Contractible Fréchet algebras

A unital Fréchet algebra is called contractible if there exists an element such that and for all where is the canonical Fréchet -bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra to be a direct sum of a finite-dimensional semisimple algebra and a contractible Fréchet algebra without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fréchet -algebra is contractible if, and only if, it is algebraically and topologically isomorphic to for some . On the other hand, we show that a Fréchet algebra, that is, a locally -algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.

     

Maximality theorems for Fréchet algebras

The algebra of unbounded holomorphic functions that is contained in the algebra is studied. For in but not in , we show that the algebra generated by and is dense in for all .

     

Reflexivity and sets of Fréchet subdifferentiability

We show that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space are Borel if and only if is reflexive. This answers a question of L. Zajíček.

     

On subsymmetric bases in Fréchet spaces

It is proved that any non-normable Fréchet space with a semi-symmetric absolute basis is isomorphic to the space of all scalar sequences. A similar result is shown for quasi-homogeneous absolute bases. It is also proved that any nuclear Fréchet space with a semi-subsymmetric basis is isomorphic to .

     

Hypercyclic subspaces in Fréchet spaces

In this note, we show that every infinite-dimensional separable Fréchet space admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes the earlier work of several authors.

     

Projective boundedness and convolution of Fréchet measures

Fréchet measures of order ( -measures) are the measure-theoretic analogues of bounded -linear forms on products of spaces. In an LCA setting, convolution of -measures is always defined, while there exist -measures whose convolution cannot be defined. In a three-dimensional setting, we demonstrate the existence of an -measure which cannot be convolved with arbitrary -measures.

     

Pseudocompact Whyburn spaces need not be Fréchet

We prove in ZFC that there exists a Tychonoff pseudocompact scattered AP-space of uncountable tightness. We give some sufficient and necessary conditions for a -space to be AP as well as a characterization of AP-property in linearly ordered topological spaces.

     

Generalized stable ranks for commutative Banach algebras

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra A(mathbbD){A(mathbb{D})}, many of its subalgebras, and the algebra H ^∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of A(mathbbD){A(mathbb{D})} and H ^∞ is bigger than 2.     

Weak spectral synthesis in commutative Banach algebras

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Generalizing the notion of spectral sets in Δ(A), the considerably larger class of weak spectral sets was introduced and studied in [C.R. Warner, Weak spectral synthesis, Proc. Amer. Math. Soc. 99 (1987) 244-248]. We prove injection theorems for weak spectral sets and weak Ditkin sets and a Ditkin-Shilov type theorem, which applies to projective tensor products. In addition, we show that weak spectral synthesis holds for the Fourier algebra A(G) of a locally compact group G if and only if G is discrete.     

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 $\text{∥x}{}^{\mathrm{n}}\text{∥}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ 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.     

Star ring homomorphisms between commutative Banach algebras

We consider a -ring homomorphism from a commutative Banach algebra with an involution to a commutative Banach algebra with a symmetric involution. We give the Gelfand transform of the -ring homomorphism image.

     

Riesz multipliers on commutative semisimple Banach algebras

This work was supported in part by the C.N.R. (Fondi ministeriali 40% e 60%).     

Generalized stable ranks for commutative Banach algebras

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra \({A(\mathbb{D})}\), many of its subalgebras, and the algebra H ^∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of \({A(\mathbb{D})}\) and H ^∞ is bigger than 2.     

Metrizability vs. Fréchet-Uryshon property

In metrizable spaces, points in the closure of a subset are limits of sequences in ; i.e., metrizable spaces are Fréchet-Uryshon spaces. The aim of this paper is to prove that metrizability and the Fréchet-Uryshon property are actually equivalent for a large class of locally convex spaces that includes - and -spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Fréchet-Urysohn property. We provide applications of our results to, for instance, the space of distributions . The space is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.

     

Power boundedness in Fourier and Fourier-Stieltjes algebras and other commutative Banach algebras

We study power boundedness in the Fourier and Fourier-Stieltjes algebras, A(G) and B(G), of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A(G) and B(G) and also the structure of the Gelfand transform of a single power bounded element.     

A sum-product theorem in semi-simple commutative Banach algebras

The following analogue of the Erdös-Szemerédi sum-product theorem is shown. Let A=f₁,?,f_N be a finite set of N arbitrary distinct functions on some set. Then either the sum set f_i+f_j or the product set has at least N^1+c elements, where c>0 is an absolute constant. We use Freiman's lemma and Balog-Szemerédi-Gowers Theorem on graphs and combinatorics.As a corollary, we obtain an Erdös-Szemerédi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂R is a finite set, |A| large enough, then

|A+A|+|A.A|>|A|1+c,