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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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