Relatively uniform Banach lattices

Sequential relative uniform and norm convergence agree in a Banach lattice, if and only if it is equivalent to an M space. Let (E, 0, An E R, An I 0, such that Ifn f K 0, such that given any E > O there exists a0 E A with If a f aO. Proposition 1. E is net relatively uniform if and only if E contains a strong unit. Proof. Suppose E is net relatively uniform. If a, f8 E E\tO0 write a 11jf> 1 Let fa= a, then 11 |-lim fa = 0, so ru-lim fa= 0. Hence E contains a strong unit. Conversely suppose E contains a strong unit e. The norms and II. lie are equivalent, by Birkhoff [1, Lemma 5, p. 367], where I|f I e inf tA: fl Ae}. O From now on we only consider sequential convergence. The following result is contained, implicitly, in Leader [3]. Lemma 1 (Leader). If E is relatively uniform then it is equivalent to an M space. Proof. By [3, Theorem 4] it follows that | is equivalent to an M norm if and only if, for every sequence tp(n) of positive integers, and for every sequence fIfni such that llfnll0, we have (1) iltfnl v Ifn+?1 V V.. v Ifn+P(n ""l O Received by the editors June 19, 1974. AMS (MOS) subject classifications (1970). Primary 46A40.