Hello, The next Banach spaces webinar is on Friday July 2 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Title: Closed ideals in the algebra of compact-by-approximable operators Speaker. Henrik Johannes Wirzenius (University of Helsinki) Abstract: In this talk I will present various examples of non-trivial closed ideals of the compact-by-approximable quotient algebra $\mathfrak A_X=\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. Here $\mathcal K(X)$ denotes the algebra of compact operators $X\to X$ and $\mathcal A(X)=\overline{\mathcal F(X)}$ is the uniform norm closure of the bounded finite rank operators $\mathcal F(X)$. The examples include: (i) If $X$ has cotype 2, $Y$ has type 2, $\mathfrak A_X\neq\{0\}$ and $\mathfrak A_Y\neq\{0\}$, then $\mathfrak A_{X\oplus Y}$ has at least 2 (and in some cases up to 8) closed ideals. (ii) For all $4\lt p\lt \infty$ there are closed subspaces $X\subset\ell^p$ and $X\subset c_0$ such that $\mathfrak A_X$ has a non-trivial closed ideal. (iii) A Banach space $Z$ such that $\mathfrak A_Z$ contains an uncountable lattice of closed ideals. The talk is based on a recent preprint [arXiv:2105.08403] together with Hans-Olav Tylli (University of Helsinki). For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari