ORDER AND CLOSURE PROPERTIES OF ORDER CONTINUOUS OPERATORS ON THE SECOND DUAL OF THE SPACE OF CONTINUOUS FUNCTIONS
Abstract
Lex X be a compact Hausdorff space and let C(X) be the space of all real valued continuous functions on X. C'(X) and C''(X) are the first and second norm duals of C(X). (For brevity, we often refer to these as C, C', and C''.) We study the spaces L('b)(C,C) and L('c)(C'',C''), respectively the set of order bounded operators from C to C and the set of order continuous operators from C'' to C''. L('r)(C,C) is the subspace of L('b)(C,C) consisting of all differences of positive operators. L('b)(C,C) and L('r)(C,C) may be imbedded in L('c)(C'',C''). In chapter II we characterize operators in L('c)(C'',C'') which map C to lsc, the subspace of C'' consisting of all suprema of subsets of C. Every element of L('c)(C'',C'') which is the supremum of a subset of L('b)(C,C) maps C to lsc. When X is a metric space, the converse is true. We also show that if T(ELEM)L('c)(C'',C'') maps C to U, the subspace of C'' consisting of all order limits of nets in C, then there is a net {T(,(omega))}(L-HOOK)L('b)(C,C) such that T(,(omega))f order converges to Tf for all f(ELEM)C. In Chapter III, we show that L('r)(C,C) is order dense in the band of L('c)(C'',C'') which it generates. If X is a metric space, L('b)(C,C) is order dense in the band of L('c)(C'',C'') which it generates. Let L('r)(C,U) denote the subspace of L('c)(C'',C'') consisting of all differences of positive operators which map C to U. If X is a metric space, L('r)(C,U) is order dense in L('c)(C'',C''). Let C(,(mu))(''') be the band of C'' dual to C(,(mu))(''), which is the band of C' generated by a fixed (mu)(ELEM)C'. If f(ELEM)C'', f(,(mu)) is the projection of f onto this band. C(,(mu)) is the subspace of C(,(mu))(''') consisting of projections of elements of C. In chapter IV, we show that, for each T(ELEM)L('c)(C(,(mu))('''),C'') in the band generated by L('r)(C(,(mu)),C), there is an operator(' )T(ELEM)L('c)(C(,(mu))('''),C'') which takes values in Bo, such that(' )(Tf)(,(mu)) = (Tf)(,(mu)) for each f(ELEM)C''. Bo represents the space of Borel elements of C'', the smallest subspace of C'' which contains lsc and is closed under order convergence of sequences. Bo is isomorphic with the set of Borel functions on X. C(,(mu))(''') is isomorphic with the space L('(INFIN))((mu)).
Degree
Ph.D.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.