Characterizing C*-algebras of Compact Operators by Generic Categorical Properties of Hilbert C*-modules

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. In 1997 B. Magajna obtained the equivalence of the property of the category of Hilbert C*-modules over a certain C*-algebra A that any Hilbert C*-submodule is automatically an orthogonal summand with the property of the C*-algebra A of coefficients to admit a faithful ∗-representation in some C*-algebra of compact operators on some Hilbert space, cf. Theorem 2.1. In 1999 J. Schweizer was able to sharpen the argument replacing the Hilbert C*-module property of B. Magajna by the property of the category of Hilbert C*-modules over a certain C*-algebra A that any Hilbert C*-submodule which coincides with its biorthogonal complement is automatically an orthogonal summand, cf. Theorem 2.1. Later on in 2003 M. Kusuda published further results which indicate that in the majority of situations the Hilbert C*-module property can be weakened merely requiring the KA(M)-Asubbimodules of the Hilbert C*-modules M to be always orthogonal summands, see [11, 12] for the details. Studying the work of B. Magajna and J. Schweizer C*-algebras A of the form A = c0∑ α⊕K(Hα) become of special interest, where the symbol K(Hα) denotes the C*-algebra of all compact operators on some Hilbert space Hi, and the c0-sum is either a finite block-diagonal sum or a block-diagonal sum with a c0-convergence condition on the C*-algebra 1991 Mathematics Subject Classification. Primary 46L08 ; Secondary 46H25.