On a Cauchy-Jensen functional equation and its stability

In this paper, we obtain the general solution and stability of the Cauchy-Jensen functional equation

     

Isomorphisms between C-ternary algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in C^∗-ternary algebras and of derivations on C^∗-ternary algebras for the following Cauchy-Jensen additive mappings:

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Spectral invariant subalgebras of reduced groupoid <Emphasis Type="Italic">C</Emphasis>*-algebras

We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S ₂ ^l (G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra C _r ^* (G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G ⁰ of G, gives rise to a canonical map τ _c on the algebra C _c (G) of complex continuous functions with compact support on G. We show that the map τ _c can be extended continuously to S ₂ ^l (G) and plays the same role as an n-trace on C _r ^* (G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C _r ^* (G).     

Quasidiagonal representations of nilpotent groups

We show that every unitary representation of a discrete solvable virtually nilpotent group G is quasidiagonal. Roughly speaking, this says that every unitary representation of G   approximately decomposes as a direct sum of finite dimensional approximate representations. In operator algebraic terms we show that C^?(G)$C^{?}(G)$ is strongly quasidiagonal.     

LINEAR ＊-DERIVATIONS ON JB^＊-ALGEBRAS

It is shown that for a derivation f(x1o…oxj-1oxj 1o…xk)=k∑j=1x1o…oxu-1oxj 1o…oxkof(xj) on a JB*-algebra β, there exists a unique C-linear *-derivation D : β→β near the derivation.     

The stable rank of full corners in C*-algebras

We give a treatment of Rieffel's theory of stable rank for C*-algebras in terms of left invertibility of generalized nonsquare matrices, and prove that if is a full projection in a unital C*-algebra , then the stable rank of the corner is at least as large as the stable rank of .

     

Isometric isomorphisms in proper <Emphasis Type="Italic">CQ</Emphasis>*-algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ＊-algebras for the following Cauchy-Jensen additive mapping： 2f[（x1＋x2）/2＋y]=f（x1）＋f（x2）＋2f（y） The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias＇ stability theorem that appeared in the paper： On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 （1978）, 297-300. This is applied to investigate isometric isomorphisms between proper CQ＊-algebras.     

Homomorphisms between Poisson JC*-Algebras

It is shown that every almost linear mapping of a unital Poisson JC*-algebra to a unital Poisson JC*-algebra is a Poisson JC*-algebra homomorphism when h(2 ⁿ uy) = h(2 ⁿ u) h(y), h(3 ⁿ u y) = h(3 ⁿ u) h(y) or h(q ⁿ u y) = h(q ⁿ u) h(y) for all , all unitary elements and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.*This work was supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.     

On almost periodic mild solutions for stochastic functional differential equations

In this paper, a class of stochastic functional differential equations given by

     

Existence and decay of solutions of an abstract second order nonlinear problem

In this paper we consider the following nonlinear evolution problem with damping:

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Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

We consider the Sobolev inner product

     

The monotone method for periodic differential equations with the non well-ordered upper and lower solutions

In this paper, we consider the following periodic problem:

     

Equivalence Bimodule between Spherical Noncommutative Tori

Let n _i ,m _j > 1. In [5], the sperical noncommutative torus was defined by twisting in by a totally skew multiplier p on for T ^pd a pd-homogeneous C*-algebra over . It is shown that is strongly Morita equivalent to . This work is supported by Grant No. 1999-2-102-001-3 from the interdisciplinary research program year of the KOSEF     

Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the second order differential equation

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Iterative equations in Banach spaces

Let X be a Banach space, let

     

Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems

We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes:

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Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Asymptotic behavior results for nonlinear neutral delay difference equations

This paper is concerned with the nonlinear neutral delay difference equation

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