Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.     

Description of Real AW*-Factors of Type I

In the paper, real AW*-algebras are considered, i.e., real C*-algebras which are Baer *-rings. It is proved that every real AW*-factor of type I (i.e., having a minimal projection) is isometrically *-isomorphic to the algebra B(H) of all bounded linear operators on a real or quaternionic Hilbert space H and, in particular, is a real W*-factor. In the case of complex AW*-algebras, a similar result was proved by Kaplansky.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

On Inductive Limits for Systems of <Emphasis Type="Italic">C</Emphasis>*-Algebras

We consider a covariant functor from the category of an arbitrary partially ordered set into the category of C*-algebras and their *-homomorphisms. In this case one has inductive systems of algebras over maximal directed subsets. The article deals with properties of inductive limits for those systems. In particular, for a functor whose values are Toeplitz algebras, we show that each such an inductive limit is isomorphic to a reduced semigroup C*-algebra defined by a semigroup of rationals. We endow an index set for a family of maximal directed subsets with a topology and study its properties. We establish a connection between this topology and properties of inductive limits.     

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.     

Model theory of operator algebras II: model theory

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ? are isomorphic even when the Continuum Hypothesis fails.     

Weak-local derivations and homomorphisms on C*-algebras

We prove that every weak-local derivation on a C*-algebra is continuous, and the same conclusion remains valid for weak*-local derivations on von Neumann algebras. We further show that weak-local derivations on C*-algebras and weak*-local derivations on von Neumann algebras are derivations. We also study the connections between bilocal derivations and bilocal*-automorphism with our notions of extreme-strong-local derivations and automorphisms.     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.     

A Dixmier-Douady theorem for Fell algebras

We generalise the Dixmier-Douady classification of continuous-trace C^?-algebras to Fell algebras. To do so, we show that C^?-diagonals in Fell algebras are precisely abelian subalgebras with the extension property, and use this to prove that every Fell algebra is Morita equivalent to one containing a diagonal subalgebra. We then use the machinery of twisted groupoid C^?-algebras and equivariant sheaf cohomology to define an analogue of the Dixmier-Douady invariant for Fell algebras, and to prove our classification theorem.     

Non-Pointed Strongly Protomodular Theories

We give criterions for strong protomodularity and prove that the strong protomodularity of an algebraic theory is inherited by its models in a category with finite limits. We give examples of strongly protomodular theories with several constants: C ^*-algebras, rings, Heyting algebras and Boolean algebras.     

Restricted Lie 2-algebras

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.     

Omitting types and AF algebras

We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.     

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.     

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the -algebra of a CKT family, and prove a universality theorem for -algebras generated by these families. In both cases, the -algebras generated by quiver algebras play the universal role.

     

Limit Automorphisms of the <Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Isometric Representations for Semigroups of Rationals

We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are C*-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these C*-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold-Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E_∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E_∞-operads to E_∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.     

PrO-C*-代数的顺从性和核性

研究了Pro—C^*-代数的顺从性和核性．主要证明了(1)顺从Pro—C^*代数的闭理想是顺从的；(2)核Pro—C^*代数类对归纳极限封闭；(3)交换σ-C^*-代数和核C^*-代数都是核，σ-C^*-代数并且核σ-C^*-代数类对于商运算、张量积运算和可数逆向极限封闭．进一步得到核，σ-C^*-代数的扩张保持核性的条件。     

Completely Positive Definite Maps on σ-C*-algebras

Recently,there has been increased interest[1-8]in topological*-algebras that are inverse