IF A, B ARE TWO C *-ALGEBRAS AND M IS A HILBERT A-MODULE ONE CAN INDUCE, BY MEANS OF A *-ISOMORPHISM J : A ® B, A HILBERT B-MODULE STRUCTURE OVER M. IN THIS PAPER WE SHOW THAT IF M IS A FULL HILBERT A-MODULE, THEN THE INDUCED HILBERT B-MODULE IS ALSO FULL AND THEIR INNER PRODUCTS WILL BE EQUIVALENT.

THIS ARTICLE, WE START WITH A COLLECTION  M OF MORPHISMS OF A SMALL CATEGORY X, THAT SATISFIES CERTAIN CONDITIONS AND WE CONSTRUCT AN UNIVERSAL CLOSURE OPERATION BY GENERAL METHOD. NEXT DESCRIBE CLOSED SUB OBJECTS RELATIVE TO THIS UNIVERSAL CLOSURE BY A DIFFERENT VIEW POINT.

SUPPOSE THAT E AND F ARE HILBERT C∗-MODULES OVER C∗-ALGEBRAS A AND B, RESPECTIVELY. LET Φ: A → B BE A MORPHISMOF C∗-ALGEBRAS AND F: E → F BE A Φ-MORPHISM OF HILBERT C∗-MODULES. IN THIS TALK, WE INVESTIGATE UNDER WHICH CONDITIONS THE Φ-MORPHISMS OF HILBERT C∗-MODULES PRESERVE BIRKHOFF ORTHOGONALITY.

LET A, B, C BE LOCALLY C*-ALGEBRAS SUCH THAT A IS AN IDEAL IN B AND LET E BE A HILBERT C-MODULE. WE SHOW THAT EVERY NONDEGENERATE CONTINUOUS * -MORPHISM Φ: A ®.LC(E) CAN BE EXTENDED UNIQUELY TO A CONTINUOUS * -MORPHISM `Φ: B ®LC (E).

LET G BE A FINITE P-GROUP, WHERE P IS A PRIME NUMBER AND AUT(G) BE THE AUTOMORPHISM GROUP OF G. WE PROVE THAT IF AUT(G) IS ISOMORPHIC TO GL(N, P) FOR SOME POSITIVE INTEGER N, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PN. MOREOVER WE PROVE THAT IF G IS A FINITE ABELIAN P-GROUPS WITH |AUT(G)| = |GL(D, P)|, THEN G IS AN ELEMENTARY ABELIAN GROUP OF ORDER PD, WHERE D IS THE MINIMAL NUMBER OF GENERATORS OF G.