On a functional equation for symmetric linear operators on $C^{*}$ algebras

Document Type: Research Paper


Faculty of Mathematics and Computer Science‎, ‎Damghan University‎, ‎Damghan‎, ‎Iran.


‎Let $A$ be a $C^{*}$ algebra‎, ‎$T‎: ‎A\rightarrow A$ be a linear map which satisfies the functional equation $T(x)T(y)=T^{2}(xy),\;\;T(x^{*})=T(x)^{*} $‎. ‎We prove that under each of the following conditions‎, ‎$T$ must be the trivial map $T(x)=\lambda x$ for some $\lambda \in \mathbb{R}$: ‎‎
‎i) $A$ is a simple $C^{*}$-algebra‎.
‎ii) $A$ is unital with trivial center and has a faithful trace such that each‎ ‎zero-trace element lies in the closure of the span of commutator elements‎.
‎iii) $A=B(H)$ where $H‎$‎ is a separable Hilbert space‎. 
‎For a given field $F$‎, ‎we consider a similar functional equation {$ T(x)T(y) =T^{2}(xy), T(x^{tr})=T(x)^{tr}, $} where $T$ is a linear map on $M_{n}(F)$ and‎ ‎"tr"‎ ‎is the transpose operator‎. ‎We prove that this functional equation has trivial solution for all $n\in \mathbb{N}$ if and only if $F$ is a formally real field‎.


Main Subjects

E‎. ‎T‎. ‎Bell‎, ‎Partitions polynomials‎, Ann‎. ‎of Math‎. ‎(2) 29 (1927)‎, ‎no‎. ‎1-4‎, ‎38--46‎.

‎J‎. ‎Chmielinski‎, ‎Linear mappings approximately preserving orthogonality‎, J‎. ‎Math‎. ‎Anal‎. ‎Appl. 304 (2005)‎, ‎no‎. ‎1‎, ‎158--169‎.

‎P‎. ‎Civin and B‎. ‎Yood‎, ‎The second conjugate space of a Banach algebra as an algebra‎, Pacific J‎. ‎Math. 11 (1961) 847--870‎.

K‎. ‎I‎. ‎Davidson‎, ‎$Csp *$-algebras by example‎, ‎Fields Institute Monographs‎, ‎6‎, ‎American Mathematical Society‎, ‎Providence‎, ‎1996‎.

D‎. ‎E‎. ‎Evans and Y‎. ‎Kawahigashi‎, ‎Quantum Symmetries on Operator Algebras‎, ‎Oxford Mathematical Monographs‎, ‎Oxford Science Publications‎. ‎The Clarendon Press‎, ‎Oxford University Press‎, ‎New York‎, ‎1998‎.

P‎. ‎Fan and C‎. ‎K‎. ‎Fong‎, ‎Which operators are the self commutators of compact operators‎, Proc‎. ‎Amer‎. ‎Math‎. ‎Soc. 80 (1980)‎, ‎no‎. ‎1‎, ‎58--60‎.

B‎. ‎E‎. ‎Johnson‎, ‎An introduction to the theory of centralizers‎, Proc‎. ‎Lond‎. ‎Math‎. ‎Soc‎. ‎(3) 14 (1964) 299--320‎.

N‎. ‎E‎. ‎Wgge-Olsen‎, ‎K-Theory and $C^*$ Algebras‎, ‎Oxford University Press‎, ‎Oxford‎, ‎1994‎.

Volume 42, Issue 5
September and October 2016
Pages 1169-1177
  • Receive Date: 19 November 2014
  • Revise Date: 24 July 2015
  • Accept Date: 30 July 2015