Operators on C*-algebras and Hilbert modules

Welcome to the webpage of the Croatian Science Foundation project IP-2016-06-1046 titled Operators on C^*-algebras and Hilbert modules whose research topics include:

Derivations, automorphisms and elementary operators on C^*-algebras
Elementary operators on Hilbert C^*-modules
Finite rank operators and approximations by such operators
Orthogonality preservers
Bicircular and n-circular projections
Frame theory

In the proposed project we shall study some classes of operators on C*-algebras and Hilbert C*-modules. First, we will study the sets of derivations, automorphisms and elementary operators on unital C*-algebras; in particular, the interrelations of the closures of these sets in the operator and completely bounded (cb) norm. A related part of the project is the analysis of elementary operators on Hilbert C*-modules. We aim to find appropriate generalizations of formulae (known for C*-algebras) for the norm and cb-norm of such operators in terms of their coefficients. We also plan to study some other classes of operators on C*-algebras and Hilbert C*-modules: ''finite rank'' operators and approximations by such operators, n-circular projections, and unitary/anti-unitary operators (in connection with generalizations and analysis of the stability of Wigner's equation). A related topic which we aim to explore is the class of orthogonality preserving maps on C*-algebras and Hilbert C*-modules, with respect to various concepts of orthogonality.

In our study of operators on Hilbert C*-modules we plan to use frames and outer frames for such modules as a pivotal tool. Recently, outer frames for Hilbert C*-modules are introduced and recognized as a natural generalization of frames that bridges some gaps in the modular frame theory. We plan to use systematically the existence of frames and outer frames in all countably generated modules and their reconstruction property to obtain a new insight into the properties of transformations of Hilbert C*-modules and into the structure of sets of some classes of such maps. Finally, we aim to address some open questions in the modular frame theory itself such as the correspondence of bounded operators and Bessel sequences, perturbations of frames and description (together with construction methods) of dual frames with special properties.