On certain multiplier projections

TL;DR

This paper investigates the properties of certain diagonal multiplier projections in the algebra of continuous functions into compact operators over an infinite product of two-spheres, revealing conditions under which the generated ideal is proper.

Contribution

It characterizes when the ideal generated by specific multiplier projections in the algebra is proper, linking it to stable finiteness of the projection.

Findings

Multiplier projections of a special form have values outside the compact operators.

The ideal generated by such a projection can be proper.

Properness of the ideal is equivalent to the projection being stably finite.

Abstract

Let $\MCZK$, denote the multiplier algebra over $\CZK$, the algebra of continuous functions into the compact operators with spectrum the infinite product of two-spheres. We consider multiplier projections in $\MCZK$ of a certain diagonal form. We show that, while for each multiplier projection $Q$ of the special form, we have that $Q(x)\in\BH\setminus \KK$ for all $x\in \prod_{j=1}^\infty S^2$, the ideal generated by $Q$ in $\MCZK$ might be proper. We further show that the ideal generated by a multiplier projection of the special form is proper if and only if the projection is stably finite.