Aspects of commutative Banach algebras

Date of Completion

January 1990






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. ^