I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even mention what part of the book I should be looking at. Also, since I have no interest in vector spaces over fields (or division rings) other than ℂ, I'm not particularly eager to search for the relevant stuff in Baer's book. So I'm wondering if someone recognizes this theorem, and can give me better reference.(adsbygoogle = window.adsbygoogle || []).push({});

I think I will only need a simplified version of the theorem: Let V be a finite-dimensional inner product space over ℂ. Let L be the lattice of subspaces of V, partially ordered by inclusion. For each T:V→V, we define ##\alpha_T:L\to L## by ##\alpha_T(M)=T(M)## for all M in L.

(a) If T:V→V is bijective, and either linear or conjugate linear, then ##\alpha_T## is an automorphism of L.

(b) If ##\alpha:L\to L## is an automorphism of L, then there's either a linear bijection T:V→V such that ##\alpha=\alpha_T##, or a conjugate linear bijection T:V→V such that ##\alpha=\alpha_T##. This T is unique up to multiplication by a complex number.

Part (a) is probably easy to prove. I just haven't tried it yet. I expect (b) to be a bit tricky though.

Edit:The thread title should say "linear or conjugate linear operators", not "unitary/antiunitary operators". I was half-way through this post before I realized that the theorem doesn't even mention an inner product.

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# Linear or conjugate operators and automorphisms on the lattice of subspaces

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