Language：English   Publishing type：Research paper (scientific journal)   Publisher：Academic Press Inc.  

In this note we investigate the form of a class of idempotents, called generalized bi-circular idempotents on spaces of continuously differentiable functions defined on the interval [0,1] and with values in the complex numbers.

DOI： 10.1016/j.jmaa.2018.05.022

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：University of Szeged  

We present a unified framework to study isometries, with respect to various norms, on C1([0, 1]), the space of continuously first differentiable functions on the unit interval [0, 1]. We also discuss continuous deformations of isometry groups induced by perturbations of norms on C1([0, 1]).

DOI： 10.14232/actasm-017-331-0

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We study surjective isometries between subspaces of continuous functions containing all constant functions and separating the points of the underlying spaces. In many contexts, every such isometry is represented by a combination of a weighted composition operator and its complex conjugate, called the canonical form, while there exists an isometry which does not take such a form ([14]). We seek a topological condition on compact Hausdorff spaces such that every surjective isometry on function spaces over the spaces has the canonical form. Also we extend the construction of [14] to show that, if a compact metrizable space X admits a semi free action of the circle group with a global section, then there exists an isometry of a function space on X which does not take the canonical form. (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2017.05.002

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATHEMATICAL SOC REP CHINA  

A necessary and sufficient condition for a Lau type binary operation defined by two mappings to be an algebra-operation is given in terms of multipliers. Also a characterization of multipliers of a Lau algebra constructed by semisimple commutative Banach algebras is given in terms of multipliers of original Banach algebras.

DOI： 10.11650/tjm.20.2016.7386

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We characterize mappings S (i) and T (i) , not necessarily linear, from sets , i=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products: sigma (pi) (S (1)(a)S (2)(b))aS,sigma (pi) (T (1)(a)T (2)(b)) and sigma (pi) (S (1)(a)S (2)(b))a (c)sigma (pi) (T (1)(a)T (2)(b))not equal a..., , . As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

DOI： 10.1007/s11512-014-0210-y

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：AMER MATHEMATICAL SOC  

Let X be a compact Hausdorff space. We denote by C(X) the Banach space of all continuous complex valued functions on X with respect to the supremum norm. A function space A on X is a normed linear subspace of C(X) that contains the constant function 1 and separates the points of X. We give a necessary and sufficient condition for a surjective distance preserving map between two function spaces be represented by a weighted composition operator.

DOI： 10.1090/conm/645/12926

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER INTERNATIONAL PUBLISHING AG  

We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.

DOI： 10.1186/1029-242X-2014-228

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let X and Y be locally compact Hausdorff spaces. Let A and B be complex-linear subspaces of C-0(X) and C-0(Y), respectively. Suppose that for each triple of distinct points x, x', x" epsilon X, there exists f epsilon A such that vertical bar f(x)vertical bar not equal vertical bar f(x')vertical bar and f (x") = 0. Also suppose that for each pair of distinct points y, y' epsilon Y. there exists g epsilon B such that vertical bar g(y)vertical bar not equal vertical bar g(y')vertical bar. For such A and B, we prove the following statement: If T is a real-linear isometry of A onto B, then there exist an open and closed subset E of ChB, a homeomorphism phi of ChB onto Ch A and a unimodular continuous function omega on ChB such that Tf = omega(f o phi) on E and Tf = omega(f o phi) on ChB\E for all f epsilon A, where Ch A and ChB are the Choquet boundaries for A and B, respectively. Moreover, we remark that the separation condition on A cannot be omitted in the above result. (C) 2013 Elsevier Inc. All right's reserved.

DOI： 10.1016/j.jmaa.2013.11.050

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：VERSITA  

We describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

DOI： 10.2478/s11533-013-0282-0

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELEMENT  

We will consider the Banach space valued differential equation eta'(t) = A eta(t), where A is an n x n complex matrix. We give a necessary and sufficient condition in order that the equation have the Hyers-Ulam stability. As a Corollary, we prove that the Banach space valued linear differential equation with constant coefficients y((n))(t) + a(n-1)y((n-1))(t) + ... + a(1)y'(t) + a(0)y(t) = 0 has the Hyers-Ulam stability if and only if Re lambda not equal 0 for all the solutions lambda of the equation z(n) + a(n-1)z(n-1) + ... + a(1)z + a(0) = 0.

