Real structure occurs naturally and crucially in very many areas of mathematics. Besides a pair of papers of Ruan from 2003 there was very little theory of real operator spaces and (possibly nonselfadjoint) algebras for almost two decades. Together with students we have recently developed this to a somewhat mature level. We begin by describing this theory and how standard constructions interact with the complexification. For example, we develop the real case of the theory of operator space multipliers and the operator space centralizer algebra, and discuss it connects with the complexification. We use this to characterize real structure in complex operator spaces, and characterize some of the most important objects in the subject. We also mention a couple of remaining and seemingly hard questions. Generalizing further, joint work with Mehrdad Kalantar gives a novel framework that contains the operator space complexification, as well as the less-studied quaternification, as special cases. It also may be viewed as the appropriate variant of Frobenius and Mackey's induced representations for the category of operator spaces.

For the spectral action consisting of the average number associated to the gas of free $q$-particles (including Bose, Fermi and classical ones corresponding to $q=\pm 1$ and $0$, respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off given by (a function of) the inverse temperature. We treat both relevant situations relative to massless and non relativistic massive particles, where the natural cut-off is $1/\beta =k_{\mathrm{B}}T$ and $1/\sqrt{\beta }$, respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also treat in some detail the relativistic massive case for which the natural cut-off is again $1/\beta$. We then consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium, by also discussing the appearance of condensation phenomena occurring for Bose-like $q$-particles, $q\in (0,1]$. The situations relative to the finite volume (discrete spectrum) and the infinite volume (continuous spectrum) are studied and compared. The more singular situation corresponding to the massive case is handled by using the theory of distributions associated to a very particular class of test-functions, the last having connections with the Riemann $\zeta$-function.

The present talk is based on: F. Ciolli and F. Fidaleo Spectral actions for $q$-particles and their asymptotics, J. Phys. A (Math. Theor.) 55 (2022), 424001 (19 pp).

The Lipschitz free space $ℱ(X)$ over a metric space $X$ is a canonical Banach space containing a copy of $X$ isometrically. Naturally, a principal theme is to investigate the relationship between geometric properties of X and linear-geometric properties of $ℱ(X)$. A well-studied problem within this theme is to classify those $X$ for which $ℱ(X)$ is isomorphic to $L_1$. In this talk, we’ll discuss our recent result that $ℱ(X\cup Y)$ is isomorphic to $L_1$ whenever $X,Y$ are subsets of some ambient metric space with finite Nagata dimension and both $ℱ(X)$ and $ℱ(Y)$ are isomorphic to $L_1$. Based on joint work with David Freeman.

We investigate the problem of when a unital Banach algebra has (uniformly) open multiplication. For commutative C*-algebras, we show that this is precisely so when the covering dimension of the maximal ideal space is at most one. We find sufficient conditions for a commutative Banach *-algebra to have open multiplication that require an embedding into a commutative Banach algebra. Joint work with N. Maślany.

We discuss small-ball probability estimates of the smallest singular value of a rather general ensemble of random matrices which we call “inhomogeneous”. One of the novel ingredients of our family of universality results is an efficient discretization procedure, applicable under unusually mild assumptions, while another new ingredient is the notion of the so-called randomized Least Common Denominator of a vector and of a matrix, and a double-counting method. Most of the talk will focus on explaining the ideas behind the proof of the first ingredient. Partially based on the joint work with Tikhomirov and Vershynin, and an ongoing joint work with Fernandez and Tatarko.

Let ${𝔽}_n,2\le n\le \mathrm{\infty }$, be the non-abelian free group of $n$-free generators, and ${𝔽}_n^{(i)}$ be the subsets of ${𝔽}_n$ consisting of reduced words starting with the $i$-th generator. The geometrically paradoxical decomposition ${𝔽}_n={\cup }_{1\le i\le n}{𝔽}_n^{(i)}\cup \{e\}$ implies the well-known nonameanablility of ${𝔽}_n$. In a recent joint work with E. Ricard, we show that this decomposition is unconditional with respect to the noncommutative $L^p$-norm. This implies that the group von Neumann algebra of ${𝔽}_{\mathrm{\infty }}$ admits a $L^p$-unconditional decomposition with infinitely many components that satisfy a geometrical paradoxical property. It is a mystery whether the group von Neumann algebra of ${𝔽}_2$ ( or ${𝔽}_n$ for any finite $n$) admits such a decomposition. In this talk, I wish to introduce recent progress in this direction. Part of the talk is based on joint works with Z. Liu, E. Ricard, Q. Xu, and S. Yin.

For a Banach space $X$ denote by $ℒ(X)$ the algebra of bounded linear operators on $X$ and by $𝒦(X)$ the compact operator ideal on $X$. The quotient $ℒ(X)/𝒦(X)$ is called the Calkin algebra of $X$. We prove that $ℒ(X)/𝒦(X)$ can be reflexive, even isomorphic to a Hilbert space. More precisely, for every Banach space $U$ with an unconditional basis satisfying certain non-trivial lower estimates asymptotically, we construct a Banach space ${𝔛}_U$ and a sequence of “pairwise orthogonal” projections ${(I_s)}_{s=1}^{\mathrm{\infty }}$ on ${𝔛}_U$, in the sense that $I_s{I}_t=0$, for $s\ne t$, such that $ℒ({𝔛}_U)=𝒦({𝔛}_U)\oplus [{(I_s)}_{s=1}^{\mathrm{\infty }}]\oplus ℂI$ and ${(I_s)}_{s=1}^{\mathrm{\infty }}$ is equivalent to ${(u_s)}_{s=1}^{\mathrm{\infty }}$. In particular, $ℒ(X)/𝒦(X)$ is isomorphic, as a Banach algebra, to the unitization of $U$ with coordinate-wise multiplication. Banach spaces $U$ meeting these criteria include ${\mathrm{\ell }}_p$, with the unit vector basis, $L_p$, , with the Haar system, the asymptotic-${\mathrm{\ell }}_1$ Tsirelson space with its usual basis and many others. In the lecture, we discuss the steps of the solution of a simplified version of this problem, namely finding a “pairwise orthogonal” sequence of projections equivalent to ${(u_s)}_{s=1}^{\mathrm{\infty }}$ on some Banach space. If time permits, we will describe how this conceptual framework is combined with powerful construction techniques such as the Bourgain-Delbaen and Argyros-Haydon methods to achieve the final result.

This lecture is based on joint work with Anna Pelczar-Barwacz

This talk is based on joint work with Krystian Kazaniecki (J. Kepler University Linz)

We identify those filtered probability spaces $(\mathrm{\Omega },ℱ,\left({ℱ}_n\right),\mathbb{P})$ that determine already the martingale type of Banach space $X$. We do this by isolating intrinsic conditions on the filtration $({ℱ}_n)$