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Register a new userStructure theory of homologically trivial and annihilator locally C*-algebras
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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 $sigma$-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.
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In the present paper we obtain an intrinsic characterization of real locally C*-algebras (projective limits of projective families of real C*-algebras) among complete real lmc *-algebras, and of locally JB-algebras (projective limits of projective families of JB-algebras) among complete fine Jordan locally multiplicatively-convex topological algebras.
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