Equivalence of commutation relations

We consider C*-algebras of commutation relations over the fields ℚ _p, p = 2, 3, 5, …, ∞. We describe all the irreducible separable representations of these algebras. We prove that the algebras are not isomorphic at different p. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 406–412, December, 2008.     

Extremal maximal sectorial extensions of sectorial relations

The extremal maximal sectorial extensions of a not necessarily densely defined sectorial relation (multivalued linear operator) in a Hilbert space are characterized in terms of a construction which goes back to Sebestyén and Stochel. In particular the two extreme maximal sectorial extensions, namely the Friedrichs extension and the Kreĭn extension, are characterized. For this purpose a survey is given of the connection between closed sectorial forms and maximal sectorial relations.     

Equivalence systems with finitely many relations

We consider a setX with a finite totally ordered setE of equivalence relations onX. We describe the automorphism group of this system, that is, the group of all those permutations ofX that leave each relation inE invariant.     

On proper accretive extensions of positive linear relations

A linear relation S is called a proper extension of a symmetric linear relation S if S S S^*. As is well known, an arbitrary dissipative extension of a symmetric linear relation is proper. In the present paper, we establish criteria for an accretive extension of a given positive symmetric linear relation to be proper.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 723–730, June, 1995.     

Equivalence relations and distances between Hilbert frames

We study some equivalency relations between Hilbert frames and closed subspaces of . We define also a distance between frames and we establish the geometric meaning of this metric. Finally we find the closest and respectively the nearest tight frame to a given frame.

     

Extremal extensions for the sum of nonnegative selfadjoint relations

The sum of two nonnegative selfadjoint relations (multi-valued operators) and is a nonnegative relation. The class of all extremal extensions of the sum is characterized as products of relations via an auxiliary Hilbert space associated with and . The so-called form sum extension of is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with and . Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on and .

     

Self adjoint extensions satisfying the Weyl operator commutation relations

Sans résumé Dedicated to the memory of our friend Karel deLeeuw Research partially supported by grants from the National Science Foundation.     

Normal extensions of subnormal linear relations via quaternionic Cayley transforms

A quaternionic Cayley transform for linear relations is introduced and some of its properties are exhibited. We emphasize the role played by the linear relations whose quaternionic Cayley transforms are unitary operators, which happen to be normal relations, and investigate the class of those linear relations which extend to such normal relations.     

Equivalence relations among inequalities for some relative operator entropies

Positivity - The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz...     

Equivalence relations in semihypergroups and the corresponding quotient structures

In this paper, we introduce and study two equivalence relations in semihypergroups, for which the corresponding quotient structures are monoids and commutative monoids.     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

The lattice of extensions of the modal logic of two equivalence relations has the cardinality of the continuum

A continuum of different logics over S5⋇S5 is constructed. This proves that the cardinality of the lattice of all normal extensions of the logic of two equivalence relations Ext(S5⋇S5) is continuum.     

Baer extensions of rings and stone extensions of semigroups

Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e₁ ε P and e₂ ε Ann(P) such that x→(e₁x, e₂x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed. An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico, Mayaguez, Puerto Rico, March 9–13, 1970.     

Equivalence relations in the approximations for the M/G/s/s + r queue

This paper investigates some equivalence relations among previously established approximations for the steady-state distribution in an M/G/s queue with finite waiting spaces. The focus is on four approximations developed by Hokstad [1], Tijms and van Hoorn [2], Miyazawa [3] and Kimura [4]. These approximations have been obtained by completely different approaches and they have different expressions. Equivalence theorems show conditions under which some of the approximations coincide.