On the unit sphere of positive operators

Antonio M. Peralta

Published 2017-11-15Version 1

Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite dimensional and separable. Let $E$ and $P$ be subsets of a Banach space $X$. The unit sphere around $E$ in $P$ is defined as the set $$Sph(E;P) :=\left\{ x\in P : \|x-b\|=1 \hbox{ for all } b\in E \right\}.$$ In a first result we establish a geometric characterization of the projections in $B(H)$ by showing that an element $a\in S(B(H_1)^+)$ is a projection if and only if $$Sph^+_{B(H)} \left( Sph^+_{B(H)}(a) \right) =\{a\}.$$ The same characterization holds when $B(H_1)$ is replaced with $K(H_3)$. This characterization is applied to establish a positive variant to Tingley's problem by showing that every surjective isometry $\Delta : S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $\Delta : S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$).