Contractivity theorems in real ordered Banach spaces with applications to relative operator bounds, ergodic projections and conditional expectations

This paper provides various "contractivity" results for linear operators of the form I C where C are positive contractions on real ordered Banach spaces X: If A generates a positive contraction semigroup in Lebesgue spaces L p (), we show (M. Pierre's result) that A(A) 1 is a "contraction on the positive cone", i.e. A(A) 1 x kxk for all x 2 L p + () (> 0); provided that p > 2: We show also that this result is not true for 1 p < 2: We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X + : We deduce from this result that, in such spaces, I C is a contraction on X + for any positive projection C with norm 1: We give also a direct proof (by E. Ricard) of this last result if additionally the norm is smooth on the positive cone. For any positive contraction C on base-norm spaces X (e.g. in real L 1 () spaces or in preduals of hermitian part of von Neumann algebras), we show that N (u Cu) kuk 8u 2 X where N is the canonical half-norm in X: For any positive contraction C on order-unit spaces X (e.g. on the hermitian part of a C algebra), we show that I C is a contraction on X + : Applications to relative operator bounds, ergodic projections and conditional expectations are given