DOI： 10.7153/mia-16-54

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Language：English   Publishing type：Research paper (scientific journal)  

Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called d-preserving maps between subsets of groups and apply them in several directions. We consider d-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group T. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.

DOI： 10.3836/tjm/1358951327

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PERGAMON-ELSEVIER SCIENCE LTD  

We prove that, if f is an approximate solution of the Banach space valued Chehyshev differential equation (1 - x(2))y ''(x) - xy'(x) + n(2)y(x) = 0 for x is an element of (-1, 1), then f is near to an exact solution. (C) 2012 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.aml.2012.03.012

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：VERSITA  

Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A -> B is a surjective real-linear isometry, then there exist a continuous function kappa: ChB -> {z a a",: |z| = 1}, a (possibly empty) closed and open subset K of ChB and a homeomorphism phi: ChB -> ChA such that T(f) = kappa(f a similar to phi) on K and T(f) = kappa (f.phi) over bar on ChB \ K for all f a A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.

DOI： 10.2478/s11533-011-0044-9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Mathematical Society of the Rep. of China  

J. Arhippainen and V.M̈uller showed that ||a + λe|| 1 ≤ 3 ||a + λe|| op for all a + λe || A ̃, where A ̃ is the unitization of an arbitrary non-unital Banach algebra A with regular norm. In this note, we present a necessary and sufficient condition for the equality of this estimate and its related problem.

DOI： 10.11650/twjm/1500406167

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Rocky Mountain Mathematics Consortium  

Let A and B be unital commutative Banach algebras. Suppose that A is semi-simple. Let ρ : A → A and τ : B → B be bijections. If T : A → B is a surjection with, for some α ∈ C \\ {0}, r ( T (f) r ( T ( s ) ) - α) = r ( f ρ( g ) - α) for all f,g ∈ A, then B is semi-simple and r ( T ( f ) T ( g )-1 - 1) = r(fg-1 - 1) for every f ∈ A and G ∈ A-1. As a consequence, T ( 1 ) is invertible and T ( 1 )-1 T is a real-algebra isomorphism. If, in addition, T(1)-1T(i) = i, then T ( 1 )T is a complexalgebra isomorphism. This result unifies and generalizes [3, Theorem 7.4] and [4, Theorem 3.2 and 6.2]. Copyright © 2011 Rocky Mountain Mathematics Consortium.

DOI： 10.1216/RMJ-2011-41-5-1675

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：AMER MATHEMATICAL SOC  

A spectral preserver is a map between Banach algebras such that some properties or objects connected with the spectra of algebra elements are invariant under its action. Though such maps have been studied extensively in the operator algebra setting, only during the last decade have they been systematically analyzed in commutative Banach algebras. This article is a survey of recent investigations on spectral preserver problems in commutative Banach algebras, including some history, basic results, future directions and open problems.

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Language：English   Publishing type：Research paper (scientific journal)  

Let A and B be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: A → B with S(1) = T(1) = 1 and α ∈ ℂ\\ {0} satisfy r(S(a)T(b) - α)= r(ab- α) for all a,b ∈ A then S = T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σπ (S(p)T(q)) ⊂ σπ (ρ(p) τ(q)) for all p,q ∈ I, where σπ (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions "enough". There exists a homeomorphism φ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(φ(y)) and T(p)(y)= τ(p)(φ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A. © 2010 Springer-Verlag Italia.

DOI： 10.1007/s12215-010-0013-3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：VERSITA  

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 is an element of A, then there exist a homeomorphism phi : M(B) -> M(A) and a closed and open subset K of M(B) such that
<(T(a))over cap>(y) = {<(T(e))over cap>(y)(a) over cap(phi(y)) y is an element of K <(T(e))over cap>(y)(a) over cap(phi(y)) y is an element of M(B)\K
for all a is an element of A, where e is unit element of A. If, in addition, <(T(e))over cap> = 1 and <(T(ie))over cap> = i on M(B), then T is an algebra isomorphism.

