Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

This is a review article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau )$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Pólya, Köthe duality, the spaces $L_p(\mathcal{M},\tau )$, $1\le p<\mathrm{\infty }$, the identification of $C_E$ and $G(B(H),\mathrm{tr})$ for some symmetric function space $G$, the commutative case when $E$ is identified with $E(\mathcal{N},\tau )$ for $\mathcal{N}$ isometric to $L_{\mathrm{\infty }}$ with the standard integral trace, trace preserving $\ast$-isomorphisms between $E$ and a $\ast$-subalgebra of $E(\mathcal{M},\tau )$, and a general method for removing the assumption of non-atomicity of $\mathcal{M}$. The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, $k$-extreme points and $k$-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec−Klee properties, Banach−Saks properties, Radon−Nikodym property and stability in the sense of Krivine−Maurey. We also state some open problems.