HERE WE STUDY THE CLASS OF OPERATORS SO CALLED ALMOSTDUNFORD PETTIS COMPLETELY CONTINUOUS (DPACC) OPERATORS ON BANACH LATTICES AND WE GIVE SOME PROPERTIES OF THEM RELATED TO SOMEWELL-KNOWN CLASSES OF OPERATORS.

A BANACH LATTICE E HAS THE SCHUR PROPERTY, IF EVERY WEAKLY NULL SEQUENCE IN E IS NORM NULL. ALSO A BANACH LATTICE E HAS THE DUNFORD-PETTIS PROPERTY, IF FOR EACH WEAKLY NULL SEQUENCES (XN) IN E AND (X′N) IN THE DUAL E′ OF E, ONE HAS X′N(XN) ® 0 AS N®¥. HERE WE OBTAIN SOME CLASSES OF BANACH LATTICES WITH SCHUR OR DUNFORD-PETTIS PROPERTY.

FOLLOWING THE INTRODUCING OF THE CONCEPT OF L-LIMITEDSETS IN DUAL BANACH SPACES INTRODUCED BY SALIMI AND MOSHTAGHIOUN, WE DEFINE L-DUNFORD–PETTIS SETS AND L-DUNFORD–PETTIS PROPERTY.WE OBTAIN SOME CHARACTERIZATIONS OF THEM WITH RESPECT TO SOME WELLKNOWN GEOMETRIC PROPERTIES OF BANACH SPACES, SUCH AS, DUNFORDPETTIS PROPERTY, RELATIVELY COMPACT DUNFORD-PETTIS PROPERTY ANDDUNFORD-PETTIS COMPLETELY CONTINUOUS OPERATORS ON SUCH BANACHSPACES.

FOLLOWING THE CONCEPT OF L-LIMITED SETS IN DUAL BANACH SPACES, WE INTRODUCE THE CONCEPT OF ALMOST L-DUNFORD-PETTIS SETS IN DUAL BANACH LATTICES. THEN BY A CLASS OF OPERATORS ON BANACH LATTICES, SO CALLED DISJOINT DUNFORD-PETTIS COMPLETELY CONTINUOUS OPERATORS, WE CHARACTERIZE BANACH LATTICES WITH THE POSITIVE RELATIVELY COMPACT DUNFORD-PETTIS PROPERTY.

IN THIS PAPER, WE PRESENT A CHARACTERIZATION OF THE CLOSED IDEAL OF A TRIVOLUTIVE BANACH ALGEBRA (A,T).THE MAIN RESULTS ARE: A) THE CLOSED IDEAL I=KER T OF A UNITAL BANACH ALGEBRA A IS PRINCIPLE. B) FOR A COMMUTATIVE TRIVOLUTIVE ALGEBRA A, THERE IS AN IDEMPOTENT UÎA** SUCH THAT KER T =IU IN WHICH IU={A Î A; AU=0}. AFTER THAT WE INVESTIGATE TRIVOLUTIONS ON THE FOURIER-STIELTJES ALGEBRA B (G): MOREOVER, WE STUDY TRIVOLUTIONS ON THE ALGEBRA A´QB WITH Q-LAU PRODUCT IN RELATION TO THE CORRESPONDING ONE OF A AND B.

IN THIS PAPER, WE INVESTIGATE SOME THE ALGEBRAIC STRUCTURE OF TU(¦) = U * ¦, SO-CALLED LAMBERT OPERATOR ON L2(Å) SPACES, THAT U WILL BE ANY MEASURABLE AND CONDITION ABLE FUNCTION.