A note on the algebra of bounded functions. II
Kenneth G. Wolfson · Proceedings of the American Mathematical Society · 1956
Let K be a commutative 2?*-algebra with identity 1 (with ||e*¿||=p||2 for all kEK, and ||l|| = l).Then K is equivalent (isomorphic in a norm and * preserving manner) to the algebra C(M) of all continuous complex-valued functions on the compact Hausdorff space M (its structure space) [l; 2], In [S] we have given necessary and sufficient conditions that K be equivalent to B(X), the ring of all bounded complex-valued functions on the discrete space X.These conditions were ideal-theoretic, involving the annulets (annihilating ideals) of K, and did not depend on the representation C(M).Using the representation C(M) two further characterizations of B(X) are given in [3], one involving the properties of the space M, and the other the notion of projection.The characterizations in [3 ] are derived independently of the one in [5], and in fact no attempt is made in [3 ] to relate directly the ideal-theoretic conditions with the notions used there.In this note, we show how the characterizations in [3] can be derived from the characterization in [5] by relating directly the ideal-theoretic properties of K with the properties of the structure space M, and with the idea of projection.In particular, the lattice of annulets of K is anti-isomorphic to the lattice of regular open sets in M (Lemma 1).Another characterization of B(X) (Theorem 4) is a byproduct of our procedure.2. The notation will follow that in [5].If G^K then R(G) is the set of all functions kEK = C(M) such that kg = 0 for all g£G.Such ideals were called annulets.N(G) is the set of y EM such that g(y) =0 for all gEG.If S^M, then A(S) is the set of functions/ such that f(x) =0 for all x£5.Since N(G) =f\aç=.g N(g), it is a closed set.Now by following the arguments of Lemma 1 of [5], and using the fact that if 0 is open in a compact space and x£0, there exists a function fEK with/(x) = 1 and f(0') =0, we have(1) R(G)=A[N(G)'] for G^K;(2) R [A (S) ] =A(S') if 5 is a closed subset of M;(3) N[A(S)] = Sil Sisa closed subset of M.Now (1) and (2) show that an annulet of K is the set of all functions vanishing on an open set of M and conversely.Since A(S)=A(S)