This is an announcement for the paper “On a second numerical index for Banach spaces” by Sun Kwang Kim, Han Ju Lee, Miguel Martin and Javier Meri.
Abstract: We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces$\ldots$ which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollob$'$as property for numerical radius is given.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.06198