Three observations regarding Schatten p classes ∗

The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation of a new complemented subspace of Cp in the reflexive range (and p ̸= 2). This construction answers a question of Arazy and Lindestrauss from 1975. The second result relates to tight embeddings of finite dimensional subspaces of Cp in C n p with small n and shows that l k p nicely embeds into Cn p only if n is at least proportional to k (and then of course the dimension of Cn p is at least of order k 2). The third result concerns single elements of Cn p and shows that for p > 2 any n × n matrix of Cp norm one and zero diagonal admits, for every ε > 0, a k-paving of Cp norm at most ε with k depending on ε and p only.