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, 31.03.2016) - contains about 1984 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.

- 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*indexed by a set_{i}*I*such that the equality*x=SUM*is valid for every_{i in I}(x,x_{i}) x_{i}*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*l*as an orthogonal direct summand for some index set_{2}(A,I)*I*? Partial 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 9, 2016

(Home) /