ar X iv : f un ct - a n / 96 05 00 1 v 1 4 M ay 1 99 6 Diagonalizing operators over continuous fields of C ∗ - algebras

It is well known that in the commutative case, i.e. for A = C(X) being a commutative C *-algebra, compact selfadjoint operators acting on the Hilbert C *-module H A (= continuous families of such operators K(x), x ∈ X) cannot be diagonalized inside this module but it becomes possible if we pass to a bigger module over a bigger W *-algebra L ∞ (X) = A ⊃ A which can be obtained from A by completing (on bounded sets) it with respect to the weak topology in the natural representation of A on the Hilbert space L 2 (X) where the norm is defined by a finite exact trace (measure) on A. Unlike the " eigenvectors " , which have coordinates from A, the " eigenvalues " are continuous, i.e. lie in the C *-algebra A. We discuss here the non-commutative analog of this well-known fact. When we pass to non-commutative C *-algebras the " eigenvalues " are defined not uniquely but in some cases they can also be taken from the initial C *-algebra instead of the bigger W *-algebra. We prove here that such is the case for some continuous fields of real rank zero C *-algebras over a one-dimensional manifold and give an example of a C *-algebra A for which the " eigenvalues " cannot be chosen from A, i.e. are discontinuous. The main point of the proof is connected with a problem on almost commuting operators. We prove that for some C *-algebras (including the matrix ones) if h ∈ A is a selfadjoint, u ∈ A is a unitary and if the norm of their commutant [u, h] is small enough then one can connect u with the unity by a path u(t) so that the norm of the commutant [u(t), h] would be also small along this path. 0 Introduction Let X be a locally compact Hausdorff space and let {A(x), x ∈ X} be a family of unital C *-algebras with exact finite traces τ x , τ x (1 x) = 1. Denote by x∈X A(x) the set of functions a = a(x) defined on X and such that a(x) ∈ A(x) for any x ∈ X. x∈X A(x) be a subset with the following properties: i) A is a *-subalgebra in x∈X A(x), ii) for any x ∈ X the set {a(x), a ∈ A is dense in the …