A dynamical hierarchy of Banach algebras — Matthias Neufang <mneufang@math.carleton.ca>

The concept of stability in the sense of Krivine--Maurey has proven very useful in Banach space geometry. We introduce and study a corresponding notion in the setting of Banach algebras, which we call multiplicative stability. As we shall see, the algebra of Schatten $p$-class operators on a separable Hilbert space is multiplicatively stable, where $p \in [1, \infty)$, while no infinite-dimensional $C^*$-algebra is. We also explore stronger and weaker versions of this concept, using the rich structure of spaces of functions defined on topological semigroups, including almost periodic, weakly almost periodic, and tame functions. This leads us to a novel classification of Banach algebras, providing a dynamical hierarchy. In this context, we also investigate further important classes of Banach algebras, such as group algebras and Fourier algebras, algebras of compact operators on Banach spaces, and algebras of differentiable functions. The talk is based on joint work with (my former PhD student) Narjes Alabkary, (my former postdoctoral fellow) Reza Esmailvandi, and Stefano Ferri.

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1. Plenary

Dynamics of unit groups of von Neumann's continuous rings — Friedrich Martin Schneider <martin.schneider@math.tu-freiberg.de>

In the 1930s, John von Neumann developed a continuous-dimensional analogue of finite-dimensional projective geometry. Inspired by conversations with Garrett Birkhoff as well as his collaboration with Francis Murray on rings of operators, von Neumann introduced and studied the notion of a continuous geometry, which is a complete complemented modular lattice possessing a certain continuity property. Among other remarkable results, von Neumann proved that every continuous geometry of order at least four can be coordinatized by some (up to isomorphism unique) ring, and that the continuous rings (i.e., rings corresponding to continuous geometries via this coordinatization theorem) are precisely those irreducible, regular rings which admit a complete rank function. The necessarily unique rank function of a continuous ring gives rise to a compatible metric and thus furnishes the ring with a natural topology. Unit groups of such continuous rings, equipped with the relative topology, constitute an interesting family of topological groups with many peculiar dynamical properties. The talk will provide an introduction to von Neumann's continuous geometry and discuss some of the latest advances concerning topological dynamics of unit groups of continuous rings.

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Extension of maps into equivariant hulls of convex sets — Sergey Antonyan <antonyan@unam.mx>

We will establish the following equivariant extension theorem. Let $G$ be a compact Lie group, $L$ a locally convex metrizable linear $G$-space, and $V$ a closed convex subset of $L$. Denote $G(V):=\{gv\mid g\in G, v\in V\}$ -- the equivariant hull of $V$. Then any $G$-equivariant map $f:A\to G(V)$ defined on a closed invariant subset of a metrizable $G$-space $X$, extends to a $G$-equivariant map $F:U\to G(V)$ over some invariant neighborhood $U$ of $A$ in $X$. If, in addition, $V$ contains a $G$-fixed point, the extension can be taken over the whole space, i.e. $U=X$. In particular, any continuous map $f:A\to G(V)$ from a closed subset of a metrizable space $X$, extends to a continuous map $F:U\to G(V)$ over some neighborhood $U$ of $A$ in $X$. Several applications will be discussed.

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Frechet derivative is of first Baire class — Eva Kopecka <eva.kopecka@uibk.ac.at>

Let $X$ and $Y$ be Banach spaces, $G\subset X$ an open set and $f:G \to Y$ a mapping. We show that the Fr\'echet derivative $f'$ of $f$ is of first Baire class on the (possibly empty) set $D\subset G$ where it is defined.

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1. Plenary

From group theory to topological data analysis: asymptotic dimension and the Gromov–Hausdorff distance — Nicolò Zava <nicolo.zava@ist.ac.at>

In his seminal work on finitely generated groups, Gromov established that such groups possess a well-defined large-scale metric structure induced by the word metric of a finite generating set. This perspective transformed geometric group theory by introducing quasi-isometric invariants, a prominent example of which is the asymptotic dimension—a large-scale analogue of the Lebesgue covering dimension. A parallel milestone in this geometric framework was the proof of Gromov's polynomial growth theorem, which characterises groups with polynomial growth, utilising the Gromov–Hausdorff distance to quantify dissimilarities between metric spaces. Almost a decade later, Topological Data Analysis (TDA), a field at the interplay of computational geometry, computer science, and algebraic topology, emerged to study the shape of data. The main tools are topology-inspired invariants, such as persistent homology, used to extract geometric features from datasets. Within this framework, both classic metric notions found new, independent utilities. The Gromov–Hausdorff distance became a standard tool for comparing datasets and evaluating the stability of invariants. The asymptotic dimension was used to analyse the spaces of these invariants, thereby bounding the unavoidable information loss incurred during their vectorisation, a necessary step to integrate them into statistical and machine learning pipelines. In this talk, we discuss how the asymptotic dimension and the Gromov–Hausdorff distance, originally introduced in the realm of topological methods to study algebraic structures, have gained a crucial role in TDA, and present recent results that bridge these notions by determining the asymptotic dimension of the Gromov–Hausdorff space.

