Imaginaries in Hilbert spaces

Abstract

Abstract. We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.

An equivalence of categories

“…In the case of Hilbert spaces, it is known [12] that for any imaginary h, we have cl(h) = cl(e) with e coding a finite dimensional subspace E of H. (Note that the lemma refers to hyperimaginaries of H itself, not to the induced structure from the piecewise-interpretatoin.) Moreover, E ⊆ cl(h).…”

“…For readability we will write omit the variable letter x, writing φ(123) for φ(x 1 , x 2 , x 3 ), φ(124) for φ ′ (x 1 , x 2 , x 4 ), L(123) for the measure algebra of formulas in x 1 , x 2 , x 3 over M , L (12,23,13) for the join of the measure algebras L(ij) (1 ≤ i, j ≤ 3), S (12,23,13) for the corresponding (measured) Stone spaces. We view formulas φ as {0, 1} valued (so conjunction is the same as multiplication), or more generally valued in a bounded interval of R (so multiplication is still defined.…”

“…The case n = 4 is representative. We want then to prove independence of L(123) from L(124, 134, 234) over L (12,13,14).…”