Non-Commutative Stochastic Independence: Algebraic and Analytic Aspects
Final Report Abstract
In order to classify non-commutative independences in quantum probability Philipp Varšo developed in his thesis the new concept of m-coloured Universal Classes of Partitions (m-UCPs). This new ideas could not be foreseen in the original plans for the PhD programme of Philipp Varšo. The results of the thesis already have and certainly will have a great impact on the research of the field of quantum probability and also of abstract algebra. In particular, noticing the possibilities of continuous deformation significantly influenced the paper of Gerhold, Hasebe and Ulrich, which in combination with Varšo’s work brings us close to a complete classification of the positive symmetric 2-u.a.u.-products, and in a way that could lead to a classification in an operator algebraic setting in the future. Of course, there are many open mathematical questions left. However, this is because and why the thesis is the starting point for further interesting results, for example new types of quantum independence, their classification, and a mathematically more satisfying theoretical approach to a classification of positive quantum independence. A paper of Gerhold and Philipp Varˇo on s further steps towards a classification of positive multi-faced independences is in preparation. Much of this research has been done within the project. Diehl, Gerhold and Gilliers extended the shuffle algebra approach of Ebrahimi-Fard and Patras to the two-faced setting. Gerhold and Shalit developed a new perspective on dilation of Hilbert space operators, namely to use them as a tool to measure their distance as abstract operators (i.e. not in a fixed representation). This idea proved to be very fruitful with many applications beyond dilation theory, in particular in the study of continuity of rotation C*-algebras. Gerhold and Skeide developed a general theory of interacting Fock spaces which contains and extends earlier approaches and also links interacting Fock spaces with subproduct systems. For the case of finite-dimensional subproduct systems they they could prove a number of new results for the corresponding dimension sequences. Franz, Schürmann and Monica Varšo developed a theory of Brownian motions and Lévy processes on braided involutive bialgebras, i.e., involutive bialgebras which are objects of a braided monoidal category. The main examples are braided spaces which are generated by a finite number of primitive elements satisfying certain quadratic commutative relations coming from an R-matrix. A further paper introduces several interesting new classes of quantum stochastic processes. In dimension one (i.e., with one self-adjoint primitive generator) there is only one R-matrix which leads to a braided involutive space in our setting, the 1×1 matrix R = (q) with q ∈ R\{0}. In this case the braiding can be defined via a group, and one obtains, e.g., the Azéma process, whose surprising properties have been studied. But in higher dimension there are many possibilities to choose an R-matrix with a compatible involution, and most of them cannot be obtained from groups. As an example, the Brownian motions (i.e., Lévy processes with quadratic generator) on the braided spaces associated to the sl2- and sl3 -R-matrix are classified. Malte Gerhold succeeded in proving the Schoenberg correspondence in the general multi-faced case. This means that in the most general case of a quantum Lévy process on a dual semi-group we have that this class of processes is described by the class formed by the conditionally positive linear functionals on the underlying dual semi-group. This had been a long-time conjecture but before only could be shown for the single-face case by using the Muraki classification of non-commutative independences. Gerhold’s proof applies an approximation argument which before had only been used for a fixed independence. It is the new idea that the same argument goes through for arbitrary and for different notions of independences. Since Schoenberg correspondence was proved to hold in the tensor case some time ago, the old tensor result could be used to approximate the general independence case.
Publications
- Interacting Fock spaces and subproduct systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23(03), 2020
M. Gerhold and M. Skeide
(See online at https://doi.org/10.1142/S0219025720500174) - Subproduct systems and cartesian systems: New results on factorial languages and their relations with other areas. Journal of Stochastic Analysis, 1(4), 2020
M. Gerhold and M. Skeide
(See online at https://doi.org/10.31390/josa.1.4.05) - Dilations of unitary tuples. Journal of the London Mathematical Society, 104(5):2053–2081, 2021
M. Gerhold, S. K. Pandey, O. M. Shalit, and B. Solel
(See online at https://doi.org/10.1112/jlms.12491) - On the matrix range of random matrices. Journal of Operator Theory, 85(2):527–545, 2021
M. Gerhold and O. M. Shalit
(See online at https://doi.org/10.7900/jot.2019dec04.2277) - Schoenberg correspondence for multifaced independence. arXiv
M. Gerhold
(See online at https://doi.org/10.48550/arXiv.2104.02985) - Studies on positive and symmetric two-faced universal products. PhD thesis, Greifswald, 2021
P. Varšo
- Towards a classification of multi-faced independence: A representation-theoretic approach. arXiv
M. Gerhold, T. Hasebe, and M. Ulrich
(See online at https://doi.org/10.48550/arXiv.2111.07649) - Brownian motion on involutive braided spaces. Rev. Un. Mat. Argentina
U. Franz, M. Schürmann, and M. Varšo
- Categorial independence and Lévy processes. SIGMA, 18(075):(27 pp.), 2022
M. Gerhold, S. Lachs, and M. Schürmann
(See online at https://doi.org/10.3842/SIGMA.2022.075) - Dendriform algebras and noncommutative probability for pairs of faces. arXiv
J. Diehl, M. Gerhold, and N. Gilliers
(See online at https://doi.org/10.48550/arXiv.2201.11747) - Dilations of q-commuting unitaries. International Mathematics Research Notices, 85(1):63–88, 2022
M. Gerhold and O. M. Shalit
(See online at https://doi.org/10.1093/imrn/rnaa093)
