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-(kB-convex) subALGEBRAS AB of A. In the case where A, in addition, is sequentially BA-complete (sequentially advertibly complete), then every subalgebraAB is a LOCALLY m-(kB-convex) Frechet algebra (respectively, an advertibly complete metrizable LOCALLY m-(kB-convex) algebra) for some kB Î (0, 1]. 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 BA-complete and advertibly complete, the statements (a)–(j) of Proposition 3.2 are equivalent.

Let G be a LOCALLY compact group with a fixed left Haar measure λ and ω be a weight function on G; that is a Borel measurable function ω : G→ (0,  ) with ω (x, y)≤ ω (x) ω (y) for all x, y  G...

In this paper, we generalize the construction of a pullback diagram in the framework of Hilbert modules over LOCALLY C*-ALGEBRAS and we explore some conditions under which diagrams of Hilbert modules over the corresponding C*-ALGEBRAS are pullback.

For a LOCALLY compact group $G$, $L^1(G)$ is its group algebra and $L^\infty(G)$ is the dual of $L^1(G)$. We consider on $L^\infty(G)$ the $\tau$-topology, i.e. the weak topology under all right multipliers induced by measures in $L^1(G)$. For such an arbitrary $G$ the $\tau$-topology is not weaker than the weak$^*$-topology and not stronger than the norm topology on $L^\infty(G)$. Among the other results we mention that except for discrete $G$ the $\tau$-topology is always different from the norm-topology. The properties of $\tau$ are then studied further and we pay attention to the $\tau$-almost periodic elements of $L^\infty(G)$.

LOCALLY extended affine Lie ALGEBRAS were introduced by Morita and Yoshii as a natural generalization of extended affine Lie ALGEBRAS. After that, various generalizations of these Lie ALGEBRAS have been investigated by others. It is known that a LOCALLY extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight vectors corresponding to nonisotropic roots modulo its centre. In this paper, in order to realize LOCALLY extended affine Lie ALGEBRAS of type A1, using the notion of Tits-Kantor-Koecher construction, we construct some Lie ALGEBRAS which are isomorphic to the centerless cores of these ALGEBRAS.