On an algebraic extension of A(E)

An algebraic extension of the algebra A(E), where E is a compactum in ℂ with nonempty connected interior, leads to a Banach algebra B of functions that are holomorphic on some analytic set K° ⊂ ℂ 2 with boundary bK and continuous up to bK. The singular points of the spectrum of B and their defects are investigated. For the case in which B is a uniform algebra, the depth of B in the algebra C(bK) is estimated. In particular, conditions under which B is maximal on bK are obtained