JournalsrmiVol. 20, No. 1pp. 87–105

Isometries between C*-algebras

  • Cho-Ho J. Chu

    Queen Mary University, London, UK
  • Ngai-Ching Wong

    National Sun Yet-sen University, Kaohsiung, Taiwan
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Abstract

Let AA and BB be C*-algebras and let TT be a linear isometry from AA \emph{into} BB. We show that there is a largest projection pp in BB^{**} such that T()p:ABT(\cdot)p : A \longrightarrow B^{**} is a Jordan triple homomorphism and

T(abc+cba)p=T(a)T(b)T(c)p+T(c)T(b)T(a)pT(a b^* c + c b^* a) p= T(a) T(b)^* T(c) p + T(c) T(b)^* T(a) p

for all aa, bb, cc in AA. When AA is abelian, we have T(a)p=a\|T(a)p\|=\|a\| for all aa in AA. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces {\it locally} to a Jordan triple isomorphism, by a projection.

Cite this article

Cho-Ho J. Chu, Ngai-Ching Wong, Isometries between C*-algebras. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 87–105

DOI 10.4171/RMI/381