If $E=e_i$ and $F=f_i$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i,j}$ with norm $\left|\right|a_{i,j}{\left|\right|}_{E\left(F\right)}=\left|\right|{\sum }_k\left|\right|{\sum }_l{a}_{k,l}{f}_l\left|\right|e_k\left|\right|$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

We prove, among other things, that the space C[0,ω₂] has no countably norming Markushevich basis. This answers a question asked by G. Alexandrov and A. Plichko.

We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $\left(e_k\right)$ is said to be subsequentially minimal if for every normalized block basis $\left(x_k\right)$ of $\left(e_k\right)$, there is a further block basis $\left(y_k\right)$ of $\left(x_k\right)$ such that $\left(y_k\right)$ is equivalent to a subsequence of $\left(e_k\right)$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson...