The structure of crossed productC *-algebras: A proof of the generalized Effros-Hahn conjecture

IfG is a second countable locally compact group acting continuously on a separableC ^*-algebraA, then every primitive ideal of the crossed productC ^* (G, A) is contained in an induced primitive ideal, and ifG is amenable, equality holds. Thus ifG is amenable and acts freely on Prim(A), the generalized Effros-Hahn conjecture holds: there is a canonical bijection between primitive ideals ofC ^* (G, A) andG-quasi-orbits in Prim(A). Applications to the Mackey machine for a non-regularly embedded normal subgroup of a locally compact group are discussed. The proof of the theorem is based on a local cross-section result together with Mackey's original methods.The authors were partially supported by National Science Foundation Research GrantsThe first-named author would like to thank the Department of Mathematics, University of Pennsylvania, for its warm hospitality during his 1977–78 stay, during which time this research was conducted     

Primitive Limit Algebras and C-Envelopes

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C^*-envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C^*-envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C^*-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.     

Relative K homology and C* algebras

Let A be a separable nuclear C ⁺ algebra with unit. Let be a closed two-sided ideal in A. A relative K homology group K ⁰(A,) is defined. Closely related are topological definitions of properly supported K homology and of compactly supported relative K homology. Applications are to indices of Toeplitz operators and existence of coercive boundary conditions for elliptic differential operators.     

The Lindenbaum-Tarski algebra for Boolean algebras with distinguished ideals

A complete solution is given to the problem of describing algebras with distinguished ideals, formulated by Peretyatkin. It is proven that such an algebra is isomorphic to _× , an interval algebra of the linear ordering × . I-algebras the elementary theory of each of which is axiomatizable by a single atom in some finite quotient with respect to the Frechet ideal of the Lindenbaum-Tarski algebra for the class of Boolean algebras with distinguished ideals are fully described in terms of direct summands.Translated fromAlgebra i Logika, Vol. 34, No. 1, pp. 88–116, January–February, 1995.     

Derived-tame Tree Algebras

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form _A is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of _A . Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.     

BRST Operator for Quantum Lie Algebras: Relation to the Bar Complex

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit quantum generalizations. In particular, there is a BRST operator Q (Q ²=0) that generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers, we gave and solved a recursive relation for the operator Q for quantum Lie algebras. Here, we consider the bar complex for q-Lie algebras and its subcomplex of q-antisymmetric chains. We establish a chain map (which is an isomorphism) of the standard complex for a q-Lie algebra to the subcomplex of the antisymmetric chains. The construction requires a set of nontrivial identities in the group algebra of the braid group. We also discuss a generalization of the standard complex to the case where a q-Lie algebra is equipped with a grading operator.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

Homomorphisms of unary algebras with a given quotient

For a unary algebraA with2 fundamental operations, letH(A) denote the class of all unary algebras that have a homomorphisrn intoA, and let the classQ(A) consist of all algebras havingA as one of their quotients. IfA is freely indecomposable then H(A) andQ(A) are shown to be categorically universal if and only if either class contains a rigid algebra; this, in turn, is equivalent to the absence of homomorphisms fromA into a free algebra.Presented by Ralph McKenzie.The support of the NSERC is gratefully acknowledged.     

When Is a Subspace of B(H) Ideal?

Let be a real or complex Hilbert space and let () denote the algebra of all bounded linear operators on . We show that if N is a subspace of (H) and for positive operators P₁, P₂ and every A N, P₁P₂A^* + AP₁P₂ N then N is an ideal. Furthermore if is an infinite dimensional real space then N = (H).AMS Subject Classification (1991): Primary 47B47, 47D25     

Aur la vacuité du spectre d'un élément d'une algèbre\mathcal{L}\mathcal{F}

We give an example of a complete commutative unitary and semi-simple topological algebra, which is a locally convex inductive limit of an increasing sequence of Fréchet algebras ( algebra), and which contains the field (X) of rational functions; so it contains elements which have empty spectrum and therefore does not contain any character, neither continuous nor non-continuous. This unitary algebra is not a division algebra, so it contains at least one non-trivial maximal ideal; but none of its maximal ideals is closed and they all have infinite codimension. The Gelfand-Mazur Theorem remains therefore unknown for algebras.

     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Color Lie superalgebras and hopf algebras

We give a generalization of the well-known theorem stating that the category of primitively generated Hopf algebras is equivalent to the category of (restricted) Lie algebras. In so doing, instead of Lie algebras, we consider color Lie superalgebras, and instead of a primitively generated Hopf algebra, we take a Hopf algebra H whose semigroup elements form an Abelian group G =G(H), and H is generated by its relatively primitive elements which supercommute with the elements of G. Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 420–436, July-August, 1995.Supported by the Russian Foundation for Fundamental Research, grant No. 93-01-16171.     

Z n -Orthograded Monocomposition Algebras

We study NQM algebras A having an orthogonal automorphism of finite order n 3 (called Z _n -orthograded NQM algebras). The Z ₃-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras A which are not bi-isotropic in this class, and for every algebra A, determine an automorphism group Aut,A and an orthogonal automorphism group Ortaut,A. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras.     

Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element xA is a resolvent value of a positive element yA if and only if the element x satisfies the negative principle: If aA, < 0 and xaa then xa 0.     

On trivial and non-trivial n-homogeneousC * algebras

LetA be aC ^* — algebra for which all irrèducible representations are of dimensional n. Then ([F], [TT], [V]) algebraA is isomorphic to algebra of all continuous sections of an appropriate algebraic bundle _A . The basisX of this bundle coincides with the compact of all maximal two-sided ideals ofA. We obtain some conditions which provide that _A is trivial and this yields thatA is isomorphic to the algebra of alln×n matrix functions continuous onX. In the case whenX=S ⁿ is a sphere we describe the set of algebraic bundles overX and algebraic structures on this set. Some applications to algebras generated by idempotents are suggested.     

On Flat Semimodules over Semirings

The following analog of the characterization of flat modules has been obtained for the variety of semimodules over a semiring R: A semimodule _RA is flat (i.e., the tensor product functor – A preserves all finite limits) iff A is L-flat (i.e., A is a filtered colimit of finitely generated free semimodules). We also give new (homological) characterizations of Boolean algebras and complete Boolean algebras within the classes of distributive lattices and Boolean algebras, respectively, which solve two problems left open in [14]. It is also shown that, in contrast with the case of modules over rings, in general for semimodules over semirings the notions of flatness and mono-.atness (i.e., the tensor product functor – A preserves monomorphisms) are different.     

The Structure of Frobenius Algebras and Separable Algebras

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R¹²R²³=R²³R¹³=R¹³R¹², called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang–Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) – the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B A is Frobenius if and only if A has a B-coring structure (A, , ) such that the comultiplication : A A_B A is an A-bimodule map.     

Proper uniform pseudodifferential operators on unimodular Lie groups

An algebra of proper pseudodifferential operators on an arbitrary unimodular Lie group is constructed. This algebra is a generalization of a well-known algebra of operators with uniform estimates of the symbols onR ⁿ (such operators have been investigated in detail by Kumano-go); in the general case the estimates have to be left-invariant. An L²-boundedness theorem is proved and uniform Sobolev spaces are introduced and investigated. The essential self-adjointness of uniformly elliptic operators is proved. A criterion for the coincidence of the left and right Sobolev spaces and of the corresponding algebras of operators is given: it is necessary and sufficient that the considered Lie group be a central extension of a compact group.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 74–97, 1986.