Local automorphisms of the sets of states and effects on a Hilbert space

Abstract

Abstract. We prove that every local automorphism (affine 1-local, or non-affine 2-local) of the sets of all states on a Hilbert space is an automorphism. We also present similar results concerning the various automorphisms of the set of all effects.

Linear maps on the space of all bounded observables preserving maximal deviation

“…One can find an interesting unified treatment of those automorphisms in [7]. In our recent papers [16,1] we presented some results on the local behaviour of the automorphisms in question, while in [17,18,19] we have started to study how these automorphisms can be characterized by their preserving properties.…”

“…Consequently, we can find unit vectors x n in H (n ∈ N) such that Qx n + Rx n − 2x n → 0 as n → ∞. 1 We remark that in his/her report the referee presented a more elementary proof of this lemma which uses only matrix (finite dimensional) arguments.…”

“…Lemma 3. Let P and Q be projections on H. Suppose that P is nontrivial and P − Q < 1 2 . Then P is unitarily equivalent to Q.…”

“…Proof. 1 Let R be a rank-1 subprojection of the projection P . Then the diameter of the spectrum of P + R is 2, so by Lemma 1 we have…”