Sur une classe d'algèbres topologiques

Properties ofTQ-algebras

“…In particular, when G(A) = G t (A), A is called an invertive algebra 3 (see [2], p. 14) and a topological invertible element is said to be proper (see [34], p. 323) if it is noninvertible. Properties of topologically invertible elements have been discussed in several papers, for example, in [2], [6], [8], [9], [11], [12], [15], [17], [19], [25], and [31]- [34].…”

Structure of locally idempotent algebras

“…It is known that every locally m-convex Hausdorff algebra is a bornological inductive limit (with continuous canonical injections) of metrizable locally m-convex subalgebras of A (see [9], Proposition on p. 943, or [10], Theorem II.4.3) and every complete locally m-convex algebra is a bornological inductive limit (with continuous canonical injections) of locally m-convex Fréchet subalgebras of A (see [9], p. 941, or [10], Theorem II.4.2). Later on this result was generalized to the case of a sequentially B A -complete locally m-convex Hausdorff algebra A (see [26], Theorem 2.1) and to the case of an advertibly complete locally m-convex Hausdorff algebra A (see [12], Theorem 6.2, or [15], Theorem 3.14). All these results hold in case of locally m-(k-convex) algebras as well, but not in general in the case of degenerated locally m-pseudoconvex algebras.…”

“…If, at the same time, A is sequentially B A -complete and advertibly complete (in particular, A is complete), then all the statements (a)-(j) of Proposition 3.2 are equivalent.Remark 3.5. Corollary 3.4 in case k = 1 has been partly proved in many papers (see, for example,[12], Proposition 4.3, and[26], Proposition 4.1, for complete case see[25], Proposition 3.3;[11], Theorem on the p. 61 and others).…”