an:02215247
Zbl 1108.46041
Wong, Ngai-Ching
Triple homomorphisms of \(C^\ast\)-algebras
EN
Southeast Asian Bull. Math. 29, No. 2, 401-407 (2005).
00118336
2005
j
46L05 46B04 47B48 17C65
C*-algebras; Jordan triples; isometries; disjointness preserving operators
A celebrated result of \textit{R.\,Kadison} [Ann.\ Math.\ (2) 54, 325--338 (1951; Zbl 0045.06201)] establishes that for every surjective linear isometry \(T\) between two C*-algebras \(A\) and \(B\), there exists a unitary element \(u\) in the unitization of \(B\) and a Jordan *-isomorphism \(J : A\to B\) satisfying that \(T(a) = u J(a)\) for each \(a\in A\).
A linear operator \(T\) between two C*-algebras is said to be a triple isomorphism if it preserves triple products of the form \(\{x,y,z\} := 2^{-1} (x y^* z +z y^* x)\).
Kadison's theorem was generalized by \textit{W.\,Kaup} in [Math.\ Z.\ 183, 503--529 (1983; Zbl 0519.32024)] by showing that a surjective linear operator between two C*-algebras \(A\) and \(B\) is a triple isomorphism if and only if it is an isometry. Alternative proofs of Kaup's result have been found in the last years (see, for example, [\textit{T.\,Dang, Y.\,Friedman} and \textit{B.\,Russo}, Rocky Mt.\ J.\ Math.\ 20, No.\,2, 409--428 (1990; Zbl 0738.47029); \textit{F.\,J.\,Fern??ndez--Polo, J.\,M.\,Moreno} and \textit{A.\,M.\,Peralta}, J.~Math.\ Anal.\ Appl.\ 295, No.\,2, 435--443 (2004; Zbl 1058.46033)]).
A linear operator \(T\) between two C*-algebras \(A\) and \(B\) is said to be disjointness preserving if
\[
a^* b= a b^*= 0 \text{ implies } (T(a))^* T(b) = T(a) (T(b))^* =0\;\forall a,b\in A.
\]
In his main result, the present author shows that a bounded linear operator \(T\) between two C*-algebras is a triple isomorphism if and only if \(T\) is disjointness preserving and \(T^{**} (1)\) is a partial isometry in \(B^{**}\).
Antonio M. Peralta (Granada)
Zbl 0045.06201; Zbl 0519.32024; Zbl 0738.47029; Zbl 1058.46033