Representation of Locally Convex Algebras

Abstract

ABSTRACT. We deal with the representation of locally convex algebras.On one hand as "suhalgebras" of sorne weighted space CV(X) anO on the other hand, in the case ob uniborrnly A-convex algebras, as inductive lirnits of Banach algebras. We amo study sorne questions on the spectrurn of a locally convex algebra. INTROD1JCTIONA locally convex algebra is an algebra togetber with a Hausdorff loca-fly convex topology such tha-t tite midtiplication ob E is sepa-rately continuous. We denote by M (resp. M#) the assumed nonvoid set ob a-II continuous (resp. a-lgebra-ic) citaracters of E. Endowed witit tite weak topology generated by E, M anO M# are Ha-usdorff completely regular spaces. M will be calleO tite spectrum ob E a-nO M# its algebraic spectrurn. Titis work consists obthree indepenOant sections. In [9], conditions are given under witich E is embedded aigebraicaily (anO topologica-lIy) in sorne (7(X) with tite compa-ct open topology or a-weaker topology. A.C. Cocitran sitows iii [4] titat every semisimple uniformly A-convex algebra E can be embedded continuously, via tite Gelba-nd ma-p

On functional representation of normed algebras

“…For the use of these types of weights see [2,5,9]. Now (Â, · ˆ) is an A-convex algebra and the Gelfand mapping x →x is a continuous algebra homomorphism from (A, · ) onto (Â, · ˆ).…”

“…Also if the norm · is irregular this standard representation is not preferable (since we do not know whether the weight function v · is bounded or not) and the use of weighted supremum-norm can give better description. For further information about functional representation of different kind of A-convex algebras see [1][2][3]5,9]. Then · 0 is an A-convex norm on A with m( · 0 ) = ∞ (this implies that · 0 is not complete) and r( · 0 ) = ∞ and · 1 is an m-convex complete norm on A with m( · 1 ) = 1 and r( · 1 ) = ∞.…”

“…Instead of supremum norm we will providê A with some weighted supremum norm and show that there are cases in which these weights are not necessarily bounded and that this weighted representation is more closely connected to the topological structure of the algebra (A, · ). We follow here the ideas which were first represented in [5] and later extended and sharpened in [1,2,9].…”