A VERTEX PROPERTY OF REAL FUNCTION ALGEBRAS

We investigate a chain of properties of real function algebras along the analogous proofs of the complex cases such as the fact that any real function algebra which is both maximal and essential is pervasive. And some properties of real function algebras with a vertex property will be discussed. Comparing with complex Banach algebras, the study of real Banach algebras started quite late and not many researches have been done because the structure of real algebras is much more complicated to handle with than that of complex algebras. The rst mathematician who studied real Banach algebras systematically is known to be L. Ingelstam (I2) in the early 1960's. A typical example of complex (real, respectively) Banach algebras is C(X)(CR(X); respectively), the algebra of complex-valued (real-valued, respectively) continuous functions on a compact Haus- dor space X. Usually, a real function algebra can be understood as a real subalgebra ofCR(X). In this note we will consider a function algebra which lies in the complex algebra C(X) but has a real linear space structure under certain restriction. More pre- cisely, by a complex function algebra B on X we mean a uniformly closed complex subalgebra of C(X) which separates the points of X and contains the constant functions. Here point separation means that if x1 and x2 are distinct points of X then f(x1) 6 f(x2) for some function f in B. On the other hand, by a real function algebra A on (X; ) we mean a uniformly closed real subalgebra of the real