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Quasi-states and symplectic intersections. (English) Zbl 1096.53052
A link between symplectic topology and the theory of quasi-states and quasi-measure (toplogical measure) is established.
The notion of quasi-state is orginated in quantum mechanics [J. F. Aarnes, Acta Math. 122, 161–172 (1969; Zbl 0183.14203)]. A functional $$\zeta: C(M)\to\mathbb{R}$$, where $$C(M)$$ is the Banach algebra of continuous functions on $$M$$ is called a quasi-state if $$\zeta$$ is linear on $${\mathcal A}_F= \{p\circ F|p$$ is a real polynomial}, for every $$F\in C(M)$$, and $$\zeta(F)\leq \zeta(G)$$ for $$F\leq G$$, $$\zeta(1)= 1$$. If $$M$$ is a symplectic manifold a symplectic quasi-state $$\zeta$$ is said to be symplectic, if it satisfies
1. $$\zeta(F+ G)= \zeta(F)+ \zeta(G)$$, if $$\{F, G\}= 0$$.
2. $$\zeta(F)= 0$$, if $$\text{supp\,}F$$ is displaceable.
3. $$\zeta(F)= \zeta(F\circ f)$$, $$f\in \text{Symp}_0(M)$$; the identity component of the group of symplectomorphism.
Here, a subset $$X$$ of $$M$$ is said to be displaceable, if there exists a Hamiltonian diffeomorphism $$\phi$$ such that $$\phi(X)\cap \text{Closure}(X)= \emptyset$$ (§3). If $$\zeta: C(M)\to \mathbb{R}$$ is not linear on $${\mathcal A}_F$$ or the sum of $$F$$, $$G$$, $$\{F,G\}= 0$$, but satisfies other axioms of symplectic quasi-state, and
1. $$\zeta(F_1+ F_2)= \zeta(F_1)$$, if $$\{F_1, F_2\}= 0$$ and $$\text{supp\,}F_2$$ is displaceable,
2. $$\zeta(\lambda F)=\lambda f(F)$$ for any $$F$$ and $$\lambda\in\mathbb{R}_{\geq 0}$$,
then it is called a partially quasi-state (§4).
The main results on quasi-states in this paper are Theorem 3.1: $$C(M)$$ admits a symplectic quasi-state if $$M$$ is either of $$\mathbb{C}\text{P}^n$$, a complex Grassmannian $$\mathbb{C}\text{P}^{n_1}\times\cdots\times \mathbb{C}\text{P}^{n_k}$$ with a monotone product symplectic structure, the monotone symplectic blow-up of $$\mathbb{C}\text{P}^2$$ at one point, and
Theorem 4.1: $$C(M)$$ admits a partial symplectic quasi-state, if $$M$$ is a strongly semi-positive and rational closed connected symplectic manifold.
Given a finite-dimensional Poisson-commutative subspace $${\mathcal A}\subset C^\infty(M)$$, its moment map $$\Phi_{{\mathcal A}}: M\to{\mathcal A}^*$$ is defined as $$\langle\Phi_{{\mathcal A}}(x),F\rangle= F(x)$$. Non-empty subsets of the form $$\Phi^{-1}_{{\mathcal A}}(p)$$ are called fibers of $${\mathcal A}$$. A closed subset $$X\subset M$$ is called stem, if there exists a finite-dimensional Poisson-commutative subspace $${\mathcal A}\subset C^\infty(M)$$ so that $$X$$ is a fiber of $${\mathcal A}$$ and each fiber of $${\mathcal A}$$, other than $$X$$, is displaceable. Then the following Theorem 2.1 is derived from Theorem 4.1 and Theorem 2.4 is derived from Theorem 2.1, Theorem 3.1 and the generalized Riesz representation theorem of Aarnes, which associates to each quasi-state $$\zeta$$ a quasi-measure (topological measure) $$\tau_\zeta$$ [J. F. Aarnes, Adv. Math. 86, No. 1, 41–67 (1991; Zbl 0744.46052)]. These are main results of this paper.
Theorem 2.1. Any finite-dimensional Poisson-commutative subspace of $$C^\infty(M)$$ has at least one non-displaceable fibre. Moreover, if every fibre has a finite number of connected components, there exists a fibre with a non-displaceable connected component.
Theorem 2.4. Any two stems in $$M$$ have non-empty intersection, if $$M$$ is one of the manifolds listed in Theorem 3.1.
Proofs of Theorems 3.1 and 4.1 need facts about the spectral numbers of Hamiltonian diffeomorphims [Y.-G. Oh, Prog. Math. 232, 525–570 (2005; Zbl 1084.53076)]. These are reviewed in §5. Theorem 3.1 is proved in §6 and Theorem 4.1 is proved in §7. In §8, symplectic a quasi-measure is defined and show that on a closed surface, any symplectic quasi-measure gives rise to a symplectic quasi-state (Theorem 8.1). In §9 a Clifford torus in $$\mathbb{C}\text{P}^n$$ and the monotone blow-up of $$\mathbb{C}\text{P}^2$$ at one point are shown to be a stem [cf. P. Biran, M. Entov and L. Polterovich, Commun. Contemp. Math. 6, No. 5, 793–802 (2004; Zbl 1076.53110)]. In §10, beginning from von Neumann, history and physical meaning of quasi-states are explained. Some open problems on quasi-states and symplectic intersections are presented in §8–§10.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 46L30 States of selfadjoint operator algebras 53D40 Symplectic aspects of Floer homology and cohomology
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