Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero

Details Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero

1995-01-31

arxiv.org/abs/funct-an/9501008v2

Mathematics

Operator Algebras

8 pages, LaTeX, v. 2.09, no figures, a slightly revised version

Scientific paper

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 $H_A^*\supset H_B$ can be diagonalized with diagonal entries being from the $C^*$-algebra $B$.

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