DOI： 10.2478/s11533-010-0025-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：YOKOHAMA PUBL  

Let X be a set with a binary operation circle and (Y, d) a complete metric space with a binary operation circle. Take a nonnegative function epsilon on X x X, a nonnegative function 6 on X and two mappings f, g : X -> Y. With the aid of Banach's fixed point theorem, we establish two general settings on which the following holds: If d(f (x circle x'), g(x) circle g(x')) <= epsilon(x, x') and d(f (x), g(x)) <= delta(x) for all x, x' is an element of X, then there exists a unique mapping f(infinity) : X -> Y such that f(infinity)(x circle x') = f(infinity)(x) circle f(infinity)(x'), d(f (x), f(infinity)(x)) <= A epsilon(x,x) + B delta(x) and d(g(x), f(infinity)(x)) <= A epsilon(x,x) C delta(x) for all x, x' is an element of X and some finite constants A, B and C. Moreover, we describe various concrete settings to which the above results are applicable. Some of them are the known results.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：VERSITA  

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum sigma(pi) (T) of T is defined by sigma(pi) (T) = z a sigma(T): |z| = max(wa sigma(T)) |w|. If surjective (not necessarily linear nor continuous) maps phi, I center dot: A -> B satisfy sigma(pi) (phi(S)I center dot(T)) = sigma(pi) (ST) for all S; T a A, then phi and I center dot are either of the form phi(T) = A (1) TA (2) (-1) and I center dot(T) = A (2) TA (1) (-1) for some bijective bounded linear operators A (1); A (2) of X onto Y, or of the form phi(T) = B (1) T*B (2) (-1) and I center dot(T) = B (2) T*B (-1) for some bijective bounded linear operators B (1);B (2) of X* onto Y.

DOI： 10.2478/s11533-009-0033-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER BASEL AG  

Let A and B be uniform algebras and p(z, w) = z(m)w(n) a two-variable monomial. We characterize maps T from certain subsets of A into B such that sigma(pi)(p(T(f), T(g))) subset of sigma(pi)(p(f, g)) holds for all f and g in the domain of T; peripherally monomial-preserving maps. Furthermore A and B are proved to be isometrical isomorphic as Banach algebras. If the greatest common divisor of m and n is 1, then T is extended to an isometrical linear isomorphism; a weighted composition operator. An example of peripherally monomial-preserving surjections between uniform algebras which is not linear, nor multiplicative, nor injective is given when the greatest common divisor is strictly greater than 1.

DOI： 10.1007/s00009-009-0166-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

For a compact Hausdorff space X, C(X) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C(X) is n-th root closed if, for each f epsilon C(X). there exists g epsilon C(X) such that f = g(n). It is shown that, for each integer m >= 2, there exists a compact Hausdorff space X(m) such that C(X(m)) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C(X) is algebraically closed, Pacific J. Math. 20 (1967) 433-438] et al. (C) 2008 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2008.08.015

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

Let X be a set with a binary operation. such that, for each x, y, z. X, either (x circle y) circle z = (x circle z) circle y, or z circle (x circle y) = x circle (z circle y). We show the superstability of the functional equation g(x circle y) = g(x)g(y). More explicitly, if epsilon >= 0 and f : X -> C satisfies |f(x circle y) - f(x)f(y)| <= epsilon for each x, y is an element of X, then f(x circle y) = f(x)f(y) for all x, y is an element of X, or |f(x)| <= (1 + root 1 + 4 epsilon)/2 for all x is an element of X. In the latter case, the constant (1 + root 1 + 4 epsilon)/2 is the best possible. Copyright (C) 2009 Takeshi Miura et al.

DOI： 10.1155/2009/486375

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Language：English   Publishing type：Research paper (scientific journal)  

We study maps between uniformly closed algebras of complex-valued continuous functions which vanish at infinity on locally compact Hausdorff spaces. Without assuming linearity nor multiplicativity on the maps we show that they are isometrical isomorphisms as Banach space operators if they satisfy that the peripheral range of the product of the images of any two elements coincides with the peripheral range of the product of those elements. Furthermore, if the underlying algebras contain approximate identities, then they are isometrically isomorphic as Banach algebras, which is a generalization of a recent result of Luttman and Tonev for the case of uniform algebras. On the other hand it is not the case without assuming the existence of approximate identities
An example is given. © 2009 International Academic Printing Co. Ltd. All rights reserved.