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Generalized spine algebras and their homomorphisms — Ross Stokke <r.stokke@uwinnipeg.ca>

For a locally compact group, *G*, its Fourier and Fourier--Stieltjes algebras *A(G)* and *B(G)* are Banach algebras of continuous functions on *G* that uniquely determine *G* as a topological group; when *G* is abelian, *A(G)* and *B(G)* can be identified via the Fourier--Stieltjes transform with the group and measure convolution algebras on the dual group of *G*. An old problem, solved in the abelian case by Paul Cohen in 1960, asks for a description of all homomorphisms from *A(G)* into *B(H)*. For non-abelian groups, M. Ilie, N. Spronk, M. Daws and H.L. Pham have, among others, made significant contributions to this problem. The difficulty of the problem of describing homomorphisms from *A* into *B(H)* where *A* is some other closed translation-invariant subalgebra of *B(G)* is significantly impacted by the complexity of the Gelfand spectrum of *A*. While the Gelfand spectrum of *A(G)* is just *G* and the spectrum of *B(G)* is often inaccessible, the spine of *B(G)*, *A'(G)*, is a subalgebra of *B(G)* containing *A(G)* whose spectrum is of intermediate complexity between the spectra of *A(G)* and *B(G)*. The spine algebra was introduced by J. Inoue and J. Taylor for abelian groups and by M. Ilie and N. Spronk for nonabelian locally compact groups. For any upper semilattice *D* of locally precompact topologies on *G*, we will define an associated generalized spine subalgebra *AD'(G)* of *B(G)*; when *D* is the set of all locally precompact topologies, we obtain the full spine algebra *A'(G)*. We will discuss properties of generalized spine algebras and identify their spectra as certain semilattices of topological groups. Using almost periodic compactifications, we will introduce a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. Notions of compatible fusions of homomorphisms and affine maps will be introduced and used to characterize all completely positive, completely contractive and, when *G* is amenable, completely bounded homomorphisms from a generalized spine algebra *AD'(G)* to a Fourier--Stieltjes algebra *B(H)*. These results are new, even when *AD'(G)* is the full spine algebra *A'(G)* and even when *G* and *H* are abelian. Examples illustrating the scope of these theorems will be discussed. This is joint work with Nico Spronk and Aasaimani Thamizhazhagan.

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Limits of abstraction for convergence theory — Szymon Dolecki <dolecki@u-bourgogne.fr>

Convergence theory studies relations between filters and points. Continuity assures existence of initial and final convergences. This framework enables us to define functors objectwise, like in topological constructs, but a slightly higher level of abstraction in the latter case makes the formalism much more complex. The extension of the concept of adherence to arbitrary families makes it possible to treat various reflective classes of convergence as special cases of types of compactness of families, and not only of sets. This approach enables one to see various classes of quotientness and perfection of maps as forms of compactness of corresponding relations. Our approach has the advantage to represent classical topological properties as solutions to functorial inequalities. Is there any point to consider relations between arbitrary isotone families, not only filters, and points? Greco’s theory of limitoids constitutes the affirmative answer to this question. A limitoid is a functional $T:L^X \rightarrow L$, where $L$ is a complete lattice and $X$ is a set, which is isotone, and commutes with lattice complete homomorphisms. If $L$ is completely distributive, then each limitoid can be represented as a lower limit along an isotone family, which, in general, is not a filter. As all the variational limits of De Giorgi are limitoids, Greco’s theory is a powerful tool for the latter. But as contours are, in fact, lower limits over families of sets, limitoids apply to diagonality and to regularity in convergence theory.

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Negation functions in fuzzy metric spaces: topological aspects and fixed point results — Juan-José Miñana <juamiapr@mat.upv.es>

The theory of fuzzy metric spaces, originating from the foundational work of Kramosil and Michalek, continues to be an active and relevant area of research. From a topological perspective, these spaces have been extensively investigated, while fixed point theory within this framework remains a topic of ongoing interest. A recent contribution has explored the incorporation of negation functions as a tool to develop a more general setting in fuzzy metric spaces. In particular, such functions have been proposed both to define alternative topological structures and to extend existing fixed point results. In this talk, we examine the role of negation functions from these two viewpoints. Our analysis shows that whenever an alternative way of deriving a topology can be obtained through negation functions, it coincides with the classical topology introduced by George and Veeramani. Furthermore, when negation functions are assumed to be strict or strong, the resulting classes of fuzzy contractions do not provide genuine extensions of previously known ones. Consequently, the fixed point results obtained in this context can be regarded as direct corollaries of earlier theorems.

