I shall discuss the theory of multi-norms. This has connections with norms on tensor products and with absolutely summing operators. There are many examples, some of which will be mentioned. In particular we shall describe multi-norms based on Banach lattices, define multi-bounded operators, and explain their connections with regular operators on lattices. We have new results on the equivalences of multi-norms. The theory of decompositions of Banach lattices with respect to the canonical 'Banach-lattice multi-norm' has a pleasing form because of a substantial theorem of Nigel Kalton that I shall state and discuss. I shall also discuss brie y a generalization that gives 'pmulti-norms' (for $1\leq p\leq1$) and an extension of a representation theorem of Pisier that shows that many pmulti-norms are 'sous-espaces de treillis'. The theory is based on joint work with Maxim Polyakov (deceasead), Hung Le Pham (Wellington), Matt Daws (Leeds), Paul Ramsden (Leeds), Oscar Blasco (Valencia), Niels Laustsen (Lancaster), Timur Oikhberg (Illinois), and Vladimir Troitsky (Edmonton).

multi-norms - equivalences - absolutely summing operators - tensor products[-]

I'll discuss the Banach algebra structure of the spaces of bounded linear operators on $\ell_p$ and $L_p$ := $L_p(0, 1)$. The main new results are
1. The only non trivial closed ideal in $L(L_p)$, 1 $\leq$ p < $\infty$, that has a left approximate identity is the ideal of compact operators (joint with N. C. Phillips and G. Schechtman).
2. There are infinitely many; in fact, a continuum; of closed ideals in $L(L_1)$ (joint with G. Pisier and G. Schechtman).
The second result answers a question from the 1978 book of A. Pietsch, “Operator ideals”.[-]