ON BAIRE-1 4 FUNCTIONS AND SPREADING MODELS

We prove a characterization of functions in B-1/4(K)\C(K), where K is a compact metric space in terms of c(o)-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B-1/4( K) we define a decreasing family (V-xi(K), parallel to.parallel to(xi) )(l less than or equal to xi < omega 1) of Banach spaces whose interse ction is DBSC(K) and we prove an analogous stronger property for the f unctions in V-xi(K)\C(K). Defining the s-spreading model-index, we cla ssify B-1/4(K) and we prove that s-SM[F] > xi for every F is an elemen t of V-xi(K). Also We classify the separable Banach spaces by defining the c(o)-SM-index which measures the degree to which they have sequen ces with extending spreading models equivalent to the usual basis of c (o). We give examples of Baire-1 functions and reflexive spaces with a rbitrary large indices.