This is the site of the project within the Seventh Framework Programme (FP7) EU

Description
Panopticum
Publications
Talks
Events




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B. K. Kwa¶niewski

OperaDynaDual: Operator Algebras and Single Operators via Dynamical Properties of Dual Objects



Description:

The project lies at the intersection of three fields of pure mathematics: operator algebras (specifically C*-algebras), operator theory and dynamical systems, with applications to noncommutative structures, completely positive dynamics, spectral analysis of functional operators, and potentially to the theory of functional-differential equations, quantum physics and many other fields.
The theory of operator algebras originated in connection with quantum physics. Operator algebras can be viewed as far reaching generalization of number fields where in particular multiplication need not to be commutative. They provide a rigorous mathematical structure of quantum mechanics, as well as a link with classical physics. The general principle is that observables in a quantum systems are modelled by elements in genuinely noncommutative operator algebras, and if the algebra is commutative then it is in fact an algebra of functions (classical observables) on some classical space:

noncommutativity <-----> quantum.

An important very general class of operator algebras, which are also of fundamental importance in mathematics, in particular in harmonic analysis and representation theory are called C*-algebras. The theory of dynamical system studies properties of actions (evolutions) of groups or semigroups. Group is a mathematical object that is used to describe symmetries of the system or reversible dynamics. Semigroups model evolution systems which are irreversible. The theory of operators studies single linear maps between infinite dimensional spaces. In particular, spectral properties of the so called functional operators model asymptotic and ergodic properties of evolutions of systems modeling complicated physical processes containing both dynamical contributory factors and interaction with outer media, e.g. the process of motion and transformation of particles.

The most significant achievements of the project can be divided into 5 items:
  1. CONSTRUCTION OF ALGEBRAS FROM DYNAMICAL DATA. In the “noncommutative world” dynamics is usually irreversible. Moreover, it usually is given by maps which preserve only positivity, do not preserve multiplication – completely positive dynamics. In fact, it might be given by even more general structures – C*-correspondences. These are very abstract and general notions which are hard to analyze. So far it is not known how to construct operator algebras incorporating these kind of dynamics in general. The project provides a huge contribution to this subject. A new mathematical language of right tensor C*-precategories was developed. This led to general constructions which unify and extend the most successful constructions of this sort. This includes the so called Cuntz-Pimsner algebras and Nica-Toeplitz algebras associated with semigroups of C*-correspondences. This also concerns crossed products by completely positive maps. Moreover, the structure of these algebras is deeply analyzed.
  2. NONCOMMUTATIVE GRAPHS. As we already mentioned noncommutative dynamics involve more sophisticated objects than maps. In fact it is known that C*-correspondences with commutative coefficient algebras correspond to graphs –combinatorial objects. This allows a detailed study of the corresponding C*-algebra, and the theory of graph C*-algebra is nowadays a well-established fundamental part of the theory of operator algebras. One of the main result of the project is development of the theory of graphs dual to C*-correspondences with noncommutative algebras of coefficients. This provides combinatorial technics to study these genuinely noncommutative objects.
  3. UNIQUENESS THEOREMS. Among the first important examples of operator algebras modeling quantum systems are CAR and CCR algebras. They are generated respectively by quantum anti-commutation and canonical commutation relations (they model observables in Bose-Einstein and Fermi-Dirac quantum statistics, respectively). The essential property of relations of CAR and CCR type is that whenever you have family of operators satisfying the prescribed relations the C*-algebras they generate are the same. This uniqueness property has a fundamental physical meaning, as if we had no such uniqueness, different representations would yield different physics! Similar results of this type for graph C*-algebras and more general structures are called "Uniqueness Theorems". They are powerfull tools to study the structure of the corresponding C*-algebras. Using the machinery described in item 2, general Uniqueness Theorems have been established. They unified and extended a number of previous results of these type. In particular, they were applied to study the structure algebras described in item 1. Also some new phenomena were discovered which led to results in item 4.
  4. PARADOXICALITY AND PURE INFINITENESS. The famous Banach–Tarski paradox is a theorem saying that given a solid ball (e.g. an orange) in 3 dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together using only physical motions to yield two identical copies of the original ball (two oranges).

