Diagonalizing operators over continuous fields of C*-algebras

TL;DR

This paper explores the diagonalization of operators over continuous fields of C*-algebras, extending classical results from the commutative case to non-commutative settings, and investigates conditions under which eigenvalues can be chosen within the original algebra.

Contribution

It generalizes the diagonalization of operators to non-commutative C*-algebras and identifies cases where eigenvalues can be selected from the initial algebra.

Findings

Diagonalization extends to some continuous fields of real rank zero C*-algebras.

Eigenvalues can sometimes be chosen from the original algebra, not just a bigger W*-algebra.

Counterexamples show eigenvalues may be discontinuous in certain C*-algebras.

Abstract

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.