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We obtained a "decomposition scheme" of C *-algebras. We show that the classes of discrete C *-algebras (as defined by Peligard and Zsidó), type II C *-algebras and type III C *-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C *-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C *-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal A d,1, the largest discrete essentially infinite ideal A d,∞, the largest type II finite ideal A II,1, the largest type II essentially infinite ideal A II,∞, and the largest type III ideal A III of any C *-algebra A such that A d,1 + A d,∞ + A II,1 + A II,∞ + A III is an essential ideal of A. This "decomposition" extends the corresponding one for W *-algebras. We also give a closer look at C *-algebras with Hausdorff primitive ideal spaces, AW *-algebras as well as local multiplier algebras of C *-algebras. We find that these algebras can be decomposed into continuous fields of prime C *-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above. [ABSTRACT FROM AUTHOR]

Let T : V → W be a surjective real linear isometry between full real Hilbert C ⁎ -modules over real C ⁎ -algebras A and B , respectively. We show that the following conditions are equivalent: (a) T is a 2-isometry; (b) T is a complete isometry; (c) T preserves ternary products; (d) T preserves inner products; (e) T is a module map. When A and B are commutative, we give a full description of the structure of T . [ABSTRACT FROM AUTHOR]

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions. [ABSTRACT FROM PUBLISHER]

Abstract: This paper is devoted to the study of orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups. We show that a linear bijection (resp. ) between two Fourier algebras (resp. Fourier–Stieltjes algebras) of locally compact groups will induce a topological group isomorphism between and , provided that Ψ preserves both disjointness and some kind of orthogonality. This improves earlier results of J.J. Font and M.S. Monfared, where amenability of the groups or continuity of the linear maps are assumed. We also study the structure of bounded and unbounded disjointness preserving linear functionals of Fourier algebras. In the development, general results about disjointness and orthogonality preserving linear maps between -algebras, -algebras and their preduals are obtained. [Copyright &y& Elsevier]

Abstract: We obtain several Banach–Stone type theorems for vector-valued functions in this paper. Let be realcompact or metric spaces, locally convex spaces, and a bijective linear map from onto . If preserves zero set containments, i.e., then is homeomorphic to , and is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that preserves nonvanishing functions, i.e., These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones. [Copyright &y& Elsevier]

The concept of well-posedness for a minimization problem is extended to develop the concept of well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations which includes as a special case the class of variational-hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions under which the strongly mixed variational-hemivariational inequality is stronglywell-posed in the generalized sense. On the other hand, it is also proven that under some mild conditions there holds the equivalence between the well posedness for a stronglymixed variational-hemivariational inequality and thewell-posedness for the corresponding inclusion problem. [ABSTRACT FROM AUTHOR]

By analysing the uniform attractor for multi-valued processes, we study the long-time behaviour of the solutions of a model of non-autonomous porous-medium equations. The result is obtained by using the a priori estimates and suitable compactness arguments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we try to attack a conjecture of Araujo and Jarosz that every bijective linear map θ between -algebras, with both θ and its inverse preserving zero products, arises from an algebra isomorphism followed by a central multiplier. We show it is true for CCR -algebras with Hausdorff spectrum, and in general, some special -algebras associated to continuous fields of -algebras. [Copyright &y& Elsevier]

Abstract: Let be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, a nonempty bounded closed convex subset of , a finite family of asymptotically nonexpansive self-mappings on with common sequence , be two sequences in (0, 1) such that and be a contraction on . Under suitable conditions on the sequences , we show the existence of a sequence satisfying the relation where for some unique integers and . Further we prove that converges strongly to a common fixed point of , which solves some variational inequality, provided as for . As an application, we prove that the iterative process defined by , converges strongly to the same common fixed point of . [Copyright &y& Elsevier]

Abstract: It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed. [Copyright &y& Elsevier]

Abstract: A not necessarily continuous, linear or multiplicative function θ from an algebra into itself is called a 2-local automorphism if θ agrees with an automorphism of at each pair of points in . In this paper, we study when a 2-local automorphism of a -algebra, or a standard operator algebra on a locally convex space, is an automorphism. [Copyright &y& Elsevier]

Let M be a compact connected (topological) manifold of finite- or infinite-dimension n. Let 0 ⩽ r ⩽ 1 be arbitrary but fixed. We construct in this paper a space-filling curve f from [0,1] onto M, under which M is the image of a compact set A of Hausdorff dimension r. Moreover, the restriction of f to A is one-to-one over the image of a dense subset provided that 0 ⩽ r ⩽ log|2n/log(2n + 2). The proof is based on the special case where M is the Hilbert cube [0,1]ω. [Copyright &y& Elsevier]

