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Triple homomorphisms of $$C^\ast$$-algebras. (English) Zbl 1108.46041
A 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. Fernández–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 $$B^{**}$$.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46B04 Isometric theory of Banach spaces 47B48 Linear operators on Banach algebras 17C65 Jordan structures on Banach spaces and algebras
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