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algebra homomorphisms algebra of continuous Banach algebra Banach subalgebra base the sets bounded function bounded real-valued functions Bourbaki Class of Translation closed equivalence relation closed under addition closed under continuous closed under multiplication closed under positive commutative diagram compact space compatible with r(A completely regular space containing the constants continuous at xo continuous composition continuous functions continuous real-valued functions Corollary countable at infinity discrete topology disjoint from n(A distributive lattice embedding f e L(A f is continuous f,g e L(A function f Hence homomorphism space hR(A implies Isbell lattice of functions locally compact mR(A multiplication by real obvious positive inversion Q-space real numbers result follows ring of quotients satisfies L1 set L(A set of real-valued sets Vf space of L(A sublattice subset theorem translation lattice containing translation lattice homomorphism translation lattices L(A uniformly closed weak topology defined