Rafael Payá

There is no bilinear Bishop-Phelps theorem

We answer a question posed by R. Aron, C. Finet and E. Werner, on the bilinear version of the Bishop-Phelps theorem, by exhibiting an example of a Banach spaceX such that the set of norm-attaining bilinear forms onX×X is not dense in the space of all continuous bilinear forms.

REAL BANACH SPACES WITH NUMERICAL INDEX 1

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l 1 . It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

Numerical index and renorming

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains [0,1/3[ in the real case and [e -1 ,1/2[ in the complex case. Moreover, for most Banach spaces the least upper bound of this interval is as large as possible, namely 1.

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.

A counterexample on numberical radius attaining operators

We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Our result was conjectured by C. Cardassi in 1985.

Norm attaining operators fromL1 intoL

We show that the set of norm attaining operators is dense in the space of all bounded linear operators fromL1 intoL∞.

An approximation property related to

We investigate a variant of the compact metric approximation property which, for subspaces X of c 0 , is known to be equivalent to K(X), the space of compact operators on X, being an M-ideal in the space of bounded operators on X, L(X). Among other things, it is shown that an arbitrary Banach space X has this property iff K(Y,X) is an M-ideal in L(Y,X) for all Banach spaces Y and, furthermore, that X must contain a copy of c 0 . The proof of the central theorem of this note uses a characterization of those Banach spaces X for which K(X) is an M-ideal in L(X) obtained earlier by the second author, as well as some techniques from Banach algebra theory

M

This paper gives new sucient conditions for the density of the set of norm attaining multilinear forms in the space of all continuous multilinear forms on a Banach space. The symmetric case is also discussed.