For a linear isometry T :C0(X)→C0(Y) of finite corank, there is a cofinite subset Y1 of Y such that Tf&z.sfnc;Y1=h·f∘ϕ is a weighted composition operator and X is homeomorphic to a quotient space of Y1 modulo a finite subset. When X=Y, such a T is called an isometric quasi-n-shift on C0(X). In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M(X) with exactly n joints. The T-tree is total in M(X) when T is a shift. With these tools, we can analyze the structure of T. [Copyright &y& Elsevier]

In this paper, we introduce a new class of generalized co-complementarity problems in Banach spaces. An iterative algorithm for finding approximate solutions of these problems is considered. Some convergence results for this iterative algorithm are derived and several existence results are obtained. [Copyright &y& Elsevier]

This paper is concerned with the system (NEESSVMEP) obtained by mixing a nonlinear evolutionary equation and a strong set-valued mixed equilibrium problem (SSVMEP). In our setting, the set of constraints is not necessarily compact and the problem is driven by a not necessarily monotone set-valued mapping. We show that the solution set for (SSVMEP) is nonempty, closed, convex and bounded. We then establish the upper semicontinuity and the measurability properties of the involved functions in the nonlinear evolutionary equation. Utilizing these results, we prove that the solution set for (NEESSVMEP) is nonempty and compact. [ABSTRACT FROM AUTHOR]

We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool { T λ } of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings { S n } associated to { T λ } , in the sense that the approximate fixed points of { S n } provide the common fixed points of { T λ }. Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones. [ABSTRACT FROM AUTHOR]

Let M be a semi‐finite von Neumann algebra with normal state space S(M). For any ϕ∈S(M), let Mϕ:={x∈M:xϕ=ϕx} be the centralizer of ϕ with center Z(Mϕ). We show that for ϕ,ψ∈S(M), the following are equivalent. ϕ=ψ.Z(Mψ)⊆Z(Mϕ) and ϕ|Z(Mϕ)=ψ|Z(Mϕ).ϕ,ψ have the same distances to all the closed faces of S(M). As an application, we give an alternative proof of the fact that metric preserving surjections between normal state spaces of semi‐finite von Neumann algebras are induced by Jordan ∗‐isomorphisms between the underlying algebras. We then use it to verify some facts concerning F‐algebras and Fourier algebras of locally compact quantum groups. [ABSTRACT FROM AUTHOR]

Let M n (F) be the algebra of n × n matrices over an arbitrary field F. We consider linear maps Φ : M n (F) → M r (F) preserving matrices annihilated by a fixed polynomial f (x) = (x − a 1) ⋯ (x − a m) with m ≥ 2 distinct zeroes a 1 , a 2 , ... , a m ∈ F ; namely, f (Φ (A)) = 0 whenever f (A) = 0. Suppose that f (0) = 0 , and the zero set Z (f) = { a 1 , ... , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r (F) , invertible diagonal matrices D 1 ∈ M p (F) and D 2 ∈ M q (F) , where s = r − n p − n q ≥ 0. The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f (x) in the sense that λ Z (f) ⊆ Z (f). In general, assume that Z (f) − a 1 is not an additive group. If Φ (I n) commutes with Φ (A) for all A ∈ M n (F) , or if f (x) has a unique zero multiplier λ = 1 , then Φ assumes the form (†). The above assertions follow from the special case when f (x) = x (x − 1) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n (F) → M r (F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n (F) = M 2 (Z 2). [ABSTRACT FROM AUTHOR]

Let A be an F -algebra and S (A *) be the metric semi-group of normal states of the W * -algebra A * ⁠. We show that S (A *) determines A , when A * is of type I. In particular, when A is either the Fourier algebra A (G) or the Fourier–Stieltjes algebra B (G) of a type I locally compact group G , the metric semi-group S (A *) determines G up to opposition. A remark for the general case will also be given. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L1(G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measures, weakly almost periodic functions and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A1+: = {f ∈ A: f ≥ 0 and ∥f∥ = 1} (the convex structure is not considered) is a complete invariant for G. [ABSTRACT FROM AUTHOR]

Let 1 ≤ p ≤ + ∞ . We show that the positive part of the closed unit ball of a non-commutative L p -space, as a metric space, is a complete Jordan ⁎ -invariant for the underlying von Neumann algebra. [ABSTRACT FROM AUTHOR]

