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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 \(\{T(1)\}'\).

46L05 General theory of \(C^*\)-algebras
Full Text: DOI
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