# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  DOI: 10.1016/0001-8708(91)90035-6 · Zbl 0744.46052  DOI: 10.1216/rmjm/1181071922 · Zbl 0913.46042  DOI: 10.1215/00127094-2008-036 · Zbl 1153.37029  Entov M., Comment. Math. Helv. 81 pp 75–  DOI: 10.4310/PAMQ.2007.v3.n4.a9 · Zbl 1143.53070  DOI: 10.4153/CMB-1999-035-5 · Zbl 0938.28009  DOI: 10.2140/agt.2006.6.405 · Zbl 1114.53070  DOI: 10.1007/978-3-0348-8299-6  DOI: 10.4310/MRL.2000.v7.n2.a8 · Zbl 0976.37030  DOI: 10.1090/S0002-9939-00-05541-6 · Zbl 0957.28004  DOI: 10.1215/S0012-7094-98-09207-9 · Zbl 1054.37506  DOI: 10.1016/0166-8641(95)00009-6 · Zbl 0842.28005  DOI: 10.3934/jmd.2007.1.465 · Zbl 1131.53046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.