Structure theory of homologically trivial and annihilator locally $C^{\ast}$-algebras
Volume 91 / 2010
Abstract
We study the structure of certain classes of homologically trivial locally $C^{\ast}$-algebras. These include algebras with projective irreducible Hermitian $A$-modules, biprojective algebras, and superbiprojective algebras. We prove that, if $A$ is a locally $C^{\ast}$-algebra, then all irreducible Hermitian $A$-modules are projective if and only if $A$ is a direct topological sum of elementary $C^{\ast}$-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 $\sigma$-$C^{\ast}$-algebras and describe the structure of biprojective locally $C^{\ast}$-algebras. Also, we present an example of a biprojective locally $C^{\ast}$-algebra that is not topologically isomorphic to a Cartesian product of biprojective $C^{\ast}$-algebras. Finally, we show that every superbiprojective locally $C^{\ast}$-algebra is topologically ${}^{\ast}$-isomorphic to a Cartesian product of full matrix algebras.
