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.     

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.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.     

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.     

On generalised Putnam—Fuglede theorems

LetB(H) denote the algebra of operators on the Hilbert spaceH, and letP denote the class ofAB(H) which are such that the restriction ofA to an invariant subspace is inP wheneverAP and which satisfy the property, henceforth called property (P 2), that if the restriction ofA to an invariant subspace is normal, then the subspace reducesA. GivenP-classesP ₁ andP ₂, the pair (P ₁,P ₂) is said to satisfy the (PF)-property if givenAP ₁ andB ^* P ₂ such thatAB=XB for someXB(H), we haveA ^* X=XB ^* . Generalising the (classical) Putnam—Fuglede theorem, it is shown here that the pair (P ₁,P ₂) has the (PF)-property if and only if, givenAP ₁ andB ^*P ₂ such thatAX=XB for some quasi-affinityXB(H), the following conditions hold: (i)B ^* is normal impliesA is normal; (ii)A has a normal direct summand impliesB ^* has a normal direct summand; (iii)A andB ^* pure impliesX is non-existent. An interestingP-class is the classC ₀ of contractions withC ₀ completely non-unitary parts which satisfy property (P 2). AssumingH to be separable, it is shown that ifC ₁ denotes thoseA C ₀ for which the defect operatorsD _A =(1–A^*A)^1/2 is of Hilbert—Schmidt class and for which either the pure part ofA has empty point spectrum or the eigen-values ofA are all simple, then the pair (C ₀,C ₁) has the (PF)-property. The classC ₁ defines aP-class; a crucial role in the proof of this statement is played by the interesting result that aC ₀ contraction with spectrum on the unit circle can not satisfy property (P 2). Applications of these results are considered, amongst them that ifA andB are quasi-similar hyponormal contractions such that the pure part ofA has finite multiplicity andD _B is of Hilbert —Schmidt class, then their normal parts are unitarily equivalent and their pure parts are quasi-similar.     

CertainC*-algebras with real rank zero and their corona and multiplier algebras: II

We first determine the homotopy classes of nontrivial projections in a purely infinite simpleC*-algebraA, in the associated multiplier algebraM(A) and the corona algebraM A/A in terms ofK _*(A). Then we describe the generalized Fredholm indices as the group of homotopy classes of non-trivial projections ofA; consequently, we determine theK _*-groups of all hereditaryC*-subalgebras of certain corona algebras. Secondly, we consider a group structure of *-isomorphism classes of hereditaryC*-subalgebras of purely infinite simpleC*-algebras. In addition, we prove that ifA is aC*-algebra of real rank zero, then each unitary ofA, in caseA it unital, each unitary ofM(A) and ofM(A)/A, in caseA is nonunital but -unital, can be factored into a product of a unitary homotopic to the identity and a unitary matrix whose entries are all partial isometries (with respect to a decomposition of the identity).Partially supported by a grant from the National Science Foundation.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

On radicals of triangular operator algebras

SupposeB is a type IC ^*-algebra admitting a diagonalD in the sense of Kumjian, and letE be the conditional expectation fromB ontoD. A subalgebraA ofB is called triangular with diagnoalD ifA∩A*=D. Theorem: Under the above assumptions the Jacobson radical ofA equals the intersection ofA with the kernel of the conditional expectationE. Although the statement of the theorem is coordinate free, the proof requires the use of coordinates in essential ways. A theorem by Kumjian allows us to represent everyC ^*-algebra admitting a diagonal as theC ^*-algebra of a certain groupoid. This enables us to apply the techniques of topological groupoids as developed by Renault and Muhly. A very convenient way of expressing a triangular subalgebra of theC ^*-algebra of a T-groupoid is given by the Spectral Theorem for Bimodules, due to Qui, which is a descendent of the Spectral Theorem for Bimodules due to Muhly and Solel, and to Muhly, Saito and Solel in the context of von Neumann algebras.     

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

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.     

Closed Projections and Peak Interpolation for Operator Algebras

The closed one-sided ideals of a C ^*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ^*-algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B ^** which also lies in . Although this seems quite natural, the proof requires a set of new techniques which may be viewed as a noncommutative version of the subject of peak interpolation from the theory of function spaces. Thus, the right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction, we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces.     

