E *-Stable Theories

S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being T ^*-stable, generalizing the notion of being stable. However, T ^*-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for E ^*-stable theories. This concept differs from that of being T ^*-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any P-sets in P-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over P-models.     

Extensions of <Emphasis Type="Italic">GM</Emphasis>-rings

It is shown that a ring R is a GM-ring if and only if there exists a complete orthogonal set { e ₁,...,e _n } of idempotents such that all e _i Re _i are GM-rings. We also investigate GM-rings for Morita contexts, module extensions and power series rings.This work was supported by the Natural Science Foundation of Zhejiang Province.     

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     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Embeddings in Lattice-O *-Groups

In the paper, the following results are obtained: the existence of simple divisible lattice-O ^*-groups is established (Theorem 2.1) and it is proved that any countable lattice-orderable or right-orderable group can be isomorphically embedded in a simple divisible lattice-O ^*-group (Theorem 2.2 and Corollary 2.3).     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

On C *-Extreme Maps and *-Homomorphisms of a Commutative C *-Algebra

The generalized state space of a commutative C^*-algebra, denoted , is the set of positive unital maps from C(X) to the algebra of bounded linear operators on a Hilbert space . C^*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C^*-extreme point of satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C^*-extreme maps from C(X) into , the algebra generated by the compact and scalar operators, are multiplicative. This generalizes a result of D. Farenick and P. Morenz. We then determine the structure of these maps. This paper constitutes a part of the author’s Ph.D. thesis at the University of Nebraska-Lincoln.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Extremal K -Theory and Index for C*-Algebras

We develop the general theory for a new functor K _e on the category of C ^*-algebras. The extremal K-set, K _e (A), of a C ^*-algebra A is defined by means of homotopy classes of extreme partial isometries. It contains K ₁ (A) and admits a partially defined addition extending the addition in K ₁ (A), so that we have an action of K ₁ (A) on K _e (A). We show how this functor relates to K ₀ and K ₁, and how it can be used as a carrier of information relating the various K-groups of ideals and quotients of A. The extremal K-set is then used to extend the classical theory of index for Fredholm and semi-Fredholm operators.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

Some permanence properties of C-unique groups

We will study some permanence properties of C^*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C^*-unique group. Moreover, any amenable discrete group is a retract of a discrete C^*-unique group.     

Quantum Real Projective Space, Disc and Spheres

We define the C ^*-algebra of quantum real projective space R P _q ², classify its irreducible representations, and compute its K-theory. We also show that the q-disc of Klimek and Lesniewski can be obtained as a non-Galois Z ₂-quotient of the equator Podle quantum sphere. On the way, we provide the Cartesian coordinates for all Podle quantum spheres and determine an explicit form of isomorphisms between the C ^*-algebras of the equilateral spheres and the C ^*-algebra of the equator one.     

Abelian real W*-algebras

In this paper, we point out that most results on abelian (complex)W ^*-algebras hold in the real case. Of course, there are differences in homeomorphisms of period 2. Moreover, an abelian real Von Neumann algebra not containing any minimal projection on a separable real Hilbert space is * isomorphic toL _τ ^∞ ([0, 1]) (all real functions inL ^∞([0, 1])), orL ^∞([0, 1]) (as a realW ^*-algebra), orL _τ ^∞ ([0, 1]) ⋇L ^∞ ([0, 1]) (as a realW ^*-algebra), and it is different from the complex case. Partially supported by the NNSF     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.