On inductive limits of matrix algebras of holomorphic functions

Abstract:

Let $\mathfrak {A}$ be a UHF algebra and $\mathcal {A}({\mathbf {D}})$ the disk algebra. If $\mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - }$ and $\alpha$ is a product-type automorphism of $\mathfrak {A}$ which leaves each ${\mathfrak {A}_n}$ invariant, then $\alpha$ defines an embedding \[ \mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}})\]. The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum $\Gamma (\alpha )$ is finite.