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We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool {Tλ} of nonexpansive mappings. The difficulty of handling a possibly uncountable fam...

Let $$\theta :\mathcal {A}_{sa}\rightarrow \mathcal {B}_{sa}$$ be a bijective real linear zero product preserver between the self-adjoint parts of two (complex) $$C^*$$ -algebras. Suppose $$\theta $$ preserves commutativity in two directions (i.e., $$ab=ba$$ if and only if $$\theta (a)\theta (b)=\theta (b)\theta (a)$$ ). We show that $$\theta $$ is automatically continuous. More precisely, there is an invertible central self-adjoint multiplier h of $$\mathcal {B}$$ , and a real linear Jordan isomorphism $$J: \mathcal {A}_{sa}\rightarrow \mathcal {B}_{sa}$$ such that $$\theta = hJ$$ .

This paper is concerned with the problem of finding common fixed points for a family of Bregman relatively weak nonexpansive mappings. The motivation is due to our finding of some gaps in a paper of K. S. Kim (Nonlinear Analysis, 73 (2010), 3413-3419), where the author was developing a hybrid iterative scheme for locating common fixed points of a nonlinear representation of a left reversible semigroup. After a brief discussion about the gaps and why they are fatal, we present a new approach by using Bergman type nonexpansive mappings. A correct version of Kim’s convergence theorem is given as a consequence of our new results, which also improve and extend some recent results in the literature.

Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact convex subset of a Banach space. We show that there always exists a common fixed point for any jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$. Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.

Let L be an additive map between (real or complex) matrix algebras sending n × n Hermitian idempotent matrices to m × m Hermitian idempotent matrices. We show that there are nonnegative integers p , q with n ( p + q ) = r ≤ m and an m × m unitary matrix U such that L ( A ) = U [ ( I p ⊗ A ) ⊕ ( I q ⊗ A t ) ⊕ 0 m − r ] U ⁎ , for any n × n Hermitian A with rational trace . We also extend this result to the (complex) von Neumann algebra setting, and provide a supplement to the Dye-Bunce-Wright Theorem asserting that every additive map of Hermitian idempotents extends to a Jordan ⁎-homomorphism.

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.

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.

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 Ad,1, the largest discrete essentially infinite ideal Ad,∞, the largest type II finite ideal AII,1, the largest type II essentially infinite ideal AII,∞, and the largest type III ideal AIII of any C*-algebra A such that Ad,1 + Ad,∞ + AII,1 + AII,∞ + AIII 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.

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.

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.

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 .

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.

Enlightened by the notion of perturbation of $C^{*}$ -algebras, we introduce, and study briefly in this article, a notion of closeness of groups. We show that if two groups are “close enough” to each other, and one of them has the property that the orders of its elements have a uniform finite upper bound, then these two groups are isomorphic (but in general they are not). We also study groups that are close to abelian groups, as well as an equivalence relation induced by closeness.

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&ndash, 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&ndash, 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.

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.

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ \sigma(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ 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., Comment: 11 pages

In this paper, we first introduce a broad semigroup of mappings without continuity in a Banach space which contains discrete semigroups generated by generalized nonspreading mappings and semigroups of nonexpansive mappings. Then we prove an attractive point and fixed point theorem for the semigroups of mappings without continuity. Furthermore, we establish a mean convergence theorem for the semigroups of mappings without continuity in a Banach space. Using these results, we obtain well-known and new theorems which are connected with attractive point, fixed point and mean convergence results in Banach spaces.

Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).

We study mapsϕof positive operators of the Schattenp-classes (1

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.

In this paper, using the concept of attractive points of a nonlinear mapping, we obtain a strong convergence theorem of Halpern’s type [Bull. Amer. Math. Soc. 73 (1967), 957–961] for a wide class of nonlinear mappings which contains nonexpansive mappings, nonspreading mappings and hybrid mappings in a Hilbert space. Using this result, we obtain well-known and new strong convergence theorems of Halpern’s type in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi [Nonlinear Anal. 73 (2010, 1562–1568].

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 Ψ:A(G1)→A(G2) (resp. Ψ:B(G1)→B(G2)) between two Fourier algebras (resp. Fourier–Stieltjes algebras) of locally compact groups will induce a topological group isomorphism between G1 and G2, 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 C⁎-algebras, W⁎-algebras and their preduals are obtained.

Let L be a norm closed left ideal of a C*-algebra A. Then the left quotient A/L is a left A-module. In this paper, we shall implement Tomita’s idea about representing elements of A as left multiplications: πp(a)(b + L) = ab + L. A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant πp(A) ′′ of πp(A) in B(A/L) is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of A is carried out.

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.