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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 and closed closed equivalence relation closed set 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 composition continuous functions continuous real-valued functions Corollary countable at infinity defined by setting discrete topology distributive lattice embedding extension of f f,g e L(A fL(l function f Hence homomorphism space implies Isbell lattice of functions Let g locally compact mR(A multiplication by real obvious positive inversion Q-space R-valued functions real numbers result follows ring of quotients satisfies Lx set L(A set of real-valued sets Vf sublattice theorem translation lattice containing translation lattice homomorphism translation lattices L(A uniformly closed uniformly to f valued extension weak topology defined x r(A