C*-regularity properties and their dynamical analogues

• Thesis title in Czech: Vlastnosti regularity C*-algeber a jejich dynamicke analogy
• Thesis title in English: C*-regularity properties and their dynamical analogues
• Key words: C*-algebry|Topologická dynamika|Hilbrtovy C*-moduly a bimoduly
• English key words: C*-algebras|Topological dynamics|Hilbert C*-modules and bimodules
• Academic year of topic announcement: 2022/2023
• Thesis type: dissertation
• Thesis language: angličtina
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. RNDr. Petr Somberg, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 16.09.2022
• Date of assignment: 16.09.2022
• Confirmed by Study dept. on: 07.10.2022
• Advisors: Dr. Re O'Buachalla, Dr.

Guidelines

This project would begin with learning the necessary basics of C*-algebras
(including relevant notions stemming from the C*-algebra classification program)
[6,7] and (amenable) topological dynamics [2].

The next steps are will include a study of Hilbert C*-modules and bimodules, their
crossed products [1], and how these relate to groupoid C*-algebras, twisted groupoid
C*-algebras, and C*-diagonals [5].

Understanding the role of approximately finite actions for Z-stability of crossed
products [3,4] and producing a subsequent generalization to crossed products by
Hilbert bimodules for certain amenable groups (for example, Z or Z^d actions) is a
tractable goal.

Along the way, the student can work towards establishing suitable dynamical
analogues of C*-regularity properties, possibly formulating a suitable dynamical
Toms—Winter-type conjecture.

References

[1] Abadie, Beatriz; Eilers, Søren; Exel, Ruy. Morita equivalence for crossed
products by Hilbert C*-bimodules.Trans. Amer. Math. Soc. 350 (1998), no. 8,
3043--3054.
[2] Giordano, Thierry; Kerr, David; Phillips, N. Christopher; Toms, Andrew. Crossed
products of C*-algebras, topological dynamics, and classification. Lecture notes
based on the course held at the Centre de Recerca Matemàtica (CRM) Barcelona, June
14–23, 2011. Edited by Francesc Perera. Advanced Courses in Mathematics. CRM
Barcelona. Birkhäuser/Springer, Cham, 2018. x+498 pp.
[3] Kerr, David. Dimension, comparison, and almost finiteness. J. Eur. Math. Soc.
(JEMS) 22 (2020), no. 11, 3697--3745.
[4] Kerr, David; Szabó, Gábor. Almost finiteness and the small boundary property.
Comm. Math. Phys. 374 (2020), no. 1, 1–31.
[5] Kumjian, Alexander . On C*-diagonals. Canad. J. Math. 38 (1986), no. 4,
969--1008.
[6] Murphy, Gerard J. C*-algebras and operator theory. Academic Press, Inc., Boston,
MA, 1990. x+286 pp.
[7] Strung, Karen R. An introduction to C*-algebras and the classification program.
Edited and with a foreword by Francesc Perera. Advanced Courses in Mathematics. CRM
Barcelona. Birkhäuser/Springer, Cham, [2021], ©2021. xiii+322 pp.

Preliminary scope of work in English

Dynamical systems have long been a source of interesting examples for operator
algebras. From a topological dynamical system—that is, a topological space equipped
with a group action—one can construct the C*-algebra crossed product, encoding the
function space and the action. The underlying dynamical system influences the
structural properties of the operator algebra while at the same time one hopes to
gain information about dynamical systems otherwise inaccessible without the operator
algebraist’s toolkit. Dynamical techniques, such as Rokhlin-type lemmas, are often
successfully imported into the theory of operator algebras, while operator algebraic
notions, such as K-theory, have been used to classify dynamical systems such as
Cantor minimal systems and the shifts of finite type.

In the theory of C*-algebra classification, the Toms–Winter Conjecture says that
finite dimensionality conditions (noncommutative covering dimension) should be
equivalent to a number of other regularity conditions, including absorbing tensor
products with the Jiang–Su algebra (Z-stabililty) and a comparison property called
“strict comparison of positive elements”. These properties provide good structural
behaviour and significant efforts have been put towards proving these equivalences
(see Part iii of [7] for more details). Recent advances in C*-algebraic
classification and a better understanding of how dynamical notions such as mean
dimension affect the resulting crossed product have spurred interest in finding
dynamical analogues of C*-structural properties.

This PhD project will look at topological dynamical regularity property analogues of
the C*-algebraic properties appearing in the Toms—Winter conjecture [7]. There are
various different paths one might consider. For example, just as groups can act on
C*-algebras by automorphisms, they can also act by Hilbert bimodules, which can be
seen as a generalizations of *-homomorphisms. Does almost finiteness, as defined by
Kerr [3,4], of the underlying dynamical system give us Z-stability of the bimodule
crossed product? Another, more ambitious goal, would be to try to expand on the
notion of a dynamical analogue of the “Toms—Winter conjecture”. Currently, the best
analogue for “finite nuclear dimension” seems to be “finite diagonal dimension”,
which is a refinement of the nuclear dimension that keeps track of a given maximal
abelian subalgebra called a C*-diagonal. In the case of a crossed product by a group
action on a locally compact Hausdorff space X, the C*-diagonal is C_0(X). However,
finiteness of the diagonal dimension implies that X is necessarily
finite-dimensional, and there are known examples of group actions on
infinite-dimensional spaces which give rise to crossed products with finite nuclear
dimension. So one would like to reformulate the notion of diagonal dimension to take
such things into account.