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) Frechet 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.
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