  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 $K(x)$, $x\in X$) can
be diagonalized if we pass to a bigger W*-algebra $L^\infty(X)={\bf A} \supset
A$ which can be obtained from $A$ by completing it with respect to the weak
topology. Unlike the "eigenvectors", which have coordinates from $\bf A$, the
"eigenvalues" are continuous, i.e. lie in the C*-algebra $A$. We discuss here
the non-commutative analog of this well-known fact. Here 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 if $h\in A$ is a selfadjoint, $u\in A$ 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 $[u(t),h]$
would be also small along this path.
