Isometries between \(C^ *\)-algebras. (English) Zbl 1057.46009

In this important paper, the authors describe into-isometries between \(C^\ast\)-algebras. Given such a map \(T : A \rightarrow B\), there is a largest projection \(p \in B^{\ast\ast}\) such that \(T(.)p : A \rightarrow B^{\ast\ast}\) is a triple homomorphism and \(T\{a,b,c\} = \{Ta,Tb,Tc\}p\) for all \(a,b,c \in A\). When \(A\) is abelian, one also gets \(\| (Ta)p\| = \| a\| \) for all \(a \in A\). This can be seen as an extension of a result of W. Holsztynski [Stud. Math. 26 133–136 (1966; Zbl 0156.36903)] that describes into-isometries between continuous function spaces.
Of related interest (since in the abelian case isometries are complete isometries) is the work of D. P. Blecher and D. Hay [“Complete isometries into \(C^\ast\)-algebras” (Preprint) (2002); see also Contemp. Math. 328, 85–97 (2003; Zbl 1078.46041)]. For further information on the structure of into-isometries, one may consult section 6.4 of the monograph by R. J. Fleming and J. E. Jamison [“Isometries on Banach spaces: function spaces” (Monographs and Surveys in Pure and Applied Mathematics 129, Chapman and Hall-CRC, Boca Raton) (2003; Zbl 1011.46001)].


46B04 Isometric theory of Banach spaces
46L05 General theory of \(C^*\)-algebras
46L70 Nonassociative selfadjoint operator algebras
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