# 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

**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

Contents
Full-Text PDF