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1. CJM Online first
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 |
2. CJM 2011 (vol 64 pp. 755)
Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$-algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murray-von
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.
Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra Categories:46L05, 46L80 |
3. CJM 2008 (vol 60 pp. 1001)
Isometric Group Actions on Hilbert Spaces: Structure of Orbits Our main result is that a finitely generated nilpotent group has
no isometric action on an infinite-dimensional Hilbert space with
dense orbits. In contrast, we construct such an action with a
finitely generated metabelian group.
Keywords:affine actions, Hilbert spaces, minimal actions, nilpotent groups Categories:22D10, 43A35, 20F69 |