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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 ${ℬ}_{A}$-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 ${k}_{B}\in \left(0,1\right]$. Moreover, for a commutative unital locally $m$-pseudoconvex Hausdorff algebra $A$ over $ℂ$ with pseudoconvex von Neumann bornology, which at the same time is sequentially ${ℬ}_{A}$-complete and advertibly complete, the statements (a)–(j) of Proposition 3.2 are equivalent.
##### MSC:
 46H05 General theory of topological algebras 46H20 Structure and classification of topological algebras