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Search: All articles in the CJM digital archive with keyword idempotent

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1. CJM Online first

Elduque, Alberto
Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras
Order three elements in the exceptional groups of type $G_2$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras. Over an algebraically closed field, there are two conjugacy classes of order three elements in $G_2$ in characteristic not $3$ and four of them in characteristic $3$. The centralizers in characteristic $3$ fail to be smooth for one of these classes.

Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
Categories:17A75, 14L15, 17B25, 20G15

2. CJM 2013 (vol 66 pp. 1143)

Plevnik, Lucijan; Ċ emrl, Peter
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space
Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi , \psi : {\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair $U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents
Categories:46B20, 47B49

3. CJM 2007 (vol 59 pp. 638)

MacDonald, Gordon W.
Distance from Idempotents to Nilpotents
We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\times n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

Keywords:operator, matrix, nilpotent, idempotent, projection
Categories:47A15, 47D03, 15A30

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