On non-semisplit extensions,tensor products and exactness of groupC *-algebras

Summary We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC ^*-algebraA = C _r ^* (SL ₂()), such that there is only oneC ^*-norm on the algebraic tensor productB ^op B, butB is not nuclear (even not exact). Thus the class of exactC ^*-algebras is not closed under extensions.The existence comes from a new established tensorial duality between the weak expectation property (WEP) of Lance and the local variant (LLP) of the lifting property.We characterize the local lifting property of separable unitalC ^*-algebrasA as follows:A has the local lifting property if and only if Ext (S(A)) is a group, whereS(A) is the suspension ofA.If moreoverA is the quotient algebra of aC ^*-algebra withWEP (for brevity:A isQWEP) but does not satisfyLLP then there exists a quasidiagonal extensionB of the suspensionS(A) by the compact operators such that on the algebraic tensor productB ^op B there is only oneC ^*-norm.The question if everyC ^*-algebra isQWEP remains open, but we obtain the following results onQWEP: AC ^*-algebraC isQWEP if and only ifC ^** isQWEP. A von NeumannII ₁-factorN with separable predualN _* isQWEP if and only ifN is a von Neumann subfactor of the ultrapower of the hyperfiniteII ₁-factor. IfG is a maximally almost periodic discrete non-amenable group with Haagerup's Herz-Schur multiplier constant _G =1 then the universal groupC ^*-algebraC ^*(G) is not exact but the reduced groupC ^*-albegraC _r ^* (G) is exact and isQWEP but does not satisfyWEP andLLP.We study functiorial properties of the classes ofC ^*-algebras satisfyingWEP, LLP resp. beingQWEP.As applications we obtain some unexpected relations between some open questions onC ^*-algebras.Oblatum 13-IV-92Work partially supported by DFG     

On Additive Decomposition of the Hermitian Solution of the Matrix Equation AXA* = B

The decomposition of a Hermitian solution of the linear matrix equation AXA* = B into the sum of Hermitian solutions of other two linear matrix equations A₁X₁A^*₁ = B₁{A_{1}X_{1}A^{*}_{1} = B_{1}} and A₂X₂A^*₂ = B₂{A_{2}X_{2}A^*_{2} = B_{2}} are approached. As applications, the additive decomposition of Hermitian generalized inverse C ⁻ = A ⁻ + B ⁻ for three Hermitian matrices A, B and C is also considered.     

A generalization to the non-separable case of Takesaki's duality theorem forC *-algebras

Takesaki [5] poses the question of how much information about aC ^*-algebraA is contained in its representation theory. He gives it a precise meaning in the following setting: One can furnish the set Rep (A:H) of all representations ofA in a suitable Hilbert spaceH with a topology, with an action of the unitary groupG ofB(H) on it, and with an addition. The setA ^F of operator fields Rep (A:H)B(H) commuting with the action ofG and addition, called the admissible operator fields, turn out to form aW ^*-algebra isomorphic to the bidual ofA with Arens multiplication or with the universal enveloping von Neumann algebra ofA. Takesaki shows in the separable case thatA can be identified inA ^F as the set of continuous admissible operator fields, and leaves the same question open for arbitraryC ^*-algebras. Changing the structures on Rep(A:H) slightly, it is shown here that this result obtains in the general case as well. The proof proceeds along the lines set up in [5] but makes no use of the representation theory of NGCR algebras.     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

A measuring argument for finite permutation groups

LetG be a finite transitive permutation group on a finite setS. LetA be a nonempty subset ofS and denote the pointwise stabilizer ofA inG byC _G (A). Our main result is the following inequality: [G :C _G (A)]≥|G|^|A|/|S|. This paper is a part of the author’s Ph.D. thesis research, carried out at Tel Aviv University under the supervision of Professor Marcel Herzog.     

Isolated semidefinite solutions of the continuous-time algebraic riccati equation

The set of all negative-semidefinite solutions of the CAREA ^* X+XA+XBB ^* X–C ^* C=0 is homeomorphic to a well defined set ofA-invariant subspaces provided that the purely imaginary eigenvalues ofA are controllable. Based on that homeomorphism isolated n.s.d. solutions of the CARE are characterized by properties of their kernels.Research supported by Deutsche Forschungsgemeinschaft (Wi 1219/1-1).     

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.