Kazhdan projections are usually considred objects relevant in operator algebras. In particular, they played a central part in the construction of counter-examples to the Baum-Connes conjecture.
In this talk I shall explain how, in the general setting of a family of representations on Banach spaces, one can reformulate the Kazhdan property "almost invariant implies invariant vectors" in terms of Kazhdan projections, providing also an explicit formula of the latter, using Markov operators associated to a random walk on the group. I will then explain some applications of this new approach.
This is joint work with Piotr Nowak.[-]

Motivated by considerations from quantum information theory, we study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $\mathrm{X}$ as the limit of the sequence $A_{k}^{1 / k}$, where $A_{k}$ is the equivalence constant between the projective and injective norms on $X^{\otimes} k$. We show in particular that Euclidean spaces are characterized by the property that their tensor radius equals their dimension.
Joint work with Alexander Müller-Hermes, arXiv:2110.12828[-]