DOI： 10.3836/tjm/1249648411

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Language：English   Publishing type：Research paper (scientific journal)  

A Cauchy-Euler type factorization property which is closely related with the Hyers-Ulam stability problem is introduced in the algebra of all linear self maps of a commutative algebra without order. Several examples of linear self maps with such a property are given in this note. © 2008 International Academic Printing Co. Ltd. All rights reserved.

DOI： 10.3836/tjm/1233844065

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We give the solution of the functional equation f (x + y) + lambda f (x) f (y) = phi (x, y) under some conditions. Also we show its Hyers-Ulam stability. (c) 2006 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2006.07.030

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KOSSUTH LAJOS TUDOMANYEGYETEM  

eWe give a characterization of prime ideals P of a commutative complex algebra A in order that P be the kernel of some complex ring homomorphism on A. If, in addition, A is a uniform algebra on an infinite compact metric space, then we show that there are exactly 2' complex ring homomorphisms on A, whose kernels are non-maximal prime ideals. Moreover, it turns out that ring homomorphisms on a commutative Banach algebra are deeply connected with the existence of discontinuous homomorphisms.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

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 sigma (fg) of the product of any two elements f and g in A coincides with the spectrum sigma (TfTg). 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 a (TfTg) = a (fg) 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 or (TfTg) subset of sigma (fg) holds for every f, g. (c) 2006 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2006.02.084

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm parallel to.parallel to(infinity). The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.
Also we consider continuous approximate roots of the equation z" - f = 0 with respect to z, where f is an element of C(X), and provide a topological characterization of compact Hausdorff space X with dim X <= 1 such that the above equation has an approximate root in C(X) for each f is an element of C(X), in terms of the first tech cohomology of X. (C) 2006 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2006.05.008

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Element D.O.O.  

By an elementary method, we exactly determine the best possible constant and its attaining function which satisfy ∥f∥q ≤ C q∥f′∥q (1 &lt
q &lt
∝) for certain class of continuously differentiable functions on the unit interval [0.1]. © ELEMENT.

DOI： 10.7153/mia-10-27

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Language：English   Publishing type：Research paper (scientific journal)  

Let X be a compact Hausdorff space and C(X) the Banach algebra of all complex-valued continuous functions on X. We consider the following property of C(X): for each f ∈ C(X) there exist a g ∈ C(X) and positive integers p and q such that p does not divide q and fq = gp. When X is locally connected, we give a necessary and sufficient condition for C(X) to have this property. We also give a characterization of a first-countable compact Hausdorff space X for which C(X) has the property above. As a corollary, we prove that if X is locally connected, or first-countable, then C(X) has the property above if and only if C(X) is algebraically closed. © 2007 International Academic Printing Co. Ltd. All rights reserved.

DOI： 10.3836/tjm/1202136685

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BANACH MATHEMATICAL RESEARCH GROUP  

In this paper, we will consider Hyers-Ulam-Rassias stability of multipliers and ring derivations between Banach algebras. As a corollary, we will prove superstability of ring derivations and multipliers. That is, approximate multipliers and approximate ring derivations are exact multipliers and ring derivations.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Suppose A is a Banach algebra and suppose f : A -> A is an approximate ring derivation in the sense of Hyers-Ulam-Rassias. This stability phenomenon was introduced for the first time in the subject of functional equations by Th.M. Rassias [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300]. If A has an approximate identity, or if A is semisimple and commutative, then we prove that f is an exact ring derivation. (c) 2005 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2005.06.060

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

We study the structure of the group of equivariant Lipschitz homeomorphisms of a smooth G-manifold M which are isotopic to the identity through equivariant Lipschitz homeomorphisms with compact support. First we show that the group is perfect when M is a smooth free G-manifold. Secondly in the case of C-n with the canonical U(n)-action, we show that the first homology group admits continuous moduli. Thirdly we apply the result to the case of the group L(C, 0) of Lipschitz homeomorphisms of C fixing the origin.

DOI： 10.2969/jmsj/1145287091

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Let T be a surjective mapping from a uniform algebra A on a compact Hausdor. space X onto a uniform algebra on a compact Hausdor. space Y. Suppose that fg(X) = T(f)T(g)(Y) holds for every f, g is an element of A. Then we have that T is an almost isometric isomorphism, which is a generalization of results of Molnar (2002) and Rao and Roy (2005).

