Characterization of Birkhoff-James orthogonality

The Birkhoff-James orthogonality is a generalization of Hilbert space orthogonality to Banach spaces. We investigate this notion of orthogonality when the Banach space has more structures. We start by doing so for the Banach space of square matrices moving gradually to all bounded operators on any Hilbert space, then to an arbitrary C*-algebra and finally a Hilbert C*-module.

USEFUL EXTENSIONS OF THE HENRY REACTION: EXPEDITIOUS ROUTES TO NITROALKANES AND NITROALKENES IN AQUEOUS MEDIA

The products of the Henry nitroaldol reaction from nitromethane and several aldehydes were reduced to the corresponding nitroalkanes with (n-Bu)(3)SnH in water under microwave irradiation (80 degrees C/10 min), or dehydrated to the corresponding nitroalkenes with K2CO3 in water (generally 0-5 degrees C/20 min). Both ``one-pot'' reactions occur in excellent yields across a range of aliphatic and aromatic (including heteroaromatic) substrates. It seems likely that the deoxygenation of the nitroaldols occurs via coordination of an oxygen atom of the nitro group with a tin atom, which facilitates hydride delivery in the transition state. The elimination of water from the nitroaldols in mild base is likely driven by the stability of the conjugated nitroalkene products. The elimination required workup with 2N HCl, which likely displaces a nitroalkane-nitroalkene equilibrium towards the latter. These extensions of the Henry reaction lead to products not easily obtained otherwise.