**Petr Hjek**, Czech Technical University, Prague, Czech Republic

3 lectures

We plan to deliver 3 lectures, of a survey character, on 3 topics reflecting our recent research interests. The first one concerns Markusevic bases and their applications in Banach space theory. The second one concerns Lipschitz retractions in Banach and metric spaces. The third one concerns a new characterization of a finite dimensional Hilbert space.

**Jan Hamhalter**, Czech Technical University, Prague, Czech Republic

4 lectures

There are a few important locally convex topologies on von Neumann algebras such as weak $^*$-topology, Mackey topology, $\sigma$-strong operator topology, etc.. On the other hand, order of operators induces the order topology on the set of self-adjoint operators. This topology is nonlinear in general. We shall discuss many mutual connections between the above mentioned topologies and their relations to the structure of von Neumann algebras. For example, using noncommutative Egoroff theorem, we show that the order topology coincides with the $\sigma$-strong topology on bounded subsets. The order topologies on duals of C$^*$-algebras and preduals of von Neumann algebras will also be investigated. As a consequence we obtain, among others, characterizations of finite von Neumann algebra and Type I C$^*$-algebras in terms of convergence in order topologies.

The context poset $\mathcal C(\mathcal A)$ of a C$^*$-algebra $\mathcal A$ is the set of all abelian C$^*$-subalgebras of $\mathcal A$ endowed with the set theoretic inclusion. We show that any order isomorphism between context posets of nearly all von Neumann algebras is implemented by a Jordan *-isomorphism. (Jordan $*$ - morphism is a linear map preserving the *-operation $a\to a^*$ and the squares $a\to a^2$.) Order isomorphisms of context posets of general C$^*$-algebras (and Jordan algebras) will be described in terms of partially linear Jordan maps. These results are new also in the case of abelian algebras, where they give complete lattice theoretic invariant of compact Hausdorff spaces. We discuss applications of the results above to studying preservers of the Choquet order on state spaces of C$^*$-algebras.

Classical Lyapunov theorem states that the range of a vector-valued nonatomic (finite dimensional) measure is a convex set. Linear version of this result given by Lindenstrauss says that any weak *-continuous linear map from abelian nonatomic von Neumann algebra into $\mathbb{C}^n$ maps the set of all projections onto a convex set. This result (and its variants) were then extended by Akemann and Weaver to von Neumann algebras whose "orthodimension is bounded from above". In contrast to this we show that Lyapunov theorem holds for all algebras of "big size" and quite general maps. We generalize also Lyapunov theorem to the posets of tripotents in JBW$^*$-triplets and Jordan algebras. Applications of the results to determinacy of states and normal functionals will be discussed.

**Ondej Kalenda**, Charles University, Prague, Czech Republic

4 lectures

JB*-triples form a class of complex Banach spaces with an additional algebraic structure which may be viewed as a generalization of C*-algebras. In a series of lectures I will first try to explain why it is a very natural class of spaces - the algebraic and the metric structures on them determine each other and, moreover, JB*-triples appear naturally in the investigation of infinite-dimensional analogues of the Riemann mapping theorem. Next I will focus on the structure of tripotents (which generalize projections and partial isometries). I will present both a basic background and some recent results and open problems on certain order type relations connected to products of symmetries in von Neumann algebras. Finally I will present recent results on Grothendieck inequalities for C*-algebras and JB*-triples.

**Ondej Kurka**, Czech Academy of Sciences, Prague, Czech Republic

3 lectures

Part 1: The Bossard coding of separable Banach spaces, tree spaces and their application to universality questions.

Part 2: Construction of universal spaces for analytic classes of Banach spaces, so-called amalgamations.

Part 3: Equivalence relations and Borel reductions between them, the complexity of isometry and isomorphism between Banach spaces.