    One of the results of the project is a discovery of conditions for the noncommutative (quantum) version of Banach-Tarski paradox to hold. Also its relationship to pure infiniteness of the corresponding algebras has been clarified. This analyzis is phrased in terms of Fell bundles over groups which correspond to actions of dual quantum groups on genuinely noncommutative spaces. These significant results give strong tools to construct purely infinite C*-algebras, which are of crucial importance in the theory of classification of C*-algebras.
  5. APPLICATIONS TO FUNCTIONAL OPERATORS. The fundamental spectral information of functional operators is encoded in the algebras they generate. These have a structure of generalized crossed products. As an application of the previously mentioned results of the project, spectral properties where deeply analyzed. One of fundamental tools that has been developed is a canonical dilation (extension) of an irreversible system to a reversible one. In the simplest example of a map that wraps a unit circle twice, the corresponding reversible object is a solenoid – it can be imaged as an infinitely thin solid torus wrapped infinitely many times around itself.

    Calculation of spectra of the corresponding operator involves integrating over such a complicated object, or its noncommutative counterpart.
The powerful, innovative constructions and tools described above are expected to give impulse for new lines of research. The outputs and developed innovative methods of analysis brings to the ERA a unique expertise of a great impact and scientific value, with a wide range of potential interdisciplinary applications. Thanks to the training received during the project the fellow integrated with an excellent Scandinavian scientific network of world leaders in crossing the boundaries between operator algebras and other fields. This strengthens this network and provides a new strong outside link with Poland.

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Panopticum:

The envisaged applications are broadly related to the analysis of quantum structures and evolutions of systems modeling complicated physical processes containing dynamical contributory factors as well as interaction with outer media, e.g. the process of motion and transformation of particles. The project involves computer-aided research. In particular, the following animations are relevant to the project:



The Buckethandle can be created by drawing semicircles joining the points of the Cantor Middle-Thirds Set in the above presented manner. It is homeomorphic to the inverse limit space of the Tent Map f(x)=1-|2x-1| (or if one prefers the Logistic Map g(x)=4x(1-x)). more info This is the inverse limit space arising from the inverse sequence with a single bondig map of the form f(x)=4 λ x(1-x) where λ is a little to the right from the value
¼ (1+ √ 6 ) ≈ 0.86237 .





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Publications:

These are the publications related to the project.
  1. "Pure infiniteness and ideal structure of C*-algebras associated to Fell bundles" (with W. Szymanski)
    J. Math. Anal. Appl. 445 (2017), no. 1, 898-943 ArXiv
  2. "Topological aperiodicity for product systems over semigroups of Ore type" (with W. Szymanski)
    J. Funct. Anal. 270 (2016), no. 9, 3453–3504. (pdf)
  3. "Crossed products by endomorphisms of C_0(X)-algebras"
    J. Funct. Anal. 270 (2016), no. 6, 2268–2335. (pdf)
  4. "Ideal structure of crossed products by endomorphisms via reversible extensions of C*-dynamical systems"
    Int. J. Math. 26 (2015), no. 3, 1550022 [45 pages] (pdf)
  5. "Extensions of C*-dynamical systems to systems with complete transfer operators"
    Math. Notes 98 (2015), no. 3, 419-428. (pdf)
  6. "Exel's crossed product and crossed products by completely positive maps"
    accepted to Houstan J. Math. (pdf)
  7. "Aperiodicity, topological freeness and more: from group actions to Fell bundles" (with R. Meyer)
    Studia Math. 241 (2018), 257-302. ArXiv
  8. "Nica-Toeplitz algebras associated with right tensor C*-precategories over right LCM semigroups" (with N. S. Larsen) submitted ArXiv
  9. "Nica-Toeplitz algebras associated with product systems over right LCM semigroups" (with N. S. Larsen) J. Math. Anal. Appl. 470 (2019), no. 1, 532-570 ArXiv
  10. "Topological freeness for C*-correspondences" (with T. M. Carlsen and E. Ortega) accepted in J. Math. Anal. Appl. ArXiv
  11. "Variational principles for spectral radius of weighted endomorphisms of C(X,D)" (with A. V. Lebedev) submitted ArXiv

