DIVISION THEOREMS AND THE

It is shown that H∞+C has the Shilov property. In partic-ular any function f in H∞+C vanishing in an open neighbor-hood of the zeros of another function g in H∞+C is divisible there by g. Let D be the open unit disk in the complex plane. Let L ∞ and H ∞ denote the usual algebras on the unit circle ∂D. The smallest closed subalgebra of L ∞ properly containing H ∞ is H ∞ + C, where C denotes the algebra of continuous complex valued functions on the unit circle. The algebra consisting ofH∞+C functions whose complex conjugates are also inH∞+C is denoted by QC. For any of the above algebras, denoted here by A, the maximal ideal space or spectrum of A is the space of nonzero multiplicative linear functionals on A and is denoted M(A). When M(A) is given the weak- ∗ topology, it becomes a compact Hausdorff space. Identifying each point of D with the multiplicative linear functional that is point evaluation, we think of D as