Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero
Abstract
It is known that the classical Hilbert--Schmidt theorem can be generalized to the case of compact operators in Hilbert
A
-modules
H
∗
A
over a
W
∗
-algebra of finite type, i.e. compact operators in
H
∗
A
under slight restrictions can be diagonalized over
A
. We show that if
B
is a weakly dense
C
∗
-subalgebra of real rank zero in
A
with some additional property then the natural extension of a compact operator from
H
B
to
H
∗
A
⊃
H
B
can be diagonalized with diagonal entries being from the
C
∗
-algebra
B
.