geometric aspects of the daugavet property let x be a closed subspace of a banach space y and j be the inclusion map we say that the pair x y has the daugavet property if for every rank one bounded linear operator t from x to y the following equality kj tk 1 ktk 1 holds a new characterization of the daugavet property in terms of weak open sets is given it is shown that the operators not xing

the group of isometries of a banach space and duality we construct an example of a real banach space whose group of surjective isometries has no uniformly continuous one parameter semigroups but the group of surjective isometries of its dual contains infinitely many of them other examples concerning numerical index hermitian operators and dissipative operators are also shown

an alternative daugavet property abstract we introduce a strictly weaker version of the daugavet property as follows a banach space x has this alternative daugavet property adp in short if the norm identity ade max 1 id t 1 t holds for all rank one operators t x x in such a case all weakly compact operators on x also satisfy ade we give some geometric characterizations of the alternative daugavet property in terms of the space and its successive duals we prove that the adp is stable for c0 l1 and l sums and characterize when some vector valued function spaces have the property finally we show that a c algebra or the predual of a von neumann algebra has the adp if and only if its atomic projection respectively the atomic projection of the algebra are central we also establish some geometric properties of jb triples and characterize jb triples possessing the adp and the daugavet property

the daugavet equation for operators on function spaces abstract we prove the norm identity id t 1 t which is known as the daugavet equation for weakly compact operators t on natural function spaces such as function algebras and l 1 predual spaces provided a non discreteness assumption is met we also consider c 0 factorable operators and operators on c spaces

octahedral norms and convex combination of slices in banach spaces abstract we study the relation between octahedral norms daugavet property and the size of convex combinations of slices in banach spaces we prove that the norm of an arbitrary banach space is octahedral if and only if every convex combination of w slices in the dual unit ball has diameter 2 which answers an open question as a consequence we get that the banach spaces with the daugavet property and its dual spaces have octahedral norms also we show that for every separable banach space containing l 1 and for every e 0 there is an equivalent norm so that every convex combination of w slices in the dual unit ball has diameter at least 2 e

octahedral norms in spaces of operators abstract we study octahedral norms in the space of bounded linear operators between banach spaces in fact we prove that l x y has octahedral norm whenever x and y have octahedral norm as a consequence the space of operators l l 1 x has octahedral norm if and only if x has octahedral norm these results also allow us to get the stability of strong diameter 2 property for projective tensor products of banach spaces which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of banach spaces

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

big slices versus big relatively weakly open subsets in banach spaces abstract we study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in banach spaces we show that every banach space containing c 0 isomorphically can be equivalently renormed so that every slice of its unit ball has diameter 2 and still its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small which answers an open question and stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of banach spaces

extreme differences between weakly open subsets and convex combinations of slices in banach spaces abstract we show that every banach space containing isomorphic copies of c 0 can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and however its unit ball still contains convex combinations of slices with diameter arbitrarily small which improves in an optimal way the known results about the size of this kind of subsets in banach spaces

banach spaces with the daugavet property and the centralizer we introduce representable banach spaces and prove that the class r of such spaces satisfies the following properties

the daugavet property of c algebras jb triples and of their isometric preduals a banach space x is said to have the daugavet property if every rank one operator t x x satisfies id t 1 t we give geometric characterizations of this property in the settings of c algebras jb triples and their isometric preduals we also show that in these settings the daugavet property passes to ultrapowers and thus it is equivalent to an stronger property called the uniform daugavet property

relatively weakly open sets in closed balls of banach spaces and the centralizer we prove that if the centralizer of a banach space x is infinite dimensional then every nonempty relatively weakly open subset of the closed unit ball of x has diameter equal to 2 this result together with a suitable refinement also proven in the paper contains and improves in some cases previously known facts for c algebras jb triples spaces of vector valued continuous functions and spaces of operators

weighted composition operators as daugavet centers abstract we investigate the norm identity u c t u t for classes of operators on c s where s is a compact hausdorff space without isolated point and characterize those weighted composition operators which satisfy this equation for every weakly compact operator t c s c s we also give a characterization of such weighted composition operator acting on the disk algebra a d

weakly open sets in the unit ball of the projective tensor product of banach spaces abstract a banach space is said to have the diameter two property if every non empty relatively weakly open subset of its unit ball has diameter two we prove that the projective tensor product of two banach spaces whose centralizer is infinite dimensional has the diameter two property the same statement also holds for x y if the centralizer of x is infinite dimensional and the unit sphere of y contains an element of numerical index one we provide examples of classical banach spaces satisfying the assumptions of the results if k is any infinite compact hausdorff topological space then c k y has the diameter two property for any nonzero banach space y we also provide a result on the diameter two property for the injective tensor product

slicely countably determined banach spaces we introduce the class of slicely countably determined banach spaces which contains in particular all spaces with the radon nikod m property and all spaces without copies of l 1 we present many examples and several properties of this class we give some applications to banach spaces with the daugavet and the alternative daugavet properties lush spaces and banach spaces with numerical index 1 in particular we show that the dual of a real infinite dimensional banach space with the alternative daugavet property contains l 1 and that operators which do not fix copies of l 1 on a space with the alternative daugavet property satisfy the alternative daugavet equation

the daugavet equation in uniformly convex banach spaces abstract it is shown that a continuous operator t x x on a uniformly convex banach space satisfies the daugavet equation i t 1 t if and only if the norm t of the operator lies in the spectrum of t specializing this result to compact operators we see that a compact operator on a uniformly convex banach space satisfies the daugavet equation if and only if its norm is an eigenvalue the latter conclusion is in sharp contrast with the standard facts on the daugavet equation for the spaces l 1 and l a discussion of the daugavet property in the latter spaces is also included in the paper

on the n concavity of covering spaces with parameters x t where x and t are connected complex manifolds of dimensions n m and m respectively is a proper and surjective holomorphic submersion and is a covering map thus x can be regarded as a family of n dimensional complex manifolds over t we will use the same definition of q convexity as in 1 for the precise formulation see the next section r green and h wu 7 proved that a connected non compact complex manifold is n complete in 5 m col oiu and v v j itu proved the following

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

banach spaces with the daugavet property a banach space x is said to have the daugavet property if every operator t x x of rank 1 satisfies 11 id tl 1 fltil we show that then every weakly compact operator satisfies this equation as well and that x contains a copy of t1 however x need not contain a copy of l1 we also study pairs of spaces x c y and operators t x y satisfying i i j t i i 1 4 1f t i i where j x y is the natural embedding this leads to the result that a banach space with the daugavet property does not embed into a space with an unconditional basis in another direction we investigate spaces where the set of operators with f id tll 1 il t l i is as small as possible and give characterisations in terms of a smoothness condition

non symplectic automorphisms of order 3 on k3 surfaces in this paper we study k3 surfaces with a non symplectic automorphism of order 3 in particular we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology this allows us to describe the structure of the moduli space and to show that it has three irreducible components