Homage to A. M. Gleason and I. J. Schark

Abstract

We study the maximal ideal space of , where B is the unit ball of . Following the lead of Gleason and Schark, we analyse Gleason parts, fibers, the Sĭlov boundary and other aspects of this Banach algebra. Our work here makes good use of the inner functions construction of Aleksandrov and Hakim/Løw/Sibony, particularly as formulated by Rudin (using ideas of Ryll and Wojtaszczyk).

Keywords:

• corona problem
• unit ball
• fiber
• Sĭlov boundary
• maximal ideal space
• analytic disc
• uniform algebras
• Banach algebras

AMS Subject Classifications:

• 30H80
• 30H05
• 30H50
• 32A35
• 32A37
• 32A38
• 32A65

Acknowledgements

 

Notes

1 Recall that this result says the following:

Let and and suppose that there is a homomorphisms in such that . Then there is a sequence such that and .