FOR A *-REPRESENTATION P OF A LAU *-ALGEBRA A ON A HILBERT SPACE H, THE SECOND AUTHOR HAS RECENTLY INTRODUCED AND INVESTIGATED A NEW NOTION OF AMENABILITY CALLED STRONG AMENABILITY. HERE, WE INVESTIGATE SOME HEREDITARY PROPERTIES FOR STRONGLY AMENABLE *-REPRESENTATIONS OF LAU *-ALGEBRAS.

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.