orthogonality preservers in c algebras jb algebras and jb triples abstract we study orthogonality preserving operators between c algebras jb algebras and jb triples let t a e be an orthogonality preserving bounded linear operator from a c algebra to a jb triple satisfying that t 1 d is a von neumann regular element then t a e 2 r d every element in t a and d operator commute in the jb algebra e 2 r d and there exists a triple homomorphism s a e 2 r d such that t l d r d s where r d denotes the range tripotent of d in e an analogous result for a being a jb algebra is also obtained when t a b is an operator between two c algebras we show that denoting h t 1 then t orthogonality preserving if and only if there exists a triple homomorphism s a b satisfying h s z s z h h s z s z h and t z l h r h s z 1 2 h r h s z s z r h h this allows us to prove that a bounded linear operator between two c algebras is orthogonality preserving if and only if it preserves zero triple products

the dunford pettis and the kadec klee properties on tensor products of jb triples we recall that a banach spacex satisfies the dunford pettis property dpp if every weakly compact operator t from x to another banach space is completely continuous i e t maps weakly cauchy sequences into norm convergent sequences the question if the projective tensor product of two banach spaces having the dpp also has this property has focused the attention of several researcher in 1983 m talagrand found a banach space x such that x has the schur property and l1 0 1 e does not satisfy the dpp see 34 four years latter r ryan showed that x y satisfies the dpp and contains no copies of 1 whenever x and y have both properties 33 f bombal and i villanueva prove in 7 see also 6 20 that whenever k1 and k2 are two infinite compact hausdorff spaces then the projective tensor product of c k1 and c k2 has the dpp if and only if k1 and k2 are scattered that is c k1 and c k2 contains no copies of 1 the interesting ideas developed by bombal fernandez and villanueva in 7 6 are the inspiration and the starting point for the present paper in the

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

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

weakly compact orthogonality preservers on c algebras we establish a complete description of weakly compact orthogonality preserving operators from a c algebra a or a jb algebra to a jb triple e we prove that any such operator is the sum of a series of mutually orthogonal triple homomorphisms from a to a certain peirce 2 subspace of e multiplied by a mutually orthogonal norm null sequence in e which enjoy certain operator commutativity relations

automatic continuity of derivations on c algebras and jb triples abstract we introduce the notion of banach jordan triple modules and determine the precise conditions under which every derivation from a jb triple e into a banach jordan triple e module is continuous in particular every derivation from a real or complex jb triple into its dual space is automatically continuous motivating the study which we have carried out elsewhere of weakly amenable jb triples specializing to c algebras leads to a unified treatment of derivations and jordan derivations into modules shedding light on a celebrated theorem of barry johnson

solution of the contractive projection problem in this paper we show that the class of j algebras a class of concrete jordan triple systems is stable under the action of norm one projections this result constitutes a general solution to a problem considered by the authors and others specifically our main result is theorem 2 let p be an arbitrary contractive projection defined on a j algebra m then p m is a jordan triple system in the triple product a b c a b c p f ab c t cb a for a b c e p m and p m has a faithful representation as a j algebra

geometric characterization of tripotents in real and complex jb triples abstract we establish a geometric characterization of tripotents in real and complex jb triples as a consequence we obtain an alternative proof of kaup s banach stone theorem for jb triples

blow up phenomena for the weakly dissipative dullin gottwald holm equation in this paper we study the cauchy problem for the weakly dissipative dullin gottwald holm equation we establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time

wave breaking for a periodic shallow water equation the focus of this paper is on the blow up of a recently derived one dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equations in the shallow water regime some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case

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

ergodic banach spaces abstract we show that any banach space contains a continuum of non isomorphic subspaces or a minimal subspace we define an ergodic banach space x as a space such that e 0 borel reduces to isomorphism on the set of subspaces of x and show that every banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block subspaces we also use our methods to get uniformity results we show that an unconditional basis of a banach space of which every block subspace is complemented must be asymptotically c 0 or l p and we deduce some new characterisations of the classical spaces c 0 and l p

