Multiplicative Linear Functionals Research Articles

Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth

The AMNM property for commutative Banach algebras is a form of Ulam stability for multiplicative linear functionals. We show that on any semilattice of infinite breadth, one may construct a weight for which the resulting weighted convolution algebra fails to have the AMNM property. Our work is the culmination of a trilogy started in [4] and continued in [5]. In particular, we obtain a refinement of the main result of [5], by establishing a dichotomy for union-closed set systems that has a Ramsey-theoretic flavour.

Gleason-Kahane-Zelazko Theorem in Jordan Banach algebras

We review the celebrated Gleason-Kahane-Zelazko and Kowalski-Slodkowski theorems from the setting of associative Banach algebras to the wider class of nonassociative Jordan Banach algebras. We introduce the notion of almost multiplicative linear functionals in Jordan Banach algebras and prove a theorem extending a former result of B.E. Johnson for Banach algebras by employing the more recent concept of condition spectrum. We show how to rediscover the Gleason-Kahane-Zelazko theorem for Jordan Banach algebras from the corresponding version for almost multiplicative linear functionals.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Cyclicity Preserving Operators on Spaces of Analytic Functions in $${\mathbb {C}}^n$$

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_{\alpha} \, (\alpha \in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-\.Zelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable.

Derivations on Banach algebras of connected multiplicative linear functionals

Let A and B be Banach algebras with $$\sigma (B)\ne \emptyset $$ . Let $$\theta , \phi , \gamma \in \sigma (B)$$ and $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ be the set of all linear mappings $$d: A\times B\rightarrow A\times B$$ satisfying $$d((a, b)\cdot _\theta (x, y))=d(a, b)\cdot _\phi (x, y)+ (a, b)\cdot _\gamma d(x, y)$$ for all $$a, x\in A$$ and $$b, y\in B$$ . In this paper, we characterize elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ in the case where A has a right identity. We then investigate the concept of centralizing for elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ and determine dependent elements of $$\mathrm{Der}(A\times _\theta ^{\phi , \gamma }B)$$ . We also apply some results to group algebras.

A class of certain properties of approximately n-multiplicative maps between locally multiplicatively convex algebras

‎We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps‎. ‎‎W‎e prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous‎. ‎We also study the relationship between approximately multiplicative linear functionals and approximately n-multiplicative linear functionals‎.

Almost multiplicative linear functionals and entire functions

Let $T$ be a unital, continuous linear functional defined on complex Banach algebra $A$. First, we prove an approximate version of the Gleason-Kahane-\.Zelazko theorem: given $\epsilon >0$, there exists an $M>0$ such that, if $$ T(\exp x)\neq 0,\quad x\in A,\ \|x\|\leq M, $$ then $T$ is $\epsilon $-almost multiplicative. Then, we show that this result remains true if the exponential function is replaced by a nonsurjective entire function~$F$ with $F'(0)\neq 0$.

On the range of a derivation

Let ࣛ be a Banach algebra and ܦ:ࣛ ՜ ࣛ a derivation. In this paper, it is proved, under certain conditions, that ܦሺࣛሻ ك ݎሺࣛሻ , where ݎሺࣛሻ is the Jacobson radical of ࣛ. Moreover, we prove that if ࣛ is unital and ܦ:ࣛ ՜ ࣛ is a continuous derivation, then ܦሺࣛሻ ك ⋂ � א௉ ೎ሺࣛሻ ك ⋂ � אெ ೎ሺࣛሻ ك ⋂ � ఝא஍ ࣛ ܯఝ, where ௖ሺࣛሻ denotes the set of all primitive ideals such that ࣛ is commutative, ܯ௖ሺࣛሻ denotes the set of all maximal (modular) ideals such that ஺ is commutative, and Φࣛ is the set of all non-zero multiplicative linear functionals from ࣛ into ԧ . In addition, we present several results about the range of a derivation on algebras having the property (९).

The linear functionals on fundamental locally multiplicative topological algebras

In this paper we study the dual space of fundamental locally multiplicative topological algebras and prove some results for linear and multiplicative linear functionals on these algebras. An investigation on locally compactness of the carrier space of these algebras is the last part of this note.

A distinguished real Banach algebra

We present a new and elementary approach to characterize the maximal ideals and their associated multiplicative linear functionals for a classical real Banach algebra of analytic functions.

Isomorphisms of algebras of generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the algebras of generalized functions (in the sense of Colombeau) on X, resp. Y is given by composition with a unique generalized function from Y to X. We also characterize the multiplicative linear functionals from the Colombeau algebra on X to the ring of generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.