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Talks:

  1. ``Topological aperiodicity for product systems of C*-correspondences '' (pdf)
    Seminario de Algebras de Operadores, UFSC, Florianopolis, Brasil, November 17, 2014 (screen) (link)
  2. ``From dynamical systems and operator theory to operator algebras and back'' (pdf)
    IMADA seminar, SDU, Odense, October 30, 2014 (screen)
  3. ``Purely infinite crossed products by endomorphisms of C_0(X)-algebras'' (pdf)
    Operator algebra seminar, University of Copenhagen, November 26, 2014 (screen) (link)
  4. ``Pure infiniteness and ideal structure of C*-algebras associated with Fell bundles and product systems'' (pdf)
    16th Danish-Norwegian Operator Algebra Workshop at Lysebu, Oslo, December 9, 2014.
  5. ``Topological aperiodicity for product systems of C*-correspondences'' (pdf)
    C*-algebra seminar, University in Oslo, March 4, 2015 (screen) (link)
  6. ``Topological aperiodicity for product systems of C*-correspondences'' (pdf)
    Operator Algebra Seminar: Norwegian University of Science and Technology, April 23, 2015
  7. ``A successful applicant tells his story'' (pdf)
    Marie Curie-Sklodowska Actions Workshop, SDU, May 21, 2015
  8. ``Purely infinite C*-algebras associated to Fell bundles'' (pdf)
    Workshop on C*-algebras: Geometry and Actions, Münster, July 13-17, 2015
  9. ``Purely infinite C*-algebras associated to Fell bundles over discrete groups'' (pdf)
    Banach Algebras and Applications, Toronto, August 4-12, 2015.
  10. ``Purely infinite C*-algebras associated to Fell bundles over discrete groups'' (pdf)
    YMC*A: Young Mathematicians in C*-Algebras, Copenhagen, August 17-21, 2015.
  11. ``Advances in the theory of crossed products by endomorphisms'' (pdf)
    The Annual Meeting in CEAFEL, September 21, 2015
  12. ``On crossed products by endomorphisms'' (announcement)
    IMADA seminar, SDU, Odense, Denmark October 8, 2015
  13. ``Pure infiniteness of reduced cross-sectional C*-algebras''(screen)
    Norwegian University of Science and Technology, Trondheim, Norway, May 10, 2016.
  14. ``Ideal structure and pure infiniteness of crossed products''(screen)
    Colloquium des Graduiertenkollegs, Universität Göttingen Mathematisches Institut, Goettingen, Germany, June 9, 2016.
  15. ``Uniqueness theorems for C*-algebras defined in terms of generators and relations''(screen)
    Universität Göttingen Mathematisches Institut, Goettingen, Germany, June 8, 2016.
  16. ``Crossed products by endomorphisms of C_0(X)-algebras'' (pdf)
    6th International Conference on Operator Theory, Timisoara, Romania, 27 June -2 July, 2016
  17. ``Crossed products by endomorphisms of C_0(X)-algebras'' (pdf)
    WOAT 2016: International Workshop on Operator Theory and Operator Algebras, Lisbon, Portugal, July 5-8, 2016
  18. ``Operator Algebras and Single Operators via Dynamical Properties and Dual Objects" (announcement)
    IMADA seminar, SDU, Odense, Denmark, Septemenr 22, 2016
  19. ``Paradoxicality and pure infiniteness of crossed products'' (pdf)
    C*-algebra seminar, University in Oslo, September 14, 2016

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Conference:

