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Let  C B( X ) be the Banach algebra of all continuous bounded ℂ-valued functions on a completely regular Hausdorff space  X  equipped with the supremum norm. We provide representation theorems for arbitrary closed ideals of  C B( X ). Indeed, for a closed ideal  H  of  C B( X ) we associate a subspace    of the Stone-Čech compactification of  X  such that  H  and    are isometrically isomorphic. The space   , which is locally compact, is uniquely determined by this property. As a consequence of our representation theorem we explicitly construct a dense subideal in  H  which is isometrically isomorphic with   . We further associate an open subset  U  of  X  and a (set theoretic) ideal    in  X  to  H  such that  H  is isometrically isomorphic with   . We prove that there is the smallest ideal    in  X  such that   . From this we derive several properties of closed ideals of  C B( X ). For example we show that the set of all closed ideals of a commutative  C *-algebra (with or without unit) forms a distributive lattice. As an another example we show that for every completely regular Hausdorff space  X  and every ideal    in  X , the completion of the normed ideal    of  C B( X ) is the space   .

Representation theorems for closed ideals of C B ( X )