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

local in space blow up criteria for a class of nonlinear dispersive wave equations abstract this paper is concerned with blow up phenomena for the nonlinear dispersive wave equation on the real line u t u x x t f u x f u x x x g u f u 2 u x 2 x 0 that includes the camassa holm equation as well as the hyperelastic rod wave equation f u k u 2 2 and g u 3 k u 2 2 as special cases we establish some a local in space blow up criterion i e a criterion involving only the properties of the data u 0 in a neighborhood of a single point simplifying and precising earlier blow up criteria for this equation

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

separate weak continuity of the triple product in dual real jb triples we prove that if e is a real jb triple having a predual e then e is the unique predual of e and the triple product on e is separately sigma e e continuous

aron berner extensions of triple maps with application to the bidual of jordan banach triple systems abstract by extending the notion of arens regularity of bilinear mappings we say that a bounded trilinear map on the cartesian product of banach spaces is aron berner regular when all its six aron berner extensions to the cartesian product of the bidual spaces coincide we give some results on the aron berner regularity of certain trilinear maps we then focus on the bidual e of a jordan banach triple system e and investigate those conditions under which e is itself a jordan banach triple system under each of the aron berner extensions of the triple product we also compare these six triple products with those arising from certain ultrafilters based on the ultrapower formulation of the principle of local reflexivity in particular we examine the aron berner triple products on the bidual of a jb triple in relation with the so called dineen s theorem some illuminating examples are included and some questions are also left undecided

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

isometries of real hilbert c modules abstract let t v w be a surjective real linear isometry between full real hilbert c modules over real c algebras a and b respectively we show that the following conditions are equivalent a t is a 2 isometry b t is a complete isometry c t preserves ternary products d t preserves inner products e t is a module map when a and b are commutative we give a full description of the structure of t

large time behavior for the support of momentum density of the camassa holm equation in this paper we study the large time behavior for the support of momentum density of the camassa holm equation more precisely we deduce the limit of the support of momentum density as t goes to in some sense

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

tingley s problem through the facial structure of an atomic jbw triple abstract we prove that every surjective isometry between the unit spheres of two atomic jbw triples e and b admits a unit extension to a surjective real linear isometry from e into b this result constitutes a new positive answer to tingley s problem in the jordan setting

blow up phenomena for the weakly dissipative dullin gottwald holm equation revisited in this paper we investigate the cauchy problem for the weakly dissipative dullin gottwald holm equation we first establish the local well posedness result by kato s semigroup theory then we obtain the precise blow up scenario and present some blow up results for strong solutions to the equation finally we discuss the blow up rate of the wave breaking solutions this result complements the early one in the literature such as novruzov j math phys 54 092703 2013

real contractive projections on commutative c algebras complex banach jordan triples occurred in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on complex c algebras these spaces are equipped with a ternary product the jordan triple product and are essentially geometric objects in that the linear isometries between them are exactly the linear maps preserving the triple product 13

blow up of solutions to the dgh equation in this paper firstly we find best constants for two convolution problems on the unit circle via a variational method then we apply the best constants on a nonlinear integrable shallow water equation the dgh equation to give sufficient conditions on the initial data which guarantee finite time singularity formation for the corresponding solutions finally we discuss the blow up phenomena for the nonperiodic case

the fixed point property in jb triples and preduals of jbw triples abstract we prove that for every member x in the class of real or complex jb triples or preduals of jbw triples the following assertions are equivalent 1 x has the fixed point property 2 x has the super fixed point property 3 x has normal structure 4 x has uniform normal structure 5 the banach space of x is reflexive as a consequence a real or complex c algebra or the predual of a real or complex w algebra having the fixed point property must be finite dimensional

on the zelmanovian classification of prime jb triples complex jb triples are complex banach spaces endowed with a triple product subjected to suitable conditions of algebraic and analytic nature see section 1 for the definition they were introduced by w kaup in order to provide an algebraic setting for the study of bounded symmetric domains in complex banach spaces the open unit ball of every complex jb triple is a bounded symmetric domain ka1 and every bounded symmetric domain in any complex banach space is bi holomorphically equivalent to the open unit ball of a suitable complex jb triple ka3 fundamental examples of complex jb triples are provided by complex c algebras with triple product defined by xyz x 1 2 xy z zy x 1

little grothendieck s theorem for real jb triples we prove that given a real jb triple e and a real hilbert space h then the set of those bounded linear operators t from e to h such that there exists a norm one functional varphi in e and corresponding pre hilbertian semi norm varphi on e such that