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Structure of locally idempotent algebras. (English) Zbl 1129.46037
Summary: It is shown that every locally idempotent (locally $m$-pseudoconvex) Hausdorff algebra $A$ with pseudoconvex von Neumann bornology is a regular (respectively, bornological) inductive limit of metrizable locally $m$-($k_B$-convex) subalgebras $A_B$ of $A$. In the case where $A$, in addition, is sequentially ${\cal B}_A$-complete (sequentially advertibly complete), then every subalgebra $A_B$ is a locally $m$-($k_B$-convex) Fréchet algebra (respectively, an advertibly complete metrizable locally $m$-($k_B$-convex) algebra) for some $k_B\in(0,1]$. Moreover, for a commutative unital locally $m$-pseudoconvex Hausdorff algebra $A$ over $\Bbb C$ with pseudoconvex von Neumann bornology, which at the same time is sequentially ${\cal B}_A$-complete and advertibly complete, the statements (a)--(j) of Proposition 3.2 are equivalent.
46H05General theory of topological algebras
46H20Structure and classification of topological algebras
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