Probabilistic Operator Algebras Seminar

(Northeast Normal University)

The single-generator problem asks whether every simple separable C*-algebra can be generated by a single element. It is closely related to the classical generator problem for von Neumann algebras. In this talk, I will first review the background and some of the main developments concerning this problem. I will then present recent joint work on several classes of C*-algebras. we prove that each Villadsen algebra of the first type is singly generated. In contrast, we construct a Villadsen algebra of the second kind that is not singly generated. For crossed products, we prove that if \sigma is a minimal homeomorphism of a compact metrizable space X, then C(X) \rtimes Z is singly generated. Finally, we establish that the reduced group C*-algebra \mathrm{C}_r^*(F_2) of the free group on two generators is singly generated. The results presented in this talk arise from joint work, in different collaborations with George A. Elliott, Zhuang Niu and M. Ruzicka.

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(University of Ottawa and University of Wyoming)

In Connes’ noncommutative geometry, traces on operator ideals play the role of the integral. The basic examples are Dixmier traces and, more generally, positive normalized traces on the weak trace class L_{1,\infty}. This setting is tailored to classical Weyl laws, where eigenvalues decay like j^{-1/p}. Many natural operators, however, obey nonclassical Weyl laws with logarithmic corrections, cj^{-1/p} (\log j)^q. Examples include Dirichlet Laplacians on infinite-volume domains, Dirac operators on open manifolds with cusps and assuming the Riemann Hypothesis, the Dirac operator of Connes–Moscovici whose eigenvalues are the imaginary parts of the zeros of the Riemann zeta function. Such operators lie outside L_{1,\infty}, in weak Lorentz ideals associated with regularly varying functions. This talk is about traces on these ideals. I will first show that an asymptotic additivity property for eigenvalue partial sums allows us to define Dixmier traces directly in terms of eigenvalue sequences. This leads to a spectral characterization of operators on which all Dixmier traces agree, answeing a question of Connes in this setting. I will then go beyond Dixmier traces and consider the whole class of positive normalized traces. The main tool is an extension of Pietsch’s correspondence which relates traces on weak Lorentz ideals to invariant functionals on sequence spaces and places Dixmier traces within a broader picture. It yields a spectral characterization of operators on which all postive normalized traces agree. This extends results of Pietsch and Semenov–Sukochev–Usachev–Zanin for L_{1,\infty}. Combined with an extension of the Birman–Solomyak perturbation theory, it shows that operators satisfying Weyl laws take the same value on all positive normalized traces. I will also discuss a further notion, hypermeasurability. Finally, I will illustrate these results with examples, including the Riemann Hypothesis, the logarithm of th Laplacian, and products of Laplacians. This is joint work with Yongqiang Tian (Central South Univesity, Changsha, China).

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(Universita degli Studi Aldo Moro, Bari)

We consider optimal transport between quantum dynamical systems on \mathbb{Z}_2– graded von Neumann algebras. In the usual, non-graded setting, Wasserstein distances between quantum systems can be defined in terms of transport plans on tensor products involving the commutant of one of the algebras. We briefly recall this construction and describe its extension to the graded setting, where the usual commutant and tensor product are replaced by the twisted commutant and the Fermi tensor product. The main point of the construction is a correspondence between fermionic and usual transport plans. More precisely, using cyclic representations associated with transport plans and the Klein isomorphisms induced by the gradings, we establish a one-to-one correspondence between fermionic transport plans and graded transport plans in the usual tensor product. This correspondence allows the Wasserstein distances in the fermionic setting to be related to the corresponding distances in the non-graded setting, and in particular yields their metric properties.

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(University of Vienna)

We construct a function \Phi such that the orbit under the representation of \Gamma = PSL(2;mathbb{Z}) is an orthonormal basis for the Bergman space with weight q = 12. Moreover, we show that \Phi is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2;mathbb{Z}), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The construction of the function, which can also be represented as an orthonormal wavelet in the half-line, reproduces the method used in the signal approximation practice: the off-diagonal decay of the Gramian operator allows to control its restriction to nested finite subsets of PSL(2;mathbb{Z}), where it becomes a finite matrix and can be inverted, leading to a convergent sum where all objects and bounds are computable in the cyclotomic field generated by the 12th-root of unity over the rationals. Since modular actions on the Bergman kernel enter the construction, \Phi displays Bergman and modular reminiscences, despite appearing to be modular itself. As suggested by Jones, it provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra M_12(\Gamma) and its commutant. With minor adaptations, the construction works for any cocompact Fuchsian group, but the more involved proof of computability is only executed for PSL(2;mathbb{Z}).

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(UCLA)

We will present a version of OpenAI’s proof of free group factor isomorphism. A strengthening of the proof also shows that the reduced C*-algebras of free groups have strong reduced free products absorption properties. Moreover, C*-algebras generated by n=2, 3,… or countably many semicircular elements are all isomorphic, and the same is true for reduced free products of copies of the Jiang-Su algebra. A common ingredient is a certain automorphism approximation property; a relative version of that property appeared in our recent work on the identification of von Neumann algebras of Fuchsian groups.

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(CIMAT Guanajato)

In this talk, within the framework of real infinitesimal free probability introduced by Cebron and Mingo, we will discuss the calculation of the real infinitesimal free cumulants of independent complex Wigner matrices. Our approach relies on establishing a combinatorial relation between annular noncrossing partitions and families of directed graphs. As a consequence, we show that independent complex Wigner matrices are asymptotically real infinitesimally free. In particular, under mild conditions, a complex Wigner matrix is asymptotically infinitesimally free from its transpose. This is based on joint work with James A. Mingo (arXiv:2609.24023).

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