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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]

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]

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: 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]

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]

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]

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]