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Disjointness preserving operators on $$C^*$$-algebras. (English) Zbl 0803.46069
The characterization of disjointness preserving operators on $$C(X)$$ (i.e. operators $$T$$ satisfying $$\inf(| T(f)|,| T(g)|)= 0$$ if $$\inf(| f|,| g|)= 0$$, $$f,g\in C(X))$$ is extended to noncommutative $$C^*$$-algebras. According to W. Arendt [Indiana Univ. Math. J. 32, 199-215 (1983; Zbl 0488.47016)], in the commutative case such an operator $$T$$ is of the form $$T(f)= T(1)S(f)$$, where $$S(f)= f\circ \varphi$$ is a lattice homomorphism of $$C(X)$$ into $$C_ b(\{x\mid T(1)(x)\neq 0\})$$. In the general case of $$C^*$$-algebras $$\mathcal A$$ and $$\mathcal B$$ ($${\mathcal A}$$ unital) and $$T: {\mathcal A}\to {\mathcal B}$$ with $$T(x)^*= T(x^*)$$ for $$x\in {\mathcal A}$$ and $$T(a)T(b)= 0$$ for self- adjoint elements $$a,b\in {\mathcal A}$$, a similar characterization is given $$S$$ being replaced by a Jordan $$*$$-homomorphism of $$\mathcal A$$ into the multiplier algebra of the principal ideal generated by $$T(1)$$ in the commutant $$\{T(1)\}'$$.

##### MSC:
 46L05 General theory of $$C^*$$-algebras
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##### References:
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