M-convex Algebras Research Articles

Boundedness of Multiplicative Linear Functionals

Let A be a complex sequentially complete commutative locally m-convex topological algebra which is symmetric with continuous involution. The purpose of this note is to prove that every multiplicative linear functional on A is bounded (Theorem 3). In fact, we prove a more general result for operators on real algebras (Theorem 1) from which we derive the above result.

Topological algebras and Mackey topologies

Let E be a locally m-convex algebra with dual space E ′ E’ . In a recent paper S. Warner asked if the finest locally m-convex topology on E compatible with E ′ E’ was the mackey topology. It is shown that this is not the case. A similar result is given for this question in the A-convex algebra case. For any A-convex algebra, a construction is given of an associated locally m-convex algebra. It is shown that this associated locally m-convex topology is always the compact-open topology for the space C b ( S ) {C_b}(S) with the strict topology.

On functional representations of topological algebras

The category of topological algebras we are concerned with is that of m-barreled ones for which the underlying topological vector space is not necessarily locally convex. These algebras specialize to Fréchet locally m-convex (topological) algebras (Michael) or to (locally m-convex) inductive limits of such (Warner) and they are characterized from being such that every balanced, convex, closed, idempotent and absorbing set ( m-barrel) in such an algebra to be a local neighborhood. For a topological algebra of this kind, the equicontinuous sets of its spectrum, the weakly relatively compact sets and the weakly bounded sets are the same (Corollary 2.1). As a consequence, in case of a commutative, advertibly complete, m-barreled locally m-convex algebra, equicontinuity of its spectrum is equivalent with the algebra being bounded, or even a Q-algebra, or its spectrum a weakly relatively compact subset of the respective topological dual space (Corollary 2.4). This specializes to a previous result of E. A. Michael. Now, for an m-barreled topological algebra E, whose the underlying topological vector space is a Pták locally convex one, the algebraic exactness of the sequence ▪ where g is the respective Gel'fand map, implies the sequence to be also topologically exact (: g is a topological isomorphism) in such a way that the spectrum of E is a k-space and the topology of the algebra is that of the uniform convergence on the closed equicontinuous subsets of its spectrum (Theorem 3.1). For Fréchet algebras this yields a result of S. Warner (Corollary 3.1), the spectrum of the algebra being, moreover, in this case a (Hausdorff completely regular) hemicompact k-space. Further information, but of a more technical nature, in case of a commutative, semi-simple, m-barreled topological algebra concerning a situation resembling that of the classical Gel'fand-Neumark representation theorem is provided by Theorem 3.2 and its Corollary. This extends also previous results of P. D. Morris and D. E. Wulbert. The last section contains material indicating that local equicontinuity of the spectrum of a topological algebra is not always a necessary condition as it concerns some basic formulas relating the spectrum of a topological tensor product algebra to the spectra of the factor algebras (Theorem 4.1).

On singly-generated locally M-convex algebras

On singly-generated locally M-convex algebras

On commutative locally M-convex algebras

On commutative locally M-convex algebras

On the Invariance of the Spectrum in Locally m-Convex Algebras

In this paper we consider two closely related problems concerning a complete locally m-convex (LMC) algebra A with identity. Let a be a fixed element of A, and let P(a) be the smallest closed subalgebra containing a and 1. If B is any subalgebra containing a and 1, we let σ(a; B) denote the spectrum of a as an element of B. (I) Describe the set σ(a; P(a)) in terms of σ(a; A). (II) Give necessary and sufficient conditions in order that σ (a; B) = σ(a; A) for every closed subalgebra B of A which contains a and 1.

On the spectrum of finitely-generated locally m-convex algebras

On the spectrum of finitely-generated locally m-convex algebras

Note on the tensor products and harmonic analysis

The present note aims at settling certain considerations concerning a certain type of a "generalized group algebra" considered in [4]. Given a locally compact abelian group G and a commutative complete locally m-convex algebra E, let ~:U~(G) be the space of E-valued continuous functions on G with compact support. For every continuous semi-norm p on E, x ~ p ( f ( x ) ) is a real-valued continuous function on G with compact support, for every f e off,(G), so that

Boundaries for locally m-convex algebras

Boundaries for locally m-convex algebras

On generalized topological divisors of zero in real m-convex algebras

On generalized topological divisors of zero in real m-convex algebras

Weakly topologized algebras

Let E be an algebra over the reals or complex numbers, E' a total subspace of the algebraic dual of E, o(E, E') the weak topology defined on E by E'. In Theorems 1 and 2 of [2], the author established necessary and sufficient conditions for E, equipped with topology o-(E, E'), to be a locally m-convex algebra and to be a topological algebra. Actually, these conditions may be combined as follows: