Commutative Semisimple Banach Algebra Research Articles

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ ( f g ) of the product of any two elements f and g in A coincides with the spectrum σ ( T f T g ) . Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ ( T f T g ) = σ ( f g ) holds for every f , g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ ( T f T g ) ⊂ σ ( f g ) holds for every f , g .

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.

Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to.

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

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

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

The following analogue of the Erdös–Szemerédi sum-product theorem is shown. Let A= f 1,⋯, 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 f i f j 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, where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=| H|, v= 1 N ∑ x∈H δ x . Then, for all constant c, there is a k= k( c) such that max z∈ R v ( k) ( z) < N − c , where v ( k) = v⋯ v is the k-fold convolution.

The Banach algebra induced by a double centralizer

Given a Banach algebra $A$, R. Larsen defined, in his book “An introduction to the theory of multipliers", a Banach algebra $A_{T}$ by means of a multiplier $T$ on $A$, and essentially used it in the case of a commutative semisimple Banach algebra $A$ to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra ${\mathcal A}_{R}$ induced by a bounded double centralizer $\langle L , R \rangle$ of a Banach algebra $A$. Then, our main concern is devoted to the relationships between $L$, $R$, and the algebras of bounded double centralizers ${\mathcal W}(A)$ and ${\mathcal W}({\mathcal A}_{R})$ of $A$ and ${\mathcal A}_{R}$, respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

Multipliers with closed range on commutative semisimple Banach algebras

Multipliers with closed range on commutative semisimple Banach algebras

Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_A$. In this paper, we study the norm spectra of elements of $\overline{span} Σ_A$ and present some applications. In particular, we characterize the discreteness of $Σ_A$ i

Automatic continuity of certain isomorphisms between regular Banach function algebras

AbstractLet A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if f.g = 0 implies Tf.Tg = 0, for all f, g ∈ A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional φ on a and a positive number δ satisfy |φ(ab)−φ(a)φ(b)|⩽δ‖a‖·‖b‖ for all a, b∈ a, there is a multiplicative linear functional ψ on a such that ‖φ−ψ‖=o(1) as δ→0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM. 1991 Mathematics Subject Classification 46J10.

Essential spectra through local spectral theory

Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semi-Fredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasi-affinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.

Multipliers with Natural Local Spectra on Commutative Banach Algebras

Several notions from the abstract spectral theory of bounded linear operators on Banach spaces are investigated and characterized in the context of multipliers on a semi-simple commutative Banach algebra. Particular emphasis is given to the determination of the local spectra of such multipliers in connection with Dunford's property (C), Bishop's property (β), and decomposability in the sense of Foiaş. The strongest results are obtained for regular Tauberian Banach algebras with approximate units and for multipliers whose Gelfand transforms on the spectrum of the Banach algebra vanish at infinity. The general theory is then applied to convolution operators induced by measures on a locally compact abelian groupG. Our results give new insight into the spectral theory of convolution operators on the group algebraL1(G) and on the measure algebraM(G). In particular, we identify large classes of measures for which the corresponding convolution operators have excellent spectral properties. We also obtain a number of negative results such as examples of convolution operators onL1(G) without natural local spectra, but with natural spectrum in the sense of Zafran.

Banach algebras with unique uniform norm

Commutative semisimple Banach algebras that admit exactly one uniform norm (not necessarily complete) are investigated. This unique uniform norm property is completely characterized in terms of each of spectral radius, Silov boundary, set of uniqueness, semisimple norms; and its connection with recently investigated concepts like spectral extension property, multiplicative Hahn Banach extension property and permanent radius are revealed. Several classes of Banach algebras having this property as well as those not having this property are discussed.

Operators with Finite Chain Length and the Ergodic Theorem

With a technical assumption (E-k), which is a relaxed version of the condition ${T^n}/n \to 0,n \to \infty$, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition $(X = {(I - T)^k}X \oplus \ker {(I - T)^k})$, of closedness of ${(I - T)^k}X$, and of quasi-Fredholmness of $I - T$. One consequence, still assuming (E-k), is that $I - T$ is semi-Fredholm if and only if $I - T$ is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.

SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS

Abstract We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.

Operators with finite chain length and the ergodic theorem

With a technical assumption (E-k), which is a relaxed version of the condition T n / n → 0 , n → ∞ {T^n}/n \to 0,n \to \infty , where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition ( X = ( I − T ) k X ⊕ ker ⁡ ( I − T ) k ) (X = {(I - T)^k}X \oplus \ker {(I - T)^k}) , of closedness of ( I − T ) k X {(I - T)^k}X , and of quasi-Fredholmness of I − T I - T . One consequence, still assuming (E-k), is that I − T I - T is semi-Fredholm if and only if I − T I - T is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.

Multipliers with Closed Range on Regular Commutative Banach Algebras

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then ${T^2}A$ is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.

HOMOMORPHISMS OF BANACH ALGEBRAS WITH RANGE IN C1[0, 1

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

On the Fredholm theory of multipliers

Multipliers that are Fredholm operators on certain commutative semisimple Banach algebras may be characterized by means of a quotient algebra of multipliers. Some spectral properties of multipliers of these algebras are considered

Multipliers with closed range on regular commutative Banach algebras

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then T 2 A {T^2}A is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.