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Register a new userInfinite-Dimensional Generalizations of Orthogonal Groups over Hilbert Spaces : Constructions and Properties
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Authors
Luo Jianwen
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In real Hilbert spaces, this paper generalizes the orthogonal groups $mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Theta(kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$mathrm{O}=varinjlimmathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Theta(kappa)$ is a topological and normal subgroup of $O(kappa)$; (b) $O^{(n)}(kappa) to O^{(n+1)}(kappa) stackrel{pi}{to} S^{kappa}$ is a fibre bundle where $O^{(n)}(kappa)$ is a subgroup of $O(kappa)$ and $S^{kappa}$ is a generalized sphere.
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We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(rho_beta)$, for a thermal density operator $rho_beta$ at inverse temperature $beta$. More generally, $GAP(rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(rho)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorovs theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(rho)$ depends continuously on $rho$ in the sense that convergence of $rho$ in the trace norm implies weak convergence of $GAP(rho)$.
In this paper, the $m-$order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $pi$ and $pi^frac1{m-1}$, respectively.
In this expository note we provide a proof of Artins theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned in the note but takes a somewhat different approach which seems to be of didactic value.
1. We answer Michael Gordins question providing singular spectrum for transformations with homoclinic Bernoulli flows via Poisson suspensions induced by modified Sidon rank-one constructions. 2. We give homoclinic proof of Emmanuel Roys theorem on multiple mixing of Poisson suspensions, adding new examples to Jonathan Kings ergodic homoclinic groups of special zero-entropy transformations. 3. Sasha Prikhodko found the fast decay of correlations for some iceberg automorphisms. We get similar correlations for a class of infinite rank-one Sidon transformations. This version is based on On Mixing Rank One Infinite Transformations arXiv:1106.4655
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