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We study Mazur rotations problem focusing on the metric aspects of the action of the isometry group and semitransitivity properties. [ABSTRACT FROM AUTHOR]

Abstract: We show that is the only Banach space with unconditional basis that satisfies the equation . This partially improves an old result by Kalton and Peck. We prove that the Kalton–Peck maps are strictly singular on a number of sequence spaces, including for , Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton–Peck type. [Copyright &y& Elsevier]

We exhibit a real Banach space M such that C(K, M) is almost transitive if K is the Cantor set, the growth of the integers in its Stone-Čech compactification or the maximal ideal space of L∞. For finite K, the space C(K, M) = M∣K∣ is even transitive. [ABSTRACT FROM AUTHOR]

We describe complemented copies of ℓ2 both in C( K1)⊗̂π C( K2) when at least one of the compact spaces Ki is not scattered and in L1(μ1)⊗̂ɛ L1(μ2) when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the [ABSTRACT FROM AUTHOR]