Various slicing indices on Banach spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(a) spaces, where a is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is omega (first infinite ordinal), then its weak*-dentability index is at most w 2 and that this estimate is optimal.