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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.

MSC:
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
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