Unramified forcing preserving the law of double negation

Shoenfield's unramified version of Cohen's forcing is defined in two stages: one which does not preserve double negation and the other which modifies the former so as to preserve double negation. Here we express the unramified forcing, which preserves double negation, in a single stage. Surprisingly enough, the corresponding definition of forcing for equality acquires a rather simple form. In [2] forcing ∥- is expressed in terms of strong forcing \( \Vdash * \) viap∥-Q iffp \( \Vdash * \) ¬ ¬Q for every formulaQ ofZF set theory and every elementp of a partially ordered set (P, ≦). In its turn,p \( \Vdash * \) Q is defined by the following five clauses: (1) $$p \Vdash * a \in biff(\exists c)(\exists q \geqq p)((c,q) \in b \wedge p \Vdash * a = c)$$ (2) $$\begin{gathered} p \Vdash * a \ne biff(\exists c)(\exists q \geqq p)(((c,q) \in a \wedge p \Vdash * c \notin b) \hfill \\ ((c,q) \in b \wedge p \Vdash * c \notin a)) \hfill \\ \end{gathered} $$ (3) $$p \Vdash * \neg Qiff(\forall q)(q \leqq p \to \neg (q \Vdash * Q))$$ (4) $$p \Vdash * (Q \vee S)iff(p \Vdash * Q) \vee (p \Vdash * S)$$ (5) $$p \Vdash * (\exists x)Q(x)iff(\exists b)(p \Vdash * Q(b))$$ .     

The double negation of the intermediate value theorem

In the context of intuitionistic analysis, we consider the set F consisting of all continuous functions ? from [0,1] to R such that ?(0)=0 and ?(1)=1, and the set I₀ consisting of ?’s in F where there exists x∈[0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have I₀≠F. However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema (which we call ) about intuitionistic decidability that asserts “there exists an intuitionistically enumerable set that is not intuitionistically decidable”. We also introduce the notion of strong Specker double sequence, and prove that the existence of such a double sequence is equivalent to the existence of a function ?∈F_mon where .     

Every conservative double Hausdorff matrix is a kth absolutely summable operator

Let A denote the set of kth absolutely summable double series. In this paper it is shown that every double conservative Hausdorff matrix is a bounded operator on A.     

The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator

Let (M,r) be a closed, space- and time-orientable, pseudo-Riemannian spin manifold and-let G be a compact group of orientation-preserving isometries on (M,r). If there are no isotropic directions transversal to the orbits of G, then the Dirac operator on (M,r) is transversally elliptic. In this paper we calculate its index.     

Certain Jordan operator algebras and double commutant theorems

Two operations on sets of bounded operators on Hilbert space are studied: One produces Jordan algebras and the other, its inverse in a certain sense, produces self-adjoint linear spaces. The question of which Jordan algebras arise from the first of these operations is investigated. It is shown that, in many cases, R has this property if and only if the complex associative algebra R + iR is equal to its double commutant. Thus, a by-product of the study is another approach to double commutant theorems.     

The Bergman projection as a singular integral operator

We show that the Bergman projection operator, associated to one of three classes of domains (all smoothly bounded)-a finite type domain ?²; a decoupled, finite type domain in ?ⁿ; or a convex, finite type domain in wfⁿ-may be viewed as a generalized Calderón-Zygmund operator. As an application of this observation, we show that the Bergman projector on any of these domains preserves the Lebesgue classesL ^p , 1 <p < ∞.     

Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

A sufficient condition for the local solvability of a linear partial differential operator with double characteristics

It is shown that a sufficient condition for the local solvability of an operator $\text{P(x, D}{}_{\mathrm{x}}\text{) = P}{}_{\mathrm{m}}{}^2\text{+ P}{}_{\mathrm{2m\; ?\; 1}}\text{,}\text{where}\text{P}$ is an mth-order homogeneous operator of principal type with real coefficients, is that the imaginary part of the subprincipal symbol of P has constant sign near any null-bicharacteristic curve of P_m, and that the real part of the subprincipal symbol is not equal to zero when the imaginary part is.     

Bregman divergence as relative operator entropy

The Bregman operator divergence is introduced for density matrices by differentiation of the matrix-valued function x ? x log x. This quantity is compared with the relative operator entropy of Fujii and Kamei. It turns out that the trace is the usual Umegaki’s relative entropy which is the only intersection of the classes of quasi-entropies and Bregman divergences.     

The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra of dimension attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra and a primitive root of unity q of order n² is equivalent to Lusztig's small quantum group (under some conditions on n). We also give a conceptual construction of using the notion of de-equivariantization of tensor categories.     

Perturbation of a self-adjoint operator by a subordinate symmetric operator

The following variant of Rellich's theorem is proved. Let A,B be operators in a Hilbert space, A=A^*, BB^* and D(B)D(A). We assume that (Bu,u)(Au,u), uD(A) for some> –1. Then the operator A + B with domain of definition D(A) is self-adjoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 196–198, 1985.     

Identities of semigroups of relations with an operator of reflexive double cylindrification

In the paper we find a basis of identities for the variety generated by semigroups of relations with an operation of reflexive double cylindrification.