Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
Michael Frank
TL;DR
The paper characterizes C*-algebras whose Hilbert C*-module categories admit universal orthogonal-summand structures by showing equivalences with a $c_0$-sum of compact operator algebras and with wide-ranging module-theoretic properties. It leverages the Magajna–Schweizer theorem, Wegge-Olsen’s closed-range framework, and Fröhlich–Blecher–Lance techniques to derive a comprehensive network of conditions (including adjointability of all bounded module maps and the graph-summand criterion) that identify algebras of compact operators on Hilbert spaces. Additional results on modular frames by Bakić–Guljaš and connections to open projections and multiplier algebras broaden the characterizations, highlighting the structural and categorical signatures of these algebras. The findings provide a robust, transferable set of criteria for recognizing compact-operator C*-algebras from purely categorical properties of their Hilbert modules, with implications for operator algebra theory and frame analysis in modules.
Abstract
B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*-algebras of compact operators can be characterized by the property that every norm-closed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*-module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*-modules over C*-algebras which characterize precisely the C*-algebras of compact operators.