Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group

For each integer g2 we give the complete list of groups acting as a group of dianalytic automorphisms of a real projective plane with g holes, which, in algebraic terms, correspond to birational automorphisms of real algebraic (M–1)-curves. We also determine those acting as the full group of automorphisms of such a surface. Furthermore, the conjugacy classes of the finite subgroups of its mapping class group are calculated.Mathematics 2000 Subject Classification (2000): 30F, 32G, 14H.Partially supported by BFM2002-04801.Partially supported by BFM2002-04801 and RAAG HPRN-CT-2001-00271.Partially supported by GAAR BFM2002-04797 and RAAG HPRN-CT-2001-00271     

Derivations on Real and Complex JB*-Triples

At the regional conference held at the University of California,Irvine, in 1985 [24], Harald Upmeier posed three basic questionsregarding derivations on JB*-triples: (1) Are derivations automatically bounded? (2) When are all bounded derivations inner? (3) Can bounded derivations be approximated by inner derivations? These three questions had all been answered in the binary cases.Question 1 was answered affirmatively by Sakai [17] for C*-algebrasand by Upmeier [23] for JB-algebras. Question 2 was answeredby Sakai [18] and Kadison [12] for von Neumann algebras andby Upmeier [23] for JW-algebras. Question 3 was answered byUpmeier [23] for JB-algebras, and it follows trivially fromthe Kadison–Sakai answer to question 2 in the case ofC*-algebras. In the ternary case, both question 1 and question 3 were answeredby Barton and Friedman in [3] for complex JB*-triples. In thispaper, we consider question 2 for real and complex JBW*-triplesand question 1 and question 3 for real JB*-triples. A real orcomplex JB*-triple is said to have the inner derivation propertyif every derivation on it is inner. By pure algebra, every finite-dimensionalJB*-triple has the inner derivation property. Our main results,Theorems 2, 3 and 4 and Corollaries 2 and 3 determine whichof the infinite-dimensional real or complex Cartan factors havethe inner derivation property.     

Relatively weakly open sets in closed balls of Banach spaces,and real <Emphasis Type="Italic">JB</Emphasis><Superscript>*</Superscript>-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.Mathematics Subject Classification (2000): 46B04, 46B22, 46L05, 46L70Partially supported by Junta de Andalucía grant FQM 0199.Revised version: 30 September 2003     

Von Neumann Regularity and Quadratic Conorms in JB^＊-triples and C^＊-algebras

We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

Closed stability index of excellent henselian local rings

We show that the closed stability index of an excellent henselian local ring of real dimension d2 with real closed residue field is When d=2 it is shown that the value of can be either 2 or 3 and give characterizations of each of these values in terms of the relation of A with its normalization and in terms of the real spectrum of A.Mathematics Subject Classification (2000): 14P15, 32B10, 13J15, 13J25Partially supported by DGES BFM2002-04797 and EC contract HPRN-CT-2001-00271in final form: 15 October 2003     

Convexity and <Emphasis Type="Italic">w</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-compactness in Banach spaces

Suppose X. We construct examples of bounded sets MX, such that These examples show that the previous results of the authors on quantitative versions of Kreins theorem are optimal.Mathematics Subject Classification (2000):46B20, 46B26A. S. Granero supported by DGICYT grant BFM2001-1284. P. Hájek supported by GAR 201/01/1198, A 101 92 05 and UPV grant PPI-02-02.     

4‐*GDD(6n)s and Related Optimal Quaternary Constant‐Weight Codes

Constant‐weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K‐GDD (where GDD is group divisible design) with the “star” property, denoted by K‐*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4‐*GDDs. Previously, the necessary conditions for existence were shown to be sufficient for , and also sufficient for with prime powers and . We continue to investigate the existence of 4‐*GDD(6ⁿ)s and show that the necessary condition for the existence of a 4‐*GDD(6ⁿ), namely, , is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

The Birkhoff integral and the property of Bourgain

In this paper we study the Birkhoff integral of functions f:X defined on a complete probability space (,,) with values in a Banach space X. We prove that if f is bounded then its Birkhoff integrability is equivalent to the fact that the set of compositions of f with elements of the dual unit ball Z_f={x*,f:x*B_X*} has the Bourgain property. A non necessarily bounded function f is shown to be Birkhoff integrable if, and only if, Z_f is uniformly integrable and has the Bourgain property. As a consequence it turns out that the range of the indefinite integral of a Birkhoff integrable function is relatively norm compact. We characterize the weak Radon-Nikodým property in dual Banach spaces via Birkhoff integrable Radon-Nikodým derivatives. We also point out that a recently introduced notion of unconditional Riemann-Lebesgue integrability coincides with the notion of Birkhoff integrability. Some other applications are given.Mathematics Subject Classification (2000): 28B05, 46B22, 46G10Partially supported by the research grant BFM2002-01719 of MCyT (Spain). The second author was supported by a FPU grant of MECD (Spain).Acknowledgement We gratefully thank Gabriel Vera for lively discussions about some material considered in this paper. We also thank the referee for helpful suggestions that have improved the exposition of this work.     

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.     

The Eigenvalue Problem of Singular Ergodic Control

We consider the problem of finding a real number λ and a function u satisfying the PDE Here f is a convex, superlinear function. We prove that there is a unique λ* such that the above PDE has a viscosity solution u satisfying $\lim_{|x|\rightarrow \infty}u(x)/|x|=1$ . Moreover, we show that associated to λ* is a convex solution u* with D²u^*∈ $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L^{\infty}(\R^N)$ and give two min‐max formulae for λ*. λ* has a probabilistic interpretation as being the least, long‐time averaged (ergodic) cost for a singular control problem involving f. © 2011 Wiley Periodicals, Inc.     

The fixed point property in -triples and preduals of -triples

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.

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Dunford–Pettis properties, Hilbert spaces and projective tensor products

We study complete continuity properties of operators onto ℓ₂ and prove several results in the Dunford–Pettis theory of JB^*-triples and their projective tensor products, culminating in characterisations of the alternative Dunford–Pettis property for where E and F are JB^*-triples.     

A polynomial of degree four not satisfying Rolle’s theorem in the unit ball of <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>

We give an example of a fourth degree polynomial which does not satisfy Rolles Theorem in the unit ball of l ₂.The author has been partially supported by MCyT and FEDER Project BFM2002-01423.     

On the commutativity of <Emphasis Type="Italic">C</Emphasis>* -algebras

Let be a C* -algebra. Let f be a non-constant complex-valued continuous function defined on a closed interval I. We shall show that f densely spans As an application, is commutative if f(x)f(y)=f(y)f(x) for all self-adjoint elements x and y in with spectrums contained in I.Mathematics Subject Classification (1991):Primary 46L05     

Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem

We study relatively invariant measures with the multiplicators on the space of finite adeles. We prove that for such measures are ergodic, and then deduce from this the uniqueness of KMS-states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of on the full adeles.

     

Isometries between JB*-triples

Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z||<1, we show that Open image in new window if the Möbius transform induced by T(z) preserves the unit ball of T(Z). We show further that T is, locally, a triple homomorphism via a tripotent: for any z ∈ Z, there is a tripotent u in W** such that Open image in new window for all a, b, c in the smallest subtriple Z _z of Z containing z, and also, {u,T(·),u}:Z _z →W** is an isometry.