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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 \({\mathcal B}_A\)-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 \(\mathbb C\) with pseudoconvex von Neumann bornology, which at the same time is sequentially \({\mathcal B}_A\)-complete and advertibly complete, the statements (a)–(j) of Proposition 3.2 are equivalent.

MSC:

46H05 General theory of topological algebras
46H20 Structure, classification of topological algebras
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