THE REALIZATION OF MULTIPLIER HILBERT BIMODULE ON BIDUAL SPACE AND TIETZE EXTENSION THEOREM

51. Introduction and PreliminaryThe multiplier bimodule of imprimitive bimodule and its application to coaction systemwere introduced in [if, but it is generally not an imprimitive bimodule as noted in [II by anexample. In this paper, the same concept is introduced for Hilbert bimodules which properlycontain imprimitive bimodules (for example, the same example as above), and we obtainthe realization of multiplier bimodule on a quotient of bidual space and Tietze extensiontheorem similar to t…     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

Lie triple derivations on nest algebras

Let δ be a Lie triple derivation from a nest algebra ?? into an ??‐bimodule ??. We show that if ?? is a weak* closed operator algebra containing ?? then there are an element S ∈ ?? and a linear functional f on ?? such that δ (A) = SA – AS + f (A)I for all A ∈ ??, and if ?? is the ideal of all compact operators then there is a compact operator K such that δ (A) = KA – AK for all A ∈ ??. As applications, Lie derivations and Jordan derivations on nest algebras are characterized. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

Mellin‐edge representations of elliptic operators

We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator‐valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator ??_s in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of ??_s as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices ??_s of ??_s, interpreted as Mellin‐edge representations of P. Copyright © 2005 John Wiley & Sons, Ltd.     

On weakening the Deduction Theorem and strengthening Modus Ponens

This paper studies, with techniques of Abstract Algebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen‐style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert‐style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi‐Hilbert algebras, a quasi‐variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universal Jensen＇s Equations in Banach Modules over a C^＊-Algebra and Its Unitary Group

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen‘s equations in Banach modules over a unital C^*-algebra. It is applied to show the stability of universal Jensen‘s equations in a Hilbert module over a unital C^*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital C^*-algebra.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

A note on a subvariety of linear tense algebras

In [1], Bull gave completeness proofs for three axiom systems with respect to tense logic with time linear and rational, real and integral. The associated varieties, Dens, Cont and Disc, are generated by algebras with frames {?, <, >}, {?, <, >} and {?, <, >}, respectively. In this paper we consider the subvariety ?? generated by the finite members of Disc. We prove that V is locally finite and we determine its lattice of subvarieties. We also prove that ?? = Disc ∩ Dens = Disc ∩ Cont. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equational spectrum of Hilbert varieties

We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.      

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.     

Hochschild Cohomology Ring of an Order of a Simple Component of the Rational Group Ring of the Generalized Quaternion Group

We will give an efficient bimodule projective resolution of an order Γ, where Γ is an order of a simple component of the rational group ring ? Q _{2 ^r} of the generalized quaternion 2-group Q _{2 ^r} of order 2 ^r+2. Moreover, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this bimodule projective resolution.     

Noninjectivity of the Predual Bimodule of the Measure Algebra for Infinite Discrete Groups

In this paper, it is proved that the predual bimodule of the measure algebra of an infinite discrete group is not injective despite the fact that the measure algebra of an amenable group is amenable in the sense of Connes. Thus the well-known result of Khelemskii (claiming that, for a von Neumann algebra, Connes-amenability is equivalent to the condition that the predual bimodule is injective) cannot be extended to measure algebras. Moreover, for a discrete amenable group, we give a simple formula for a normal virtual diagonal of the measure algebra. It is shown that a certain canonical bimodule over the measure algebra is not normal.     

Explicit Equivalence Bimodules for Rotation Algebras

The C*-algebras associated with irrational rotations are Morita equivalent if and only if the rotation parameters belong to the same orbit under the action of GL(2, Z). In this note, we suggest explicit type II representations such that the bimodule is dense in the corresponding Hilbert space. Bibliography: 4 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 175–188.     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.