STUDIA MATHEMATICA 201 (3) (2010) Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators by Ma lgorzata Marta Czerwi´ nska and Anna Kami´ nska (Memphis, TN) Abstract. We investigate the relationships between strongly extreme, complex ex- treme, and complex locally uniformly rotund points of the unit ball of a symmetric func- tion space or a symmetric sequence space E, and of the unit ball of the space E(M,τ ) of τ -measurable operators associated to a semifinite von Neumann algebra (M,τ ) or of the unit ball in the unitary matrix space CE. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the sym- metric space E(M,τ ) inherit these properties from their singular value function μ(x) in the unit ball of E with additional necessary requirements on x in the case of com- plex extreme points. We also obtain the full converse statements for the von Neumann algebra M with a faithful, normal, σ-finite trace τ as well as for the unitary matrix space CE. Consequently, corresponding results on the global properties such as mid- point local uniform rotundity, complex rotundity and complex local uniform rotundity follow. Let E be a symmetric sequence space, and let C E be the unitary matrix space of compact operators acting on Hilbert space, associated with E. One of the points of interest in the theory of unitary matrix spaces is to inves- tigate what properties of the symmetric sequence space E are inherited by the unitary matrix space C E [22]. It was shown by Arazy in [2] that E is isometrically embedded in C E , and that the isometry V can be chosen with respect to any compact operator x in such a way that V (s(x)) = x, where s(x)= {s n (x)} ∞ n=1 is the sequence of singular numbers of x. Therefore many geometrical properties of x ∈ C E are also satisfied by s(x) ∈ E. In the same paper Arazy showed that x ∈ C E is an extreme (resp. smooth, exposed) point of the unit ball in C E if and only if s(x) is an extreme (resp. smooth, 2010 Mathematics Subject Classification : 46B20, 46B28, 47L05, 47L20. Key words and phrases : symmetric spaces of measurable operators, unitary matrix spaces, strongly extreme points, midpoint local uniform rotundity, complex extreme points, com- plex rotundity, complex locally uniformly extreme points, complex local uniform rotundity. DOI: 10.4064/sm201-3-3 [253] c  Instytut Matematyczny PAN, 2010