Hilbert C*-Modules Home Page

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Hilbert C*-modules are an often used tool in operator theory and in operator algebra theory. They serve as a major class of examples in operator C*-module theory. Beside this, the theory of Hilbert C*-modules is very interesting on its own. Interacting with the theory of operator algebras and including ideas from non-commutative geometry it progresses and produces results and new problems attracting attention. During the last couple of years many interesting applications of Hilbert C*-module theory have been found.
At the contrary, the pieces of Hilbert C*-module theory are still rather scattered through the literature. Most publications explain only as many definitions and results as necessary for the striven for applications in the fields considered there in the main. However, there are some papers, chapters in monographs and lecture notes that give comprehensive representations of parts of the theory.
The purpose of this webpage is to give a literature list containing about 1984 items of preprints, papers, books, lecture notes, books wherein Hilbert C*-modules and their properties are described or they are successfully applied to solve problems in other research fields. The literature list starts with two guides to Hilbert C*-modules: the first one refers to mayor sources by the type of source, the second one by subject. Since the notion ''Hilbert ... modules'' is in use for at least five more or less different mathematical concepts we list basic references to the other definitions as well.

The reader has to take into account that the choice of the sources is limited by the author's research interests and linguistic profiency, as well as by the availability of sources. He apologizes for a probable insufficient representation of the work of some colleagues in the present overview. All suggestions, corrections and supplements are welcome.
I am grateful to B. Kirstein, M. A. Rieffel and E. V. Troitsky for valuable comments and suggestions complemeting this list.

Bibliography on Hilbert C*-module literature (.PDF, 31.03.2017) - contains about 2101 references, a comprehensive guide through publications on the theory and the application fields, historical remarks, statistics. Suggestions, additions and corrections are welcome.

For a quite complete literature list on operator spaces see:
What are Operator Spaces? (in German), a online lexicon on operator spaces with bibliography maintained by G. Wittstock and his colleagues at Universität des Saarlandes, Saarbrücken, Germany.

Some mayor open problems in Hilbert C*-module theory:

Does every Hilbert C*-module M over a unital C*-algebra A possess a normalized tight frame? I.e., does M admit a set of elements x_i indexed by a set I such that the equality x=SUM_{i in I} (x,x_i) x_i is valid for every x of M?
Equivalently, for every Hilbert C*-module M over a unital C*-algebra A, does there exist an isometric embedding into a standard Hilbert C*-module l₂(A,I) as an orthogonal direct summand for some index set I? Partial negative answer obtained by Hanfeng Li, see paper.
Characterize those C*-algebras with the property that for every Hilbert C*-module over them and for each of its Hilbert C*-submodules which coincides with its biorthogonal completion therein, the latter is always a topological direct summand of the former.
Whether each kernel of a surjective bounded module operator between Hilbert C*-modules is a toplogical direct summand of the domain of this operator, or not?
Prove or disprove: Each injective bounded C*-linear orthogonality-preserving mapping T on a Hilbert C*-module over a given C*-algebra A is of the form T= tU for some C*-linear isometric mapping U on the Hilbert C*-module and for some element t of the center Z(M(A)) of the multiplier C*-algebra of A which does not admit zero divisors therein.

Michael Frank, last changes: April 1, 2017

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