Topological amenability and Köthe co-echelon algebras

We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). Using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii–Sheinberg’s criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

On $$C^*$$-norms on $${{\mathbb {Z}}}_2$$-graded tensor products

We systematically investigate $$C^*$$ C ∗ -norms on the algebraic graded product of $${{\mathbb {Z}}}_2$$ Z 2 -graded $$C^*$$ C ∗ -algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $$C^*$$ C ∗ -norms. To this end, we first show that commutative $${{\mathbb {Z}}}_2$$ Z 2 -graded $$C^*$$ C ∗ -algebras enjoy a nuclearity property in the category of graded $$C^*$$ C ∗ -algebras. In addition, we provide a characterization of the extreme even states of a given graded $$C^*$$ C ∗ -algebra in terms of their restriction to its even part.

Norms of inclusions between some spaces of analytic functions

The inclusions between the Besov spaces $$B^q$$ B q , the Bloch space $$\mathcal {B}$$ B and the standard weighted Bergman spaces $$A^p_{\alpha}$$ A α p are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.