Abstract

Let $X$ be a compact Hausdorff space, $\tau$:$X\to X$ a homeomorphic involution on $X$. Denote by C(X,$\tau$) the real Banach algebra (under pointwise operations and the supremum norm) of continuous complex-valued functions $f$ on $X$ which satisfy $f$($\tau$($x$)) = $\sbsp{f(x)}{---}$ for all $x$ $\in$ $X$. A real function algebra on ($X$, $\tau$) is a uniformly closed real subalgebra $A$ of $C$($X$, $\tau$) which separates the points of $X$ and contains the real constants.

In this thesis, we study extensions to real function algebras of results already known for complex function algebras. In this setting, we first prove the following facts: (i) If Re$A$ is a ring, then $A$ = $C$($X$, $\tau$). (ii) If Re$A$ is closed under composition with a continuous "highly non-affine" function, then $A$ = $C$($X$, $\tau$).

Next, we prove: The intersection of a complex function algebra $B$ on $X$ and $C$($X$, $\tau$) is a real function algebra on ($X$, $\tau$) if $B\sp{\perp}$ $\perp$ $C$($X$, $\tau$)$\sp{\perp}$.

Finally, we examine under what conditions real function algebras satisfy reality conditions. Here, reality conditions represent in what degree real Banach algebras are apart from complex Banach algebras.