María Jesús Chasco

Some metric groups of functions

If (X,d) is a metric space and 0 is a fixed element of X,  \mathrm{Lip}_0(X,\mathbb{R}) denotes the Banach space of real-valued Lipschitz functions f defined on X and such that f(0)=0. One of its main properties is that it is isometric to the dual space of the norm-closed subspace of (\mathrm{Lip}_0(X,\mathbb{R}))^* generated by the Dirac measures. The latter space is of interest in its own right: It is the universal Banach space that contains an isometric copy of X.
Motivated by these important concepts we introduce the group of Lipschitz functions \mathrm{Lip}_0(X,\mathbb{T}) with values on the unit circle group \mathbb{T}, and the free abelian group A_d(X) on a metric space (X,d) endowed with the topology generated by the Graev extension \hat{d} of the given metric d on X. We will explore the close relation between them and derive some interesting properties of both metric groups,  {\rm Lip}_0(X,\mathbb{T}) and A_d(X).
(Joint work with X. Domínguez and M. G. Tkachenko)