Contractibility of the Maximal Ideal Space of Algebras of Measures in a Half-space

H [n] = {(t1, . . . , tn) ∈ R n \ {0} | ∀j, [tj 6= 0 and t1 = t2 = · · · = tj−1 = 0] ⇒ tj > 0} ∪ {0}. LetM(H) denote the Banach algebra of all complex Borel measures with support contained in H, with the usual addition and scalar multiplication, and with convolution ∗, and the norm being the total variation of μ. It is shown that the maximal ideal space X(M(H)) of M(H), equipped with the Gelfand topology, is contractible as a topological space. In particular, it follows that M(H) is a projective free ring. In fact, for all subalgebras R of M(H) that satisfy a certain mild condition, it is shown that the maximal ideal space X(R) of R is contractible. Several examples of such subalgebras are also given.