Imaginaries in Hilbert spaces
The paper is a contribution to the model theory of Hilbert spaces. The authors work in a ``big'' Hilbert space \(\mathcal H\) and consider it as a multi-sorted structure whose sorts are the balls \(\{v: \| v\| \leq n\}\), for \(n<\omega\). Atomic predicates are \(r\leq\| \lambda_1x_1+\dots+\lambda_kx_k\| \leq s\), where \(r,s\in\mathbb R^{>0}\), all \(\lambda_i\) are in \(\mathbb R\) or \(\mathbb C\) (depending of the ground field of \(\mathcal H\)), and all \(x_i\) are of the same sort. Positive formulas are defined to be positive Boolean combinations of atomic formulas. Then with respect to positive formulas \(\mathcal H\) is a universal domain of a compact abstract theory, in the sense of \textit{I.~Ben-Yaacov} [J. Math. Log. 3, No.~1, 85--118 (2003; Zbl 1028.03034)]. A partial type is a set of positive formulas, possibly with parameters in \(\mathcal H\). A type-definable set is a set of realizations in \(\mathcal H\) of a partial type. An imaginary is the equivalence class \(a_E\) of a (possibly infinite!) tuple \(a\) modulo a type-definable equivalence relation \(E\). The goal of the paper is to characterize imaginaries in \(\mathcal H\). A type-definable set is called bounded if its cardinality is smaller than \(| \mathcal H| \). For imaginaries \(a\) and \(b\), we say that \(b\) is definable (bounded) over \(a\) if the set of realizations of \(\text{tp}(b/a)\) is a singleton (bounded set). The set of all \(b\) definable (bounded) over \(a\) is called the definable (bounded) closure of \(a\) (in symbols, \(\text{dcl}(a)\) and \(\text{bdd}(a)\), respectively). It is shown that \(\text{bdd}(a)\) is inter-definable with a unique Hilbert space \(H\subseteq\mathcal H\). Moreover, given \(H\), there is a Galois correspondence between the imaginaries \(a\) such that \(\text{bdd}(a)\) is inter-definable with \(H\) and the compact subgroups of \(U(H)\), the unitary group of \(H\) with the strong topology. The correspondence is given by \(a\mapsto\text{Gal}(a)\) and \(H\mapsto H^G\), where \(\text{Gal}(a)\) is \(\text{Aut}(\text{bdd}(a)/a)\), and \(H^G\) is the set of fixed points in \(\text{dcl}(H)\) under the action of \(G\). For \(n<\omega\), any subgroup \(G\) of the group of unitary \(n\times n\) matrices \(U(n)\) naturally acts on \({\mathcal H}^n\). We write \(\bar{c}E_G\bar{e}\) if \(\bar{c}\) and \(\bar{e}\) are orthonormal \(n\)-tuples in \(\mathcal H\), and \(g(\bar c)=\bar e\) for some \(g\in G\). The equivalence relation \(E_G\) is type-definable if \(G\) is closed. A classical result is that for any compact subgroup \(G\) of \(U(H)\) the Hilbert space \(H\) is a Hilbert direct sum of finite-dimensional \(G\)-invariant subspaces. Together with the Galois correspondence, this implies the main result of the paper: every imaginary is inter-definable with a tuple of (finitary!) imaginaries of the form \(\bar{e}_{E_G}\), where \(\bar{e}\) is an orthonormal tuple in \(\mathcal H\) of finite length \(n\), and \(G\) is a closed subgroup of \(U(n)\).
scientific article; zbMATH DE number 2123161