DOI： 10.1090/S0002-9939-06-08500-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER INTERNATIONAL PUBLISHING AG  

We give a new interpretation of Hua's inequality and its generalization. From this interpretation, we know the best possibility of those inequalities.

DOI： 10.1155/JIA/2006/21540

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Korean Mathematical Society  

We give some necessary and sufficient conditions in order that a closed operator in a Hilbert space into another have the Hyers-Ulam stability. Moreover, we prove the existence of the stability constant for a closed operator. We also determine the stability constant in terms of the lower bound.

DOI： 10.4134/BKMS.2006.43.1.107

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

We consider continuously differentiable means, say C-1-means. As for quasi-arithmetic means Q(f)(x(1),...,x(n)), we need an assumption that f has no stationary points so that Q(f) might be continuously differentiable. Introducing quasi-weights for C-1-means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of C-1-means, we observe that a skew power mean M-t is a composition of power means if t is an integer. Copyright (C) 2006 Jun Ichi Fujii et al.

DOI： 10.1155/JIA/2006/75941

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

The inequality discovered by L. K. Hua in 1965 has been generalized in several directions. In this paper, we adopt a certain conjugate method to give a simple and fundamental inequality on two functions on a semigroup, which is the key to the proof of many generalizations of Hua's inequality.

DOI： 10.1155/JIA.2005.15

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

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 f from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to f.

DOI： 10.1155/JIA.2005.435

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DOI： 10.1155/S0161171204402324

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let X be a complex Banach space, h a complex-valued continuous function on the real line R and T-h : C-1 (R, X) --> C(R, X) the linear differential operator defined by T(h)u = u' + hu. We completely determine the Hyers-Ulam stability constant of Th. (C) 2004 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2003.12.044

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Korean Mathematical Society  

The aim of this paper is to prove the stability in the sense of Hyers-Ulam-Rassias of the Banach space valued differential equation y′ = λy, where λ is a complex constant. That is, suppose f is a Banach space valued strongly differentiable function on an open interval. If f is an approximate solution of the equation y′ = λy, then there exists an exact solution of the equation near to f.

DOI： 10.4134/JKMS.2004.41.6.995

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Language：English   Publishing type：Research paper (international conference proceedings)  

Let C(X) be the algebra of all complex-valued continuous functions on a compact Hausdorff space X. We say that C(X) is algebraically closed if each monic polynomial equation over C(X) has a continuous solution. We give a necessary and sufficient condition for C(X) to be algebraically closed for a locally connected compact Hausdorff space X. In this case, it is proved that C(X) is algebraically closed if each element of C(X) is the square of another. We also give a characterization of a first-countable compact Hausdorff space X such that C(X) is algebraically closed.

DOI： 10.1090/S0002-9939-02-06835-1

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DOI： 10.1016/S0022-247X(03)00458-X

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Language：English   Publishing type：Research paper (scientific journal)  

Let B be a strictly real commutative real Banach algebra with the carrier space φB. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism p:A→B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then p (A) is contained in the radical of B. © 2003 Hindawi Publishing Corporation. All rights reserved.

DOI： 10.1155/S0161171203302352

Scopus

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DOI： 10.1002/mana.200310088

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DOI： 10.1002/mana.200310081

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Language：English   Publishing type：Research paper (scientific journal)  

Let F be a Banach algebra. We give a necessary and sufficient condition for F to be finite dimensional, in terms of finite type n-homogeneous F-valued polynomials. © 2002 Hindawi Publishing Corporation. All rights reserved.

DOI： 10.1155/S0161171202203348

Scopus

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Language：English  

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Language：English   Publisher：JAPAN SOC APPLIED PHYSICS  

The dielectric constant, threshold voltage, and response time were measured for a nematic binary mixture of 4-cyano-4'-pentylbiphenyl (5CB) molecules and the doped electron acceptor tetracyanoethylene (TCNE). The doping of TCNE caused a large increase in the dielectric constant and dielectric anisotropy. However, no marked improvement was achieved for threshold voltage and response time. Considering that the large increase was due to mobile charges in the system, we proposed a theoretical model to discuss the singularity that appeared in experimental results obtained in an electric field. (C) 2009 The Japan Society of Applied Physics

DOI： 10.1143/JJAP.48.121404

Web of Science

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DOI： 10.1155/S0161171204304333

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9999年04月

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