LET (X, D) BE A COMPACT METRIC SPACE AND LET K BE A COMPACT SUBSET OF X. FOR A Î (0, 1], WE DENOTE BY LIP(X, K, DA) THE COMPLEX algebra OF COMPLEX-VALUED CONTINUOUS FUNCTIONS F ON X FOR WHICH PA, K(F) = SUP (FORMOLA) < ¥. IN PARTICULAR, IF K = X, WE WRITE LIP(X, DA) AND PA (F) INSTEAD LIP(X, K, DA) AND PA,K(F), RESPECTIVELY. THE algebraS LIP(X, DA) AND LIP(X, K, DA) ARE CALLED Lipschitz algebraS AND extended Lipschitz algebraS, RESPECTIVELY. IN THIS NOTE, WE STUDY UNITAL COMPACT HOMOMORPHISMS BETWEEN extended Lipschitz algebraS AND DETERMINE THE SPECTRUM OF UNITAL COMPACT ENDOMORPHISMS OF THESE algebraS.

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IN THIS PAPER WE STUDY UNITAL ENDOMORPHISMS OF EX-TENDED ANALYTIC Lipschitz algebraS WHICH ARE ALSO POWER COMPACT LINEAR OPERATORS OR QUASICOMPACT LINEAR OPERATORS. WE FIRST GIVE A SUFFICIENT CONDITION FOR A UNITAL QUASICOMPACT ENDOMORPHISM OF THESE algebraS TO BE POWER COMPACT. THEN IN CERTAIN CASES, WE SHOW THAT THIS CONDITION IS ALSO NECESSARY. AS A FINAL RESULT, WE GIVE A NECESSARY CONDITION FOR THE QUASICOMPACTNESS OF A UNITAL ENDOMORPHISM OF THESE algebraS.

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.