Prime spectra of non-commutative generalizations of MV-algebras

Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We show that GMV-algebras can be considered as special cases of Grishin algebras. As MV-algebras are algebraic models of the Łukasiewicz logic and Grishin algebras have the analogous role for the classical bilinear logic, GMV-algebras correspond to a non-commutative logic between the above logics. Further, by A. Dvurečenskij, any GMV-algebra is isomorphic to an interval of an l-group, which in general is not commutative. This generalizes D. Mundici's representation of MV-algebras by means of intervals of abelian l-groups. In the paper (using this representation) we describe the properties of prime ideal spectra of GMV-algebras and of their factor algebras and ideals and prove that the spectrum of closed ideals of any GMV-algebra is homeomorphic to that of a completely distributive GMV-algebra. Received January 4, 2001; accepted in final form May 2, 2002.     

Radical classes and weak radical mappings of GMV-algebras

We use the concept of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis; the main tool in their investigation was a truncation construction. The relations between radical classes of GMV-algebras and radical classes of lattice ordered groups are investigated in the present paper. Further, we apply the truncation construction for dealing with weak retract mappings of GMV-algebras. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence — Physics of Information (Grant I/2/2005).     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Monadic <Emphasis Type="Italic">GMV</Emphasis>-algebras

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed. The first author was supported by the Council of Czech Government, MSM 6198959214.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.     

Formations of lattice ordered groups and of <Emphasis Type="Italic">GMV</Emphasis>-algebras

A class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ₁ and ℱ₂ the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ₁ and ℱ₂ are partially ordered by the class-theoretical inclusion. We prove that ℱ₁ satisfies the infinite distributivity law and that ℱ₂ is isomorphic to a principal ideal of ℱ₁. This work was supported by VEGA grant 2/7141/27.     

On Stability of Cones in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript> with Zero Scalar Curvature

In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rⁿ⁺¹. If Mⁿ⁻¹ is a compact hypersurface of the sphere Sⁿ(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an ε for which the truncate cone C(M)_ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mⁿ⁻¹ of the sphere with zero scalar curvature and S₃ different from zero, for which all truncated cones based on M are stable. Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70.     

On finite <Emphasis Type="Italic">p</Emphasis>-groups whose automorphisms are all central

An automorphism α of a group G is said to be central if α commutes with every inner automorphism of G. We construct a family of non-special finite p-groups having abelian automorphism groups. These groups provide counterexamples to a conjecture of A. Mahalanobis [Israel J. Math. 165 (2008), 161–187]. We also construct a family of finite p-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in Group Theory, Aracne Editrice, Rome, 2002, pp. 111–127].     

Strikte Konvexität für Variationsprobleme auf demn-dimensionalen Torus

We consider a variational problem with an integrandF:R ⁿ ×R×R ⁿ →R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R ⁿ minimal solutionsu:R ⁿ →R minimising ƒF(x, u(x), u _x (x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action (whereB _r is ther-ball around 0∈R ⁿ ) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.      

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.     

Quasi-elementary <Emphasis Type="Italic">H</Emphasis>-Azumaya Algebras Arising from Generalized (Anti) Yetter-Drinfeld Modules

Let H be a Hopf algebra with bijective antipode, α, β ∈ Aut _Hopf (H) and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.     

(<Emphasis Type="Italic">o</Emphasis>)-Topology in *-algebras of locally measurable operators

We consider the topology t( M ) t\left( \mathcal{M} \right) of convergence locally in measure in the *-algebra LS( M ) LS\left( \mathcal{M} \right) of all locally measurable operators affiliated to the von Neumann algebra M \mathcal{M} . We prove that t( M ) t\left( \mathcal{M} \right) coincides with the (o)-topology in LS_h( M ) = { T ? LS( M ):T* = T } L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} if and only if the algebra M \mathcal{M} is σ-finite and is of finite type. We also establish relations between t( M ) t\left( \mathcal{M} \right) and various topologies generated by a faithful normal semifinite trace on M \mathcal{M} .     

On （α,β）-Metrics of Scalar Flag Curvature with Constant S-curvature

In this paper, we study （α,β）-metrics of scalar flag curvature on a manifold M of dimension n （n 〉 3）. Suppose that an （α,β）-metric F is not a Finsler metric of Randers type, that is, F ≠k1 V√α^2 ＋ k2β^2 ＋ k3β, where k1 〉 0, k2 and k3 are scalar functions on M. We prove that F is of scalar flag curvature and of vanishing S-curvature if metric. In this case, F is a locally Minkowski and only if the flag curvature K = 0 and F is a Berwald metric.     

Finiteness conditions and PD<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-group covers of PD<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-complexes

We show that an infinite cyclic covering space M′ of a PD _n -complex M is a PD _n-1-complex if and only if χ(M) = 0, M′ is homotopy equivalent to a complex with finite [(n−1)/2]-skeleton and π₁(M′) is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M _ν with covering group a PD _r -group under a slightly stricter finiteness condition.      

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001     

Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.