Triple homomorphisms of \(C^\ast\)-algebras.

*(English)*Zbl 1108.46041A celebrated result of 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 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, [T. Dang, Y. Friedman and B. Russo, Rocky Mt. J. Math. 20, No. 2, 409–428 (1990; Zbl 0738.47029); F. J. FernĂˇndez–Polo, J. M. Moreno and 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 \(B^{**}\).

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 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, [T. Dang, Y. Friedman and B. Russo, Rocky Mt. J. Math. 20, No. 2, 409–428 (1990; Zbl 0738.47029); F. J. FernĂˇndez–Polo, J. M. Moreno and 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 \(B^{**}\).

Reviewer: Antonio M. Peralta (Granada)