Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M _A and M _B , respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M _B → M _A and a closed and open subset K of M _B such that

Bounded and completely bounded local derivations from certain commutative semisimple Banach algebras

We show that for a locally compact group , every completely bounded local derivation from the Fourier algebra into a symmetric operator -module or the operator dual of an essential -bimodule is a derivation. Moreover, for amenable we show that the result is true for all operator -bimodules. In particular, we derive a new proof to a result of N. Spronk that is always operator weakly amenable.

     

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

     

Local spectral properties of multipliers on Banach algebras

Supported in part by NSF Grant DMS 90-96108. This work was completed while the first author visited Mississippi State University. Financial support and generous hospitality are gratefully acknowledged.     

Quasicompact and Riesz endomorphisms of Banach algebras

Let B be a unital commutative semi-simple Banach algebra. We study endomorphisms of B which are also quasicompact operators or Riesz operators. Clearly compact and power compact endomorphisms are Riesz and hence quasicompact. Several general theorems about quasicompact endomorphisms are proved, and these results are then applied to the question of when quasicompact or Riesz endomorphisms of certain algebras are necessarily power compact.     

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.     

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.     

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.

     

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.     

A structure of ring homomorphisms on commutative Banach algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that:

(i)

A ring homomorphism of a commutative -algebra onto another commutative -algebra with connected infinite Gelfand space is either linear or anti-linear.

(ii)

A ring automorphism of is either linear or anti-linear.

(iii)

, and the disc algebra are neither ring homomorphic images of nor .

     

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,     

Determining subspaces for discrete-type linear forms on commutative Banach algebras

In this paper we deal with some problems of the Korovkin-type Approximation theory concerning the convergence of nets of linear forms toward discrete-type linear forms on commutative Banach algebras. We also study the case of involutive symmetric commutative Banach algebras with identity or with bounded approximate identity. Examples and applications are presented in the context of the algebrasC ₀(X, C),C(X, C) andL ¹ (IR).