On almost commuting operators

Functional Analysis and Its Applications, Vol. 31, No. 3, 1997

O n A l m o s t C o m m u t i n g O p e r a t o r s *

v . M. M a n u i l o v UDC 517.98

In the present paper, we consider pairs of almost commuting operators and s tudy the possibility to connect one of them with unity by a path in such a way that the commuta tor norm is small along this path. In the general case, the answer is negative [4, 2]. Moreover, in the paper [2] it is shown that there exists a topological obstruction for the existence of such a homotopy. Here we present a case in which such a homotopy exists. Recall that for a C*-algebra A, the condition R R ( A ) = 0 means [1] that the elements with finite spect rum are dense in the set of all self-adjoint elements of A. Thus, without loss of generality, we can assume that if h E A is a self-adjoint element, then it is a finite linear combination of projections pi E A with eigenvalues put in the decreasing order, h = ~-~in_l ~ iP i . Recall also that the condition tsr (A) = 1 means [3] that the invertible elements are dense in A.

T h e o r e m 1. Let A be a C*-algebra with the following properties: (i) R R ( A ) = 0 and tsr(A) = 1;

(it) for every projection p e A , the unitary group of the C*-algebra pAp is connected. Let h E A be a self-adjoint operator, and let u E A be a unitary operator such that

I lu*hu- hll < ~. (1) Then there exists a constant C depending only on Ilhll and a • th connecting u with 1 such that for sufficiently small 5

Ilu*(t)hu(t)- hll < for all t .

P r o o f . Let us divide the spectrum of the operator h into small closed intervals of length ~ and number the intervals that contain at least one point of the spectrum. Then Sp(h) C Uk~=l Ak. We can assume, without loss of generality, that the points of the spectrum of h do not coincide with the ends of the intervals Ak. Note that if Ak and Ak+l have no common endpoints, then the spectrum of h has a lacuna of length at least r Let qk, ql + "'" + qm = 1, be the spectral projections of h corresponding to A k . Then we can decompose the algebra A into a direct sum corresponding to these projections,

m A = (~k=l qkA, and the elements of A can be written as matrices with respect to this decomposition, namely, a = (ai j) , where a E A and aij = qiaqi. Let /-tk be the midpoint of Ak. We define a block tridiagonal matr ix d(a) in the following way:

(i) i f j > i + 2 o r j _ < i - 2 , then d(a ) i i=O;

(ii) if j = i 4- 1 and I#i - # i l > r then d(a)ij = 0; (iii) otherwise, d ( a ) i j = a i j .

Let dk(a) be the diagonal of a lying k lines above (or below if k is negative) the main diagonal. Inequal- ity (1) means that the matr ix u is almost tridiagonal:

L e m m a 2. For sufficiently small g, one has 114- d(~)ll < 411hllC/L T h e f i rs t s t e p o f t h e h o m o t o p y . We connect the matrices u and d(u) by a linear path. This path

is not unitary, but it lies close to the unitary group U,

dist(u(t) , U) < 411hll 1/2 r (2)

and along this pa th one has Ilu(t)h - hu(t)ll < 2g + 211hl1411hll~12r

* This research was partially supported by the RFBR (grant No 96-01-00182.)

Moscow State University. Translated fi'om Funktsionallnyi Analiz i Ego Prilozheniya, Vol. 31, No. 3, pp. 80-82, July- September, 1997. Original article submitted May 7, 1996.

212 0016-2663/97/3103-0212 $15.00 (~1998 Plenum Publishing Corporation

The following lemma shows that it is possible to find a homotopy connecting the matr ix d(u) with an almost upper t r iangular matrix.

al l a 1 2 ) a qj, A • q j 2 A . Then for every r > a L e m m a 3. Let a = \ a 2 1 a22 be matr i z in 0 there ]

fO'll a'12)," where IlaLII < r ~.it~ry ~ h v(t) s~ch that , ( 0 ) = 1 a~d ,(1)~ = \~;~ %

T h e s e c o n d s t e p o f t h e h o m o t o p y . Consider the element u21 of the matr ix u. If A1 and A2 have a common endpoint , then the element u21 is sufficiently small, and by definition we have d(u)21 =

d(u)12 = 0. In this case, we set vl = 1. Otherwise, we apply the previous lemma and find a uni tary 2 x 2-matrix v l , multiplying by which we reduce the norm of the element u21 almost to zero. We define the next step of the homotopy as the path given by multiplication of d(u) by vl(t) and afterwards we connect the element U~l with zero. Then we turn to the element ua2, and in a similar way we define a matr ix v2 which differs from the unit matr ix only in the intersection of the second and the third rows and columns and such tha t after multiplication by the matr ix we can assume that u~2 = 0. Repeating this procedure, we obtain a pa th defined by the uni tary matr ix v = vm-1 " " v2vl �9 Denote the matr ix v . d(u) by ~. Obviously, this matr ix is tridiagonal and upper triangular, and along the entire constructed path the est imate (2) is valid. Since the matr ix ~ is close to the uni tary group, we see that the norm of its upper tr iangular par t is also small, and hence the matr ix ~ is almost diagonal, namely,

II ~ - / 0 ( ~ ) l ] < 2411hll 1/2~/~. (3)

Therefore, the diagonal elements of K are invertible. Moreover, since the pa th connecting d(u) with lies in the set of four-diagonal matrices (with diagonals from the - l s t to the 2nd), it follows that the commuta tor norm can be estimated:

Ilk(t) h - h~(t ) l l < 6 Ilhll ~/~ ,Y~.

T h e t h i r d s t e p o f t h e h o m o t o p y . Let us connect the matrices g and do(g) by a linear segment. It follows from (3) that d is t (u( t ) , U) < 2811h11~/~/~. After we have reduced the matr ix u to the block diagonal form d0(g), any path lying in the set of such matrices gives a small commuta tor norm: I lu(t)h- hu(t)ll < r Let us choose a path connecting the matr ix d0(g) with the unit matr ix so that this path lies at a distance at most 28 Ilhll~/2g/~ from the uni tary group.

Thus, we have constructed a pa th u( t ) which connects the initial operator u with the unit matr ix such that d is t (u( t ) , U) < 28 [Iht[~/28 and Ilu(e)h - h u ( t ) l l < 6r Therefore, there exists a uni tary path u'( t )

such that Ilu'(t) - u(t)ll < 56 Ilhllm r and along the entire path one has the desired est imate

I lu ' ( t )h - hu'(t)ll <_ I lu( t)h - hu(t)ll + 211hll lid'(t) - ~(t)ll < Cr

R e m a r k . In the saxne way, it is possible to prove a similar result for the case in which instead .of a self-adjoint h we consider a uni tary v with a lacuna of fixed length in the spectrum.

The author is grateful to A. S. Mishchenko and E. V. Troitsky for useful discussions.

R e f e r e n c e s

1. L. G. Brown and G. K. Pedersen, J. Funct. Anal., 99, 131-149 (1991). 2. R. Exel and T. A. Loring, J. Funct. Anal., 95, 364-376 (1991). 3. M. A. Pdeffel, J. Opera tor Theory, 17, 237-254 (1987). 4. D. Voiculescu, Acta Sci. Math. (Szeged), 45,429-431 (1983).

Translated by V. M. Manuilov

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