In this paper, we give some results concerning the modular function of a locally COMPACT GROUP. Among other results we prove that if G is a locally COMPACT non-unimodular topological GROUP with the left Haar measure l and E is a Borel subset of G, then: (i) G has a proper non-trivial closed normal subGROUP K such that EÇK and E−1ÇK have the same measure l.(ii) G has a non-empty non-cyclic open subsemiGROUP S such that if 0 6¹l (EÇS) <¥ then l (EÇS) > l ((EÇS)−1). If G is unimodular then E and its inverse have the same measure l.

G: A LOCALLY COMPACT GROUP.L ¥(G): BOUNDED, SCALAR-VALUED FNCTS ON G.CB(G): CONTINUOUS BOUNDED SCALAR-VALUED FNCTS ON G.C0 (G): CONTINUOUS FUNCTIONS VANISHING AT INFINITY ON G.

FIRST WE PRESENT SOME USEFUL CHARACTERIZATIONS FOR THE FUNCTION SPACES AP (G), WAP (G) AND LUC (G) ON A LOCALLY COMPACT QUANTUM GROUP G AND THEN WE STUDY SOME CONDITIONS UNDER WHICH AP (G) AND WAP (G) ARE C*-ALGEBRAS. WE ALSO INVESTIGATE THE EQUALITY LUC (G) =WAP (G).

IN THIS PAPER, WE DETERMINE THE STRUCTURE OF THOSE LOCALLY COMPACT ABELIAN (LCA) GROUPS IN WHICH THE MAXIMAL TORSION SUBGROUP IS CLOSED.

IN THIS PAPER, WE PRESENT SOME RESULTS CONCERNING DERIVATIONS AND JORDAN DERIVATIONS ON THE NONCOM MUTATIVE BANACH ALGEBRA L¥ (G)*. WE SHOW THAT THE RANGE OF ANY (SKEW-) CENTRALIZING DERIVATION ON L¥ (G)* IS EMBEDDED INTO M (L¥, C (G)), AND THE ZERO MAP IS THE ONLY SKEW-COMMUTING DERIVATION ON L¥ (G)*. WE ALSO PROVE THAT THE SUBSPACE ANNR (L¥ (G)*) OF L¥ (G)* IS INVARIANT UNDER JORDAN DERIVATIONS.

LET G BE A LOCALLY COMPACT GROUP AND P: G ® U(HP) BE A UNITARY REPRESENTATION. IN THIS ARTICLE WE WILL STUDY ON THE EXISTENCE OF AN ADMISSIBLE VECTOR FOR IRREDUCIBLE REPRESENTATIONS AND THE COMPACTNESS OF G. IN FACT, IT WILL BE SHOWN THAT IF G IS A COMPACT GROUP THEN EVERY IRREDUCIBLE REPRESENTATION OF G HAS AN ADMISSIBLE VECTOR, AND ALSO HAS A BOUNDED CYCLIC VECTOR. CONVERSELY IF G HAS PROPERTY (T) AND A FINITE IRREDUCIBLE REPRESENTATION OF G HAS AN ADMISSIBLE VECTOR, THEN G IS A COMPACT GROUP. SINCE CONTAINMENT OF AN IRREDUCIBLE REPRESENTATION L IN THE LEFT REGULAR REPRESENTATION LG IS A NECESSARY AND SUFFICIENT CONDITION FOR EXISTENCE OF ADMISSIBLE VECTOR, HENCE IT SEEMS A NATURAL OBJECT OF WEAKLY CONTAINMENT OF THESE REPRESENTATION AND THEREFORE PROPERTY (T) IS PEERED.

WE EXTEND THE CONCEPT OF STRONGLY COMPACT AND SEMI COMPACT SPACES. A SPACE (X, T) IS SAID TO BE SEMI L-COMPACT (RESP. STRONGLY L-COMPACT) IF EVERY COVER OF X BY SEMI-OPEN (RESP.PREOPEN) SETS HAS A SUBCOVER OF X WHOSE CARDINALITY IS LESS THAN L, WHERE L IS THE LEAST INFINITE CARDINAL NUMBER WITH THIS PROPERTY. WE CHARACTERIZE SEMI L-COMPACT AND STRONGLY L-COMPACT SPACES.