned linearly by the q2 monomials umv", O1. 3.3. Theorem. Let O=p/q with (p, q ) = 1. Then Sp ho has exactly 2m gaps, where q = 2 m + 1 or q = 2 m + 2 . Each gap has length at least 8 -q.

-

Proof The spectrum of ho (in Ao) is of course the union of the spectra of the images of ho in the various irreducible representations of A o. Since these are finite-dimensional (see Sect. 1), the spectra are determined by the characteristic polynomials. Our argument is based on the fundamental fact, apparently first observed by Chambers in [8; Appendix] (and rediscovered by several authors since then see e.g. [6] and [13]), that only the constant term of the characteristic polynomial of the image of ho in an irreducible representation of A o depends on the representation. Since this is a slightly different formulation of the statement from that given in [8] (where the polynomial is not defined as a characteristic polynomial), let us repeat the p r o o f here. In terms of the family of irreducible representations (re.... 2)~.... 2 ) ~ T 2 described in Sect. 1, our statement is that the characteristic polynomial of rc..... (ho) is independent of z~ and z2 except for the constant term. Since this family is a complete set of representatives - with some redundancy , it is enough to consider this family. The redundancy is that ~p ..... and ~z,,pz2 are unitarily equivalent to rr...... where p=exp2~ziO. This means that the characteristic polynomial of ~ ..... (ho) is unchanged if zl or z 2 is multiplied by p. Each coefficient of the characteristic polynomial of rc..... (ho) is, as is easily seen from the definition of rc..... (given in Sect. 1), a polynomial in z~ 1 and z~ a . Since p is a primitive qth root of unity, any polynomial in z~ 1 and z f 1 invariant under the substitution of pz~ for Zl or pz2 for Zz is a polynomial in z~ q and z2~q. In particular, each coefficient of the characteristic polynomial of ~ ..... (ho) is a polynomial in z~ q and Zz-+q. Since rCz,,z2 is of dimension q, and each matrix entry of rc..... (ho) in the standard basis is of degree at most one in z~ a and z f 1, each coefficient of the characteristic polynomial of ~ ..... (ho)

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is of degree at most q in z~ 1 and z f 1, and, except for the constant coefficient, is in fact of degree strictly less than q. This shows that alt coefficients except the constant one are constant - i.e., independent of z 1 and z2. (The constant coefficient is ( - 1)q+ l (z~ + z~ + z~-q + z2 q), but we shall not need to use this.) In addition to the fact that only the constant term of the characteristic polynomial of the image of ho in an irreducible representation depends on the representation, we shall need to use that the constant term ranges over all points of an interval. This fact follows from the continuous dependence of the characteristic polynomial of n ..... (ho) on the parameter (zl, z2), belonging to the connected space T 2. (It can be seen from the explicit form of the constant term that it ranges over the interval from - 4 to 4.) We shall also have to use that 0 e S p h 0 . This was proved in [-6] in the case that q is even (see the proof of Theorem 3 of [-6]). We shall only need the case that q is even, but by density of such 0 and by continuity of the spectrum as a function of 0 (see 4.1), this implies the case that q is odd (and also the case that 0 is irrational). Another proof, that works just as well whether q is even or odd, is as follows: The constant term in the characteristic polynomial of n ..... (ho), ( - 1)q+ 1(z~ + z~ + z~-q+ z2 q), is clearly equal to zero for certain pairs (z t,z2)~qF2; for such a pair (Zl,Zz), 0 is a root of the polynomial, i.e. 0~Sp n ..... (ho). Since 7_ 1,-a(h0) = - h o , so that Sp h o = - S p ho (a fact we shall also use), the fact that 0~Sp ho also follows from the first statement of the theorem, i.e. that Sp ho consists of exactly 2 m + 1 disjoint closed intervals. (As noted above, our proof of the theorem uses 0 e S p ho only in the case that q is even, and so in the case that q is odd, one obtains another proof of 0 ~ S p h o just by proving the theorem - it of course follows from this by continuity that 0 e S p ho also in the case that q is even.) On consideration of the preceding facts, one sees that the first statement of the theorem amounts to the following statement: For each pair (zl, z2)~7~ z, the eigenvalues of n .... 2(ho) have multiplicity one except for 0, and if 0 is an eigenvalue, then it has multiplicity one if q is odd and multiplicity two if q is even. Furthermore, the second statement of the theorem amounts to showing that, in addition, for each (zl, Z2)E'~2, distinct eigenvalues of n .... 2(ho) are at least distance 8 -q apart - except, in the case that q is even, for the two eigenvalues closest to 0 (either both equal to 0, or one on either side of 0). Fix (zl, z2)~'~ z. Consider first the case that q is odd, i.e. q = 2 m + 1. Denote by a I ~a2~

... ~_aq

the eigenvalues of n ..... (h0), with multiple eigenvalues (if any) counted according to multiplicity. Fix j = 1, 2 . . . . . q - 1 , and set I ~ ( x - a i ) = f Then f is a monic i~.j

polynomial of degree 2m, and so by Theorem 3.2, ]bnzl,z2(f(ho))]l > 1, i.e.,

~I l 8-q) apart, as desired. N o w consider the case that q is even, i.e. q = 2m + 2. N o t e that, in this case, rc..... (ho) is unitarily equivalent to - l r ..... (ho). (In fact, in Ao, the a u t o m o r p h i s m c~_ ~,_ ~, which takes ho into - h o , is inner, determined by the unitary u " + t v "+ 1.) In other words, we m a y write the eigenvalues of rc..... (ho), counted according to multiplicity, as

a_,._l (~nn -4-8 rt, i.e.,

46 + n 62 > 4e ~/n + ne 2, which is impossible if 6 = (e2 + 4e/~fn + 4/n2) ~ - 2/n, i.e.,

46 + n62=4e]/n + ne 2. In other words, with this choice of 6, for some k we have

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