Multiplicative linear functionals and ring homomorphisms of

The Hardy-Orlicz space Hϕ is the space of all analytic functions f on the open unit disk D such that the subharmonic function ϕ(|f|) has a harmonic majorant on D where Á is a modulus function. is the space of all f ϵ Hϕ such that ϕ(|f|) has a quasi-bounded harmonic majorant on D. Under certain constraints on Á we show that multiplicative linear functionals on are exactly point evaluation and ring homomorphisms of are just composition operators. Examples of such ϕ are ϕ(x)=(log(1+x))p and ϕ(x)=log(1+xp ), 0 < p ≤ 1. This generalizes the special case p = 1 where is the well known Smirnov class N +.

On some automatic continuity theorems

We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform topological algebras are extended to a larger class of algebras. Consequences and applications are given.

Unital strongly harmonic commutative Banach algebras

A unital commutative Banach algebraA is spectrally separable if for any two distinct non-zero multiplicative linear functionals ' and on it there exist a and b in A such that ab = 0 and '(a) (b)6 0: Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra A is spectrally separable if there are enough elements in A such that the corresponding multiplication operators onA have the decomposition property ( ): On the other hand, if A is spectrally separable, then for each a 2 A and each Banach left A-module X the corresponding multiplication operator La on X is super-decomposable. These two statements improve an earlier result of Baskakov.

Spectral systems with the one-way spectral mapping property

For Banach algebras we characterize spectral systems which satisfy the one-way spectral mapping property. We show that these spectral systems are determined by projective systems of multiplicative linear functionals of finitely generated subalgebras and, using this, we derive several consequences concerning multiplicative extensions of multiplicative linear functionals of subalgebras.

The spectral theory of commutative C*-algebras: The constructive spectrum

This paper introduces the notion of a commutative C*-algebra in a Grothendieck topos E and subsequently that of the spectrum MFn A of A, presented as the locale determined by an appropriate propositional theory in the topos E which describes the basic properties of a multplicative linear functional on A. Further, the locale CE of complex numbers in the topos E is defined in a similar manner and some of its basic properties are established, such as its complete regularity and the compactness of the unit square in CE. Finally, it is shown that the locale MFn A is compact and completely regular, extending the classical result that the multiplicative linear functionals on a commutative C*-algebra form a compact Hausdorff space in the weak* topology.

The spectral theory of commutative C*-algebras: The constructive Gelfand-Mazur theorem

It is shown, for a commutative C*-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C*-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses one's natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.

Interpolating sequences in the spectrum of $H^\infty $ I

We show that a sequence of trivial points in M(Hc?) is interpolating if and only if it is discrete. This answers a question of K. Izuchi. We also give a sufficient topological condition for a sequence of nontrivial points to be interpolating. Let H' be the uniform algebra of all bounded analytic functions in the open unit disk 1D3. Its spectrum, or maximal ideal space, is the space M(H?) of all nonzero multiplicative linear functionals on H' endowed with the weak-*-topology. As usual, we identify a function f in H' with its Gelfand transform, f, defined by f(m) = m(f) for m E M(H'). Moreover, we look upon DI as a subset of M(H?). A sequence (xn)nEN of points in M(H??) is said to be interpolating if for any bounded sequence (Wn)nEN of complex numbers there exists a function f E H' so that f(xn) = Wn for every n. Carleson's well known interpolation theorem describes the interpolating sequences in the unit disk (see the book [4] of Garnett for an extensive presentation of this beautiful theory). In fact, a sequence (an) in 11D is interpolating if and only if (C) iEnfII a > 6> 0. j:j:Ak 1 jak It is the aim of this paper to continue the study of the interpolating sequences in the whole spectrum of H'. For previous results in this direction we refer the reader to [1], [2], [5], [6], [7], [9], [10], [11], [14], [15]. To proceed, we need to present a few definitions. Let x and y be two points in M(H?). We define the pseudohyperbolic distance of x to y by p(x,y) = sup{If(x)I: f E H', ljfj ?o 1, f (y) = O}. Then for a, b E D we have p(a, b) = 1 b It is well known that the relation defined on M(H?) by x y X# p(x) y) < I defines an equivalence relation on M(H?). The equivalence class containing a point m E M(H?) is called the Gleason part of m and is denoted by P(m). If the Received by the editors March 6, 1998 and, in revised form, July 13, 1998. 1991 Mathematics Subject Classification. Primary 46J15. ?2000 American Mathematical Society

Multiplicative linear functionals in a commutative Banach algebra

Multiplicative linear functionals in a commutative Banach algebra

A seminorm with square property on a Banach algebra is submultiplicative

The result stated in the title is proved in a Banach algebra and is used to discuss (i) commutativity criteria in normed algebras, (ii) uniqueness of the uniform norm in uniform Banach algebras, and (iii) existence of continuous multiplicative linear functionals on topological algebras together with a simple reduction of the Michael problem in Fréchet algebras. Submultiplicativity does not imply subadditivity in the presence of the square property.