On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

Fernández-Polo, Francisco; Peralta, Antonio

doi:10.1090/btran/21

On the extension of isometries between the unit spheres of a C$^*$-algebra and $B(H)$
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by Francisco J. Fernández-Polo and Antonio M. Peralta HTML | PDF

Trans. Amer. Math. Soc. Ser. B 5 (2018), 63-80

Abstract:

Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley’s problem in the setting of $B(H)$ spaces.

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