# Journal of Operator Theory

Volume 72, Issue 1, Summer 2014  pp. 49-70.

Completeness of $n$-tuples of projections in $C^*$-algebras

Authors:  Shanwen Hu (1) and Yifeng Xue (2)
Author institution: (1) Research Center for Operator Algebras, East China Normal University, Shanghai 200241, P.R. China
(2) Department of Mathematics, Research Center for Operator Algebras and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P.R. China

Summary: Let $(P_1,\ldots,P_n)$ be an $n$-tuple of projections in a unital $C^*$-algebra $\mathcal{A}$. We say $(P_1,\ldots,P_n)$ is complete in $\mathcal{A}$ if $\mathcal{A}$ is the linear direct sum of the closed subspaces $P_1\aa,\ldots,P_n\aa$. In this paper, we give some necessary and sufficient conditions for the completeness of $(P_1,\ldots,P_n)$ and discuss the perturbation problem and connectivity of the set of all complete $n$-tuple of projections in $\mathcal{A}$.

DOI: http://dx.doi.org/10.7900/jot.2012sep10.2019
Keywords: projection, idempotent, complete $n$-tuple of projections