𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Additive mappings on von Neumann algebras preserving absolute values

✍ Scribed by M. Radjabalipour

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
135 KB
Volume
368
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

✦ Synopsis

Given Hilbert spaces H and K and a von Neumann algebra A βŠ‚ B(H ), let denote the class of all additive mappings Ο• :

The paper shows that if A contains no nonzero abelian central projection then every Ο• ∈ preserves the * -operation, the R-linear combination, and, up to a commuting operator multiple Ο•(I ) 0, the (ring) multiplication. If A contains a nonzero abelian central projection P and if the dimension of K is at least 2 or 2 rank(P ) according to whether or not P can be chosen to be minimal, then there exists an additive mapping Ο• : A β†’ B(K) such that Ο•(I ) is a projection and |Ο•(A)| = Ο•(|A|) for all A ∈ A but Ο• is neither multiplicative nor adjoint preserving. In case A = B(H ) the result was proved by MolnΓ‘r [Bull.

πŸ“œ SIMILAR VOLUMES

Nonlinear maps preserving Lie products o
Nonlinear maps preserving Lie products on factor von Neumann algebras
✍ Jian-Hua Zhang; Fang-Juan Zhang πŸ“‚ Article πŸ“… 2008 πŸ› Elsevier Science 🌐 English βš– 149 KB

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra M into another factor von Neumann algebra N is of the form A β†’ ψ(A) + ΞΎ(A), where ψ : M β†’ N is an additive isomorphism or the negative of an additive anti-isomorphism and ΞΎ : M β†’ CI is a map wi

On a positive linear map preserving abso
On a positive linear map preserving absolute values
✍ Leroy B. Beasley; An-Hyun Kim; Woo Young Lee πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 447 KB

Suppose m is an n X 12 (n 2 2) matrix algebra over a C\*-algebra g, and Q? is a C\*-algebra. If p : i?X + '23 is a positive, disjoint linear map, then p preserves absolute values. In particular, for a linear map rp : '?I + '$3 of P-algebras, p preserves absolute values if and only if it is positive