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Journal of Operator Theory

Volume 72, Issue 1, Summer 2014  pp. 257-275.

On certain multiplier projections

Authors:  Henning Petzka
Author institution: Mathematics Department, University of Toronto, Toronto, Canada

Summary: We consider multiplier projections in $\mathcal{M}(C(\prod\nolimits_{j=1}^\infty S^2,\mathcal{K}))$ of a certain diagonal form. We show that, while for each these multiplier projections $Q$, we have that $Q(x)\in\mathcal{B}(\mathcal{H})\setminus \mathcal{K}$ for all $x\in \prod\limits__{j=1}^\infty S^2$, the ideal generated by $Q$ in $\mathcal{M}(C(\prod\nolimits_{j=1}^\infty S^2,\mathcal{K}))$ 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. The results of this paper also form a basis for counterexamples to non-unital generalizations of a famous result of Blackadar and Handelman.

Keywords: $C^*$-algebra, multiplier algebra, projections

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