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Recurrence and rigidity of multipliers on commutative Banach algebras I — Enrique Jordá <ejorda@mat.upv.es>

**Definition** [Costakis, Manoussos, Parissis] Let $X$ be a Banach space. We say that a bounded linear operator $T\colon X\to X$ is 1. *Recurrent*: if for each $x\in X$ there exists an increasing sequence $(n_k)$ of natural numbers such that $T^{n_k}(x)$ converges to $x$. 2. *Rigid*: if there exists an increasing sequence of natural numbers $(n_k)$ such that $T^{n_k}(x)$ is convergent to $x$ for every $x\in X$. 3. *Uniformly rigid*: if there exists an increasing sequence $(n_k)$ of natural numbers such that $\lim_k \|T^{n_k}-I\|=0$. In these consecutive talks, we report on joint work by M.J. Beltrán, E. Jordá and J. Galindo where we study the recurrence and rigidity of multipliers on semisimple Banach algebras and analyze the case of the Fourier algebra $A(G)$ of a locally compact group. We will address the following problems: -- Let $T\colon \mathfrak{A}\to \mathfrak{A}$ be a multiplier of a semisimple Banach algebra $\mathfrak{A}$ with spectrum $\Delta(\mathfrak{A})$. Its Gelfand transform, $\widehat{T}$, then defines a multiplier $M_{_{\widehat{T}}}\,\colon C_0(\Delta(\mathfrak{A}))\to C_0(\Delta(\mathfrak{A}))$. Find the relations between the recurrence and rigidity of these multipliers and determine under which conditions they coincide. -- Describe the recurrent and (uniformly) rigid multipliers of the Fourier algebra and find examples separating these concepts. Our results show that for power-bounded multipliers on the most common algebras (including Fourier algebras), the recurrence and rigidity properties of a multiplier $T$ coincide exactly with those of $M_{_{\widehat{T}}}$; this will be the subject of Talk I. In the absence of power-boundedness this equivalence fails, and we provide counterexamples within the framework of Fourier algebras; these will be discussed in Talk II.

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Recurrence and rigidity of multipliers on commutative Banach algebras II — Jorge Galindo <jgalindo@uji.es>

**Definition** [Costakis, Manoussos, Parissis] Let $X$ be a Banach space. We say that a bounded linear operator $T\colon X\to X$ is 1. *Recurrent*: if for each $x\in X$ there exists an increasing sequence $(n_k)$ of natural numbers such that $T^{n_k}(x)$ converges to $x$. 2. *Rigid*: if there exists an increasing sequence of natural numbers $(n_k)$ such that $T^{n_k}(x)$ is convergent to $x$ for every $x\in X$. 3. *Uniformly rigid*: if there exists an increasing sequence $(n_k)$ of natural numbers such that $\lim_k \|T^{n_k}-I\|=0$. In these consecutive talks, we report on joint work by M.J. Beltrán, E. Jordá and J. Galindo where we study the recurrence and rigidity of multipliers on semisimple Banach algebras and analyze the case of the Fourier algebra $A(G)$ of a locally compact group. We will address the following problems: -- Let $T\colon \mathfrak{A}\to \mathfrak{A}$ be a multiplier of a semisimple Banach algebra $\mathfrak{A}$ with spectrum $\Delta(\mathfrak{A})$. Its Gelfand transform, $\widehat{T}$, then defines a multiplier $M_{_{\widehat{T}}}\,\colon C_0(\Delta(\mathfrak{A}))\to C_0(\Delta(\mathfrak{A}))$. Find the relations between the recurrence and rigidity of these multipliers and determine under which conditions they coincide. -- Describe the recurrent and (uniformly) rigid multipliers of the Fourier algebra and find examples separating these concepts. Our results show that for power-bounded multipliers on the most common algebras (including Fourier algebras), the recurrence and rigidity properties of a multiplier $T$ coincide exactly with those of $M_{_{\widehat{T}}}$; this will be the subject of Talk I. In the absence of power-boundedness this equivalence fails, and we provide counterexamples within the framework of Fourier algebras; these will be discussed in Talk II.

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Spectra of Beurling Algebras of locally compact abelian groups — Nico Spronk <nspronk@uwaterloo.ca>

Consider a locally compact abelian group. I will construct its universal real topological vector space and demonstrate a bijective correspondence between Gelfand spectra of Beurling algebra and weak*-closed compact convex sets of the dual of this vector space.

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