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Register a new userDefinable relations in finite dimensional subspace lattices with involution. Part II: Quantifier free and homogeneous descriptions
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For finite dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Plucker coordinates.In the latter setting, homogeneous descriptions are obtained if one allows quantification type $Sigma_1$. In absence of involution, these results remain valid.
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Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*-algebras that admit quantifier elimination in continuous logic are $mathbb{C},$ $mathbb{C}^2,$ $M_2(mathbb{C}),$ and the continuous functions on the Cantor set. We show that, among finite dimensional C*-algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate projections. Both of these predicates are definable, but not quantifier-free definable, in the usual language of C*-algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.
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We continue the study of a class of topological $mathcal{L}$-fields endowed with a generic derivation $delta$, focussing on describing definable groups. We show that one can associate to an $mathcal{L}_{delta}$ definable group a type $mathcal{L}$-definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.
For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Delta$-definable $G$-flows for any fixed set $Delta$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{Delta,M}$ and ${G^*}^{000}_{Delta,M}$, and show that $G^*/{G^*}^{00}_{Delta,M}$ is the $Delta$-definable Bohr compactification of $G$. We also note that some deeper arguments from the topological dynamics in the category of externally definable $G$-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the $Delta$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $Delta$-definable generalized Bohr compactification to the $Delta$-definable Bohr compactification of $G$. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.
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