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

TL;DR

This paper extends the diagonalization of compact operators from Hilbert modules over W*-algebras to those over certain C*-algebras of real rank zero, broadening the scope of classical spectral theorems.

Contribution

It generalizes the diagonalization of compact operators to Hilbert modules over real rank zero C*-algebras, under specific density and property conditions.

Findings

Diagonalization extends to compact operators over real rank zero C*-algebras.

Extension of operators from subalgebras can be diagonalized with entries in the subalgebra.

Generalizes classical spectral theorems to a broader algebraic context.

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