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A comparison of symplectic homogenization and Calabi quasi-states. (English) Zbl 1235.28002
Let \(X\) be a locally compact Hausdorff space, \(C_c(X)\) the space of continuous functions with compact support on \(X\). A quasi-integral on \(X\) is a functional \(\eta : C_c(X) \to \mathbb{R}\) such that a) \(\eta(f) \leq \eta(g)\) if \(f \leq g\); b) for each compact \(K \subset X\) there is a number \(N_K \geq 0\) such that \(|\eta(f)-\eta(g)| \leq N_K \cdot \sup_X |f - g|\); c) \(\eta\) is linear on every subspace \(\{\phi\circ f \mid \phi \in C({\mathbb R}), \phi(0)=0\}\), where \(f \in C_c(X)\).
Let \((M,\omega)\) be a symplectic manifold. A symplectic quasi-integral is a quasi-integral on \(M\) which is linear on any Poisson commutative subspace of \(C^\infty_c(M)\). The main result of the paper concerns symplectic quasi-integrals on \(T^*\mathbb{S}^1\) and \(\mathbb{S}^2\) with standard symplectic structures.
The authors consider a quasi-integral \(\eta_0(f) = H(f)(0)\) on \(T^*\mathbb{S}^1\), where \(H : C_c(T^*\mathbb{S}^1) \to C_c(\mathbb{R})\) is a (non-linear) operator which is monotone, Lipschitz continuous, and linear on any Poisson commutative subalgebra of \(C^\infty_c(T^*\mathbb{S}^1)\) (see [C. Viterbo, “Symplectic homogenization”, arXiv:0801.0206]). On \(\mathbb{S}^2\) there is a symplectic quasi-integral \(\zeta\) (called Calabi quasi-state; see [M. Entov and L. Polterovich, Comment. Math. Helv. 81, No. 1, 75–99 (2006; Zbl 1096.53052)]) which is invariant under Hamiltonian diffeomorphisms. For any \(r \in (0,1/2]\) there is a symplectic embedding \(j_r : U_r \subset T^*\mathbb{S}^1 \to \mathbb{S}^2\) where \(U_r = \mathbb{S}^1 \times (-r,r)\) and \(j_r(\mathbb{S}^1 \times 0)\) is the equator. Set \(\zeta_r = \zeta \circ (j_r)_*\), where \((j_r)_* : C_c(U_r) \to C(\mathbb{S}^2)\) is induced by \(j_r\). The main result of the paper is that \(\eta_0\) restricted to \(C_c(U_r)\) coincides with \(\zeta_r\) if and only if \(r \in (0,1/4]\).
The proof uses the properties of quasi-integrals and topological measures on locally compact spaces, in part, the fact that there is a one-to-one correspondence between quasi-integrals and topological measures. Also the authors give an elementary proof of the fact that a quasi-integral on a surface is symplectic.

28A25 Integration with respect to measures and other set functions
53D05 Symplectic manifolds (general theory)
53D35 Global theory of symplectic and contact manifolds
46L30 States of selfadjoint operator algebras
Full Text: DOI arXiv
[1] DOI: 10.1016/0001-8708(91)90035-6 · Zbl 0744.46052
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