Geodesic
Diagonalization of operators over continuous fields of $C^*$-algebras
Sbornik. Mathematics, Tome 188 (1997) no. 6, pp. 893-911

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A proof is given of a non-commutative analogue of the classical Hilbert–Schmidt theorem on diagonalization of a self-adjoint compact operator in a Hilbert space; namely, it is shown for a certain class of $C^*$-algebras that a self-adjoint compact operator in a Hilbert module $H_A$ over a $C^*$-algebra $A$ can be reduced to diagonal form in some larger module over a larger $W^*$-algebra, where the elements on the diagonal belong to $A$. 
@article{SM_1997_188_6_a4,
     author = {V. M. Manuilov},
     title = {Diagonalization of operators over continuous fields of $C^*$-algebras},
     journal = {Sbornik. Mathematics},
     pages = {893--911},
     publisher = {mathdoc},
     volume = {188},
     number = {6},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_6_a4/}
}
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V. M. Manuilov. Diagonalization of operators over continuous fields of $C^*$-algebras. Sbornik. Mathematics, Tome 188 (1997) no. 6, pp. 893-911. http://geodesic.mathdoc.fr/item/SM_1997_188_6_a4/