Commutative Algebras: Constructions and Classifications

Abstract

The second chapter gave the foundations of the theory of monotone complete C ^∗-algebras. We then introduced a classification semigroup and spectroid invariants. But, up to now, we have seen few concrete examples of wild monotone complete C ^∗-algebras. A good place to start is by finding commutative examples. This is what we shall do in this chapter. In Sect. 4.2 we give general constructions for commutative algebras. In Sect. 4.3 we show that $\ell^{\infty }$ has $2^{\mathbb{R}}$ subalgebras $\{A_{t}: t \in 2^{\mathbb{R}}\}$, where each A _t is a (small) wild, commutative monotone complete C ^∗-algebra. Furthermore, when r ≠ s, then A _r and A _s take different values in the semigroup $\mathcal{W}$ and have different spectroids. In a later chapter we shall use group actions on commutative algebras to construct huge numbers of small wild factors.