The "positivity phenomenon" for Bessel sequences, frames and Riesz bases $\left(u_k\right)$ are studied in $L^2$ spaces over the compacts of homogeneous (Coifman-Weiss) type $\Omega=(\Omega, \rho, \mu)$. Under some relations between three basic metric-measure dimensions of $\Omega$, we obtain asymptotics for the mass moving norms $\left\|u_k\right\|_{K R}$ (Kantorovich-Rubinstein), as well as for singular numbers of the Lipschitz and Hajlasz-Sobolev embeddings. Our main observation shows that, quantitatively, the rate of the convergence $\left\|u_k\right\|_{K R} \longrightarrow 0$ depends on an interplay between geometric doubling and measure doubling/halving exponents. The "more homogeneous" is the space, the sharper are the results.[-]

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”.[-]