Let A be a uniform algebra on a compact space X, and let M denote the maximal ideal space of A. In this chapter, we continue the line of investigation begun in Chapters 1 and 2. We will introduce and treat various classes of “quasi-subharmonic” functions. The lower semi-continuous, quasi-subharmonic functions will be the log-envelope functions introduced in Chapter 2. The upper semi-continuous, quasi-subharmonic functions will be called simply “subharmonic”. The subharmonic functions in this context correspond to the subharmonic functions on an open subset of ¢, or to the plurisubharmonic functions on an open subset of ¢n.

The main theorems of this chapter are Theorems 5.9 and 5.10, asserting that a locally subharmonic function is subharmonic, while a bounded, locally log-envelope function is a log-envelope function. Our exposition will be based on work of the author and N.Sibony[3,4].

Quasi-subharmonic Functions

Let u be a Borel function from a subset S of MA to [-∞, +∞]. We say that u is quasi-subharmonic on S if u(φ) ≤ ∫udσ for all φ ∈ S and all Jensen measures σ for φ supported on a compact subset of S It is understood implicitly that the negative part of u min(u,0), is integrable with respect to the Jensen measures σ for those φ ∈ S satisfying u(φ) > -∞.

Evidently u is quasi-subharmonic on S if and only if u is quasi-subharmonic on each compact subset of S

A function u from S to [-∞, +∞) is subharmonic if u is upper semi-continuous and quasi-subharmonic.