On Uniform Subalgebras of L∞ on the Unit Circle Generated by Almost Periodic Functions

In the present paper we introduce analogs of almost periodic functions on the unit circle. We study certain uniform algebras generated by such functions, prove corona theorems for them and describe their maximal ideal spaces. 1. Formulation of Main Results 1.1. The classical almost periodic functions on the real line first introduced by H. Bohr in the 1920s play an important role in various areas of Analysis. In the present paper we define analogs of almost periodic functions on the unit circle. We study certain uniform algebras generated by such functions. In particular, we describe in these terms some uniform subalgebras of the algebra H of bounded holomorphic functions on the open unit disk D ⊂ C, having, in a sense, the weakest possible discontinuities on the boundary ∂D. To formulate the main results of the paper we first recall the definition of almost periodic functions, see [B]. ∗Research supported in part by NSERC. 2000 Mathematics Subject Classification. Primary 30H05, Secondary 46J20.