IN THIS WORK, WE GIVE A BRIEF ACCOUNT OF THE NOTION OF INNER amenability THAT HAS BEEN EXTENSIVELY STUDIED IN THE LAST FOUR DECADES. THIS TERMINOLOGY WAS FIRST INTRODUCED BY E ROS IN 1975 FOR A DISCRETE GROUP G BASED ON THE EXISTENCE OF A NON-TRIVIAL INNER IN-VARIANT MEAN ON THE C*-ALGEBRA OF BOUNDED FUNCTIONS ON G; THAT IS, A POSITIVE LINEAR FUNCTIONAL WITH NORM ONE WHICH IS INVARIANT UNDER ALL INNER AUTOMORPHISMS.

LET (X, D) BE A COMPACT METRIC SPACE AND LET K BE A NONEMPTY COMPACT SUBSET OF X. LET A ∈ (0, 1] AND LET LIP (X, K, DA) DENOTE THE ALGEBRA OF ALL "FORMULA".IN THIS PAPER WE FIRST STUDY THE STRUCTURE OF CERTAIN IDEALS OF THE ALGEBRA LIP (X, K, DA). NEXT WE SHOW THAT IF K IS INFINITE AND INT (K) CONTAINS A LIMIT POINT OF K THEN LIP (X, K, DA) HAS AT LEASTA NONZERO CONTINUOUS POINT DERIVATION AND APPLYING THIS FACT WE PROVE THAT LIP (X, K, DA) IS NOT weakly AMENABLE AND AMENABLE.

WE SHALL CALL A MONOID S PRINCIPALLY weakly (weakly) LEFT COHERENT IF DIRECT PRODUCTS OF NON-EMPTY FAMILIES OF PRINCIPALLY weakly (weakly) FLAT RIGHT S-ACTS ARE PRINCIPALLY weakly (weakly) FLAT. SUCH MONOIDS HAVE NOT BEEN STUDIED IN GENERAL. HOWEVER, BULMAN FLEMING AND MCDOWELL PROVED THAT A COMMUTATIVE MONOID S IS (weakly) COHERENT IF AND ONLY IF THE ACT SI IS weakly FLAT FOR EACH NON-EMPTY SET I. IN THIS PAPER WE INTRODUCE THE NOTION OF FINITE (PRINCIPAL) WEAK FLATNESS FOR CHARACTERIZING (PRINCIPALLY) weakly LEFT COHERENT MONOIDS. ALSO WE INVESTIGATE MONOIDS OVER WHICH DIRECT PRODUCTS OF ACTS TRANSFER AN ARBITRARY FLATNESS PROPERTY TO THEIR COMPONENTS.

ONE OF THE NOTEWORTHY amenability PROBLEMS, IS TO EXPLORING ANY RELATIONSHIP BETWEEN THE amenability OF AN ALGEBRA STRUCTURE AND ITS BIDUAL SPACE. IN THIS PAPER WE CONSIDER THIS ISSUE IN THE CONTEXT OF TRIPLE SYSTEMS AND PROVE THAT A JB-TRIPLE IS TERNARYN-QUASI-weakly AMENABLE WHENEVER ITS BIDUAL SPACE IS TERNARY N-weakly AMENABLE.