Alling's Conjecture on Closed Prime Ideals inH∞

✍ Scribed by Pamela Gorkin; Raymond Mortini

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 305 KB
• Volume: 148
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3066

✦ Synopsis

In this paper we show that every closed prime ideal in the algebra of bounded analytic functions on the open unit disk is an intersection of maximal ideals.

1997 Academic Press

Let H be the Banach algebra of all bounded analytic functions in the open unit disk D=[z # C: |z| <1]. The maximal ideal space or space of multiplicative linear functionals on H endowed with the weak-star topology of the dual space of H is denoted by M(H ). The Shilov boundary of H will be denoted by H . Because H is a uniform algebra we may identify a function f # H with its Gelfand transform f defined on M(H ) by f (m)=m( f ), m # M(H ). As usual, we identify D with a subset of M(H ). For a function f # H , let Z( f )=[m # M(H ) : f (m)=0] be its zero set. The hull or zero set Z(I ) of an ideal I in H is defined by Z(I )= f # I Z( f ). A proper ideal P is said to be prime if for any f and g in H with fg # P one has that f # P or g # P. It is well known and easy to see that if P is a prime ideal in H whose hull is contained in D, then there exists z 0 # D such that P equals the ideal M(z 0 ) of all functions vanishing at z 0 . In particular, P is a maximal ideal. In [12, p. 224] it is shown that whenever P is a closed prime ideal whose hull meets the Shilov boundary article no. FU963066