I was a co-organiser of the conference: Workshop on the Thompson Groups, August 15-17, 2016, Odense, Denmark, University of Southern Denmark (website) (group photo) (group photo)

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Events and research visits:

These are the events that I took an active part in thanks to and in connection with the project:
  1. Research visit in Universidade Federal de Santa Catarina, Florianopolis, Brasil, November 9-22, 2014
  2. 16th Danish-Norwegian Operator Algebra Workshop at Lysebu, Norway, December 8 - 11, 2014 (program)
  3. Research visit in University of Oslo, Norway March 1-7, 2015.
  4. Research visit in Norwegian University of Science and Technology, Trondheim, Norway, April 19-25, 2015 (screen) (link)
  5. Groups, boundary actions and group C*-algebras, Copenhagen, April 13-17, 2015 (screen) (link)
  6. ErikFest: Conference on Operator Algebras and Applications in celebration of Erik Christensen and his work, University of Copenhagen, May 4-8, 2015 (screen) (link)
  7. Marie Curie-Sklodowska Actions Workshop, SDU, Odense, May 19-21, 2015 (program)
  8. Workshop on C*-algebras: Geometry and Actions, Münster, July 13-17, 2015 (screen) (link)
  9. Banach Algebras and Applications, Toronto, August 4-12, 2015. (screen) (link)
  10. YMC*A: Young Mathematicians in C*-Algebras, Copenhagen, August 17-21, 2015. (screen) (link)
  11. The Annual Meeting in CEAFEL 2015, Instituto Superior Técnico, Lisbon University, Portugal, September 20-23, 2015 (screen)
  12. Lecture series on quasidiagonality of nuclear C*-algebras, Copenhagen, Denmark, 2015, November 10-12, The University of Copenhagen (screen)
  13. Uffe Haagerup Memorial Symposium, November 24, 2015, Odense, Denmark, University of Southern Denmark (screen)
  14. Classification of operator algebras: complexity, rigidity, and dynamics, Stockholm (Djursholm), Sweden, 2016, Jan. 17-31, Institut Mittag-Leffler (screen) (invitation letter)
  15. Masterclass: Expanders and rigidity ofgroup actions, Copenhagen, Denmark, 2016, May 2-16, The University of Copenhagen (screen) (group photo)
  16. Research visit in Norwegian University of Science and Technology, Trondheim, Norway, May 7-13, 2016. Collaboration with Eduard Ortega
  17. Research visit in Universität Göttingen Mathematisches Institut, Goettingen, Germany, June 6-11, 2016. Collaboration with Suliman Albandik and Ralf Meyer (screen)
  18. Symposium: The mathematical legacy of Uffe Haagerup, Copenhagen, Denmark, 2016, June 24-26, The University of Copenhagen (poster)
  19. The 26th International Conference in Operator Theory, June 27 -July 2, 2016, Timisoara, Romania, Institute of Mathematics Simion Stoilow of the Romanian Academy and the West University in Timisora. (poster)
  20. WOAT 2016 International Workshop on Operator Theory and Operator Algebras Instituto Superior Técnico, Lisbon, Portugal July 5-8, 2016 (poster)
  21. Operator Algebras and Mathematical Physics, August 1-12, 2016, AIMR, Tohoku University, Sendai, Japan (poster)
  22. Workshop on the Thompson Groups, August 15-17, 2016, Odense, Denmark, University of Southern Denmark (website) (group photo) (group photo)
  23. Research visit in University of the Faroe Islands, Torshavn, Faroe Islands, Septemeber 5-11, 2016. Collaboration with Toke Carlsen
  24. Research visit in University of Oslo, Norway Septemeber 12-17, 2016. Collaboration with Nadia Larsen

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Teaching:

  1. Introduction to operator algebras, spring 2015 (together with Uffe Haagerup) (blackboard scan)
  2. Measure and Integration and Banach spaces, spring 2016 (together with Selcuk Barlak) (blackboard scan)
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