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 $\u2102$ 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 and classification of topological algebras |