In this paper, we give a complete description of the structure of zero product and orthogonality preserving linear maps between W*-algebras. In particular, two W*-algebras are *-isomorphic if and only if there is a bijective linear map between them preserving their zero product or orthogonality structure in two directions. It is also the case when they have equivalent linear and left (right) ideal structures.

In this paper, we introduce a broad class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings, hybrid mappings and contractive mappings. Then we prove fixed point theorems for the class of such mappings. Using these results, we prove well-known and new fixed point theorems in a Hilbert space.

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (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.

Let A be an operator ideal on LCSs. A continuous seminorm p of a LCS X is said to be Acontinuous if Qp Ainj (X, Xp), where Xp is the completion of the normed space Xp= X/p 1 (0) and Qp is the canonical map. p is said to be a Groth (A)seminorm if there is a continuous seminorm q of X such that p q and the canonical map Qpq: Xq Xp belongs to A (Xq, Xp). It is well-known that when A is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infraSchwartz) space if and only if every continuous seminorm p of X is Acontinuous if and only if every continuous seminorm p of X is a Groth (A)seminorm. In this paper, we extend this equivalence to arbitrary operator ideals A and discuss several aspects of these constructions which are initiated by A. Grothendieck and D. Randkte, respectively. A bornological version of the theory is obtained, too.

For a linear isometry T: C 0 (X) C 0 (Y) of finite corank, there is a cofinite subset Y 1 of Y such that Tf| Y 1= h f is a weighted composition operator and X is homeomorphic to a quotient space of Y 1 modulo a finite subset. When X= Y, such a T is called an isometric quasi-n-shift on C 0 (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.