We study the structure of certain classes of homologically trivial locally C*-algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C*-algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C*-algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically modular annihilator) topological algebra. We characterize all annihilator σ-C*-algebras and describe the structure of biprojective locally C*-algebras. Also, we present an example of a biprojective locally C*-algebra that is not topologically isomorphic to a Cartesian product of biprojective C*-algebras. Finally, we show that every superbiprojective locally C*-algebra is topologically *-isomorphic to a Cartesian product of full matrix algebras.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-17,
author = {Alexei Yu. Pirkovskii and Yurii V. Selivanov},
title = {Structure theory of homologically trivial and annihilator locally C*-algebras},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {279-313},
zbl = {1216.46043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-17}
}
Alexei Yu. Pirkovskii; Yurii V. Selivanov. Structure theory of homologically trivial and annihilator locally C*-algebras. Banach Center Publications, Tome 89 (2010) pp. 279-313. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-17/