Let P w ( E n ; F ) be the space of all continuous n -homogeneous polynomials from a Banach space E into another F , that are weakly continuous on bounded sets. We give sufficient conditions for the weak sequential completeness of P w ( E n ; F ) . These sufficient conditions are also necessary if both E ⁎ and F have the bounded compact approximation property. We also show that the weak sequential completeness and the reflexivity of P w ( E n ; F ) are equivalent whenever both E and F are reflexive. [ABSTRACT FROM AUTHOR]

Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved. [ABSTRACT FROM PUBLISHER]

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set. [ABSTRACT FROM AUTHOR]

Abstract: Let be standard operator algebras on complex Banach spaces , respectively. For , let be a sequence with terms chosen from , and define the generalized Jordan product on elements in . This includes the usual Jordan product , and the triple . Assume that at least one of the terms in appears exactly once. Let a map satisfy that whenever any one of has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained. [Copyright &y& Elsevier]

Abstract: In this note, we study geometric unitaries of JB-algebras (in particular, self-adjoint parts of -algebras). By using order theoretical arguments, we show that the geometric unitaries of a JB-algebra are precisely the central algebraic unitaries (or central symmetries). [Copyright &y& Elsevier]

Abstract: Inspired by the recent work of Bekka, we study two reasonable analogues of property for not necessarily unital -algebras. The stronger one of the two is called “property ” and the weaker one is called “property .” It is shown that all non-unital -algebras do not have property (neither do their unitalizations). Moreover, all non-unital σ-unital -algebras do not have property . [Copyright &y& Elsevier]

Abstract: Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type , where f is a mapping of class from a bounded open set Ω in a reflexive Banach space X into its dual , and T is a maximal monotone mapping from X into . This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type . By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings. [Copyright &y& Elsevier]

Abstract: We study disjointness preserving (quasi-)n-shift operators on , where X is locally compact and Hausdorff. When admits a quasi-n-shift T, there is a countable subset of equipped with a tree-like structure, called φ-tree, with exactly n joints such that the action of T on can be implemented as a shift on the φ-tree. If T is an n-shift, then the φ-tree is dense in X and thus X is separable. By analyzing the structure of the φ-tree, we show that every (quasi-)n-shift on can always be written as a product of n (quasi-)1-shifts. Although it is not the case for general as shown by our counter examples, we can do so after dilation. [Copyright &y& Elsevier]

Let (Ω , A , μ) and (Γ , B , ν) be two arbitrary measure spaces, and p ∈ [ 1 , ∞ ]. Set S (L p (μ)) + : = { f ∈ L p (μ) : ‖ f ‖ p = 1 ; f ≥ 0 μ -a.e. } , that is, the positive part of the unit sphere of L p (μ). We show that every surjective isometry Φ : S (L p (μ)) + → S (L p (ν)) + can be extended (necessarily uniquely) to an isometric order isomorphism from L p (μ) onto L p (ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ , B , ν) is localizable (in particular, when it is σ -finite). On the other hand, we show that for compact Hausdorff spaces X and Y , if Φ is a surjective isometry from the positive part of the unit sphere of C (X) to that of C (Y) , then there is a homeomorphism τ : Y → X satisfying Φ (f) (y) = f (τ (y)) for f ∈ S (C (X)) + and y ∈ Y. [ABSTRACT FROM AUTHOR]

The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general Banach space setting. In this paper, after introducing the notion of Bregman generalized hybrid sequences in a reflexive Banach space, we prove (with using the Bregman–Opial property instead of the Opial property) convergence theorems for such sequences. We also provide new fixed point theorems for Bregman generalized hybrid maps defined on an arbitrary but not necessarily convex subset of a reflexive Banach space. We end this paper with a brief discussion of the existence of Bregman absolute fixed points of such maps. [ABSTRACT FROM AUTHOR]

In this paper, we study bijective linear maps θ : A → B between A W ⁎ -algebras preserving either zero products or range orthogonality. Such a map is automatically continuous, and provides an algebra or a ⁎-algebra isomorphism π ( ⋅ ) = θ ( ⋅ ) θ ⁎ ⁎ ( 1 ) − 1 from A onto B . Our results extend previous works for the case of W ⁎ -algebras, and also cover such maps between C ⁎ -algebras satisfying an extra assumption on preserving abelian C ⁎ -subalgebras. [ABSTRACT FROM AUTHOR]

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $$\chi _\alpha ^2$$ -divergence for some $$\alpha \in [0,1]$$ . We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the generalized-D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized-D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples. [ABSTRACT FROM AUTHOR]

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if σ ( A B ) ⊆ W ( A ) W ( B ) ¯ , for all (rank one) operators B . An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold. [ABSTRACT FROM AUTHOR]