We define type A, type B, type C as well as C*-semi-finite C*-algebras.

In this paper, using the concept of attractive points of a nonlinear mapping, we obtain a strong convergence theorem of Halperns type [<i>Bull. Amer. Math. Soc</i>. <b>73</b> (1967), 957961] 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 Halperns type in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi [<i>Nonlinear Anal</i>. <b>73</b> (2010, 15621568].

Abstract 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.

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.