Radical Banach Algebra Research Articles

Multipliers of radical Banach algebras of power series

Multipliers of radical Banach algebras of power series

A nonstandard ideal of a radical Banach algebra of power series

A nonstandard ideal of a radical Banach algebra of power series

Closed ideals and bi-orthogonal systems in radical Banach algebras of power series

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then

Rates of decrease of sequences of powers in commutative radical Banach algebras

Every element 6 of a radical Banach algebra & satisfies l iπv^ ||6||=0. We are concerned here with the existence of a lower bound for the rate of decrease of the sequence (||δ||) under various assumptions over b and ^ , when b is not nilpotent and & commutative. If the nilpotents are dense in & then for every sequence (λn) of positive reals there exists a nonnilpotent b e & such that lim inf*-«, ||6||/ϋ»=0. A stronger result holds if & possesses furthermore a bounded approximate identity. On the other direction if & has no nilpotent element and if some element of & which is not a divisor of zero acts compactly on & then there exists a sequence (2n) of positive reals such that \immfn-+0O\\b n\\lλn— + °o for every nonzero be&. Also there exists universal lower bounds for the rate of decrease of ll<z|| if (a) is an analytic semigroup over the positive reals or over some open angle. Such lower bounds do not exist for infinitely differentiable semigroups over the positive reals.

Lower bounds for commutative radical Banach algebras

One says a commutative radical Banach algebra A has a lower bound if there is a lower growth condition on ∥x n∥ 1 n for all nonzero elements x in A. If A is a separable algebra we give necessary and sufficient conditions for A to possess a lower bound.

Irregularity of the Rate of Decrease of Sequences of Powers in the Volterra Algebra

G. R. Allan and A. M. Sinclair proved in [1] that if a commutative radical Banach algebra possesses bounded approximate identities then for every sequence (αn) of real numbers such that limn→∞αn = 0 there exists such thatIn the other direction it is shown in [6] that if is separable and if the nilpotents are dense in then for every sequence (βn) of positive reals there exists such that(This result was given in [2] for the Volterra algebra.)

Automatic Continuity and Topologically Simple Radical Banach Algebras

Automatic Continuity and Topologically Simple Radical Banach Algebras

On constructing multiplications on Banach spaces

This paper deals with the general problem of determining what kinds of multiplications can be put on a Banach space to make it into a Banach algebra. It has been shown that for a unital Banach algebra, the identity must be a vertex and a point of local uniform convexity [I ; 2, pp. 33-381. This implies that many Banach spaces (e.g., Hilbert space) cannot be made into unital Banach algebras. Grabiner has shown that any separable Banach space can be made into a radical Banach algebra of power series [6] or a semisimple Banach algebra of sequences with termwise multiplication [5]. In this paper we point out that many of the problems in the theory of strictly cyclic operator algebras as studied by Embry [4], Herrero [7], and Lambert [8] are closely related to the problem of putting multiplications on Banach spaces. In this paper we prove that any Banach space with a weak-* separable dual can be made into a radical Banach algebra of power series of any finite or countable number of indeterminates, commutative or not; a radical Banach algebra of strictly lower (or strictly upper) triangular matrices; or a radical Banach algebra of Dirichlet series. We prove that given a Banach space B with a weak* separable dual and a complemented subspace with an unconditional basis, and given an incidence algebra I(P) on a countable locally finite partially-ordered set P (as defined by Doubilet, Rota and Stanley in [3, pp. 269-271]), we can give B a separately continuous multiplication so that it is algebraically isomorphic to a dense subalgebra of I(P). As a corollary of this theorem, we get that any Banach space which satisfies the conditions of the theorem can be made into a Banach algebra of lower (or upper) triangular matrices. In this paper B denotes an infinite-dimensional real or complex Banach