Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Morita Equivalence for Locally C*-Algebras

The concept of Morita equivalence is generalized to the contextof locally C^*-algebras. This generalizes a well-known theoremof Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.2000 Mathematics Subject Classification 46L08, 46L05.     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

Cohomology of Hopf C*-Algebras and Hopf Von Neumann Algebras

We define two canonical cohomology theories for Hopf C^*-algebrasand for Hopf von Neumann algebras (with coefficients in theircomodules). We then study the situations when these cohomologiesvanish. The cases of locally compact groups and compact quantumgroups are considered in more detail. E-mail: c.k.ng{at}qub.ac.uk2000 Mathematical Subject Classification: primary 46L05, 46L55;secondary 43A07, 22D25.     

Hopf C*-Algebras

In this paper we define and study Hopf C^*-algebras. Roughlyspeaking, a Hopf C^*-algebra is a C^*-algebra A with a comultiplication: A M(A A) such that the maps a b (a)(1 b) and a (a 1)(b)have their range in A A and are injective after being extendedto a larger natural domain, the Haagerup tensor product A _hA. In a purely algebraic setting, these conditions on are closelyrelated to the existence of a counit and antipode. In this topologicalcontext, things turn out to be much more subtle, but neverthelessone can show the existence of a suitable counit and antipodeunder these conditions. The basic example is the C^*-algebra C₀(G) of continuous complexfunctions tending to zero at infinity on a locally compact groupwhere the comultiplication is obtained by dualizing the groupmultiplication. But also the reduced group C^*-algebra of a locally compact group with thewell-known comultiplication falls in this category. In factall locally compact quantum groups in the sense of Kustermansand the first author (such as the compact and discrete ones)as well as most of the known examples are included. This theory differs from other similar approaches in that thereis no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.     

On the Morita Equivalence of Tensor Algebras

We develop a notion of Morita equivalence for general C^*-correspondencesover C^*-algebras. We show that if two correspondences are Moritaequivalent, then the tensor algebras built from them are stronglyMorita equivalent in the sense developed by Blecher, Muhly andPaulsen. Also, the Toeplitz algebras are strongly Morita equivalentin the sense of Rieffel, as are the Cuntz–Pimsner algebras.Conversely, if the tensor algebras are strongly Morita equivalent,and if the correspondences are aperiodic in a fashion that generalizesthe notion of aperiodicity for automorphisms of C^*-algebras,then the correspondences are Morita equivalent. This generalizesa venerated theorem of Arveson on algebraic conjugacy invariantsfor ergodic, measure-preserving transformations. The notionof aperiodicity, which also generalizes the concept of fullConnes spectrum for automorphisms, is explored; its role inthe ideal theory of tensor algebras and in the theory of theirautomorphisms is investigated. 1991 Mathematics Subject Classification:46H10, 46H20, 46H99, 46M99, 47D15, 47D25.     

From Trace States to States on the K0 Group of a Simple C*-Algebra

It is shown that any continuous affine surjection from a metrizableChoquet simplex onto a compact convex set occurs as the restrictionmap from the tracial state space onto the state space of theK₀ group of a separable unital simple C^*-algebra which is theinductive limit of a sequence of subhomogeneous C^*-algebras     

Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topologyresults of Chas and Sullivan, and of Cohen and Jones. For afinite simplicial complex X and k 1, I construct a spectrumMaps(S^k, X)^S(X), which is obtained by taking a generalizationof the Spivak bundle on X (which however is not a stable spherebundle unless X is a Poincaré space), pulling back toMaps(S^k, X) and quotienting out the section at infinity. I showthat the corresponding chain complex is naturally homotopy equivalentto an algebra over the (k + 1)-dimensional unframed little diskoperad C_{k + 1}. I also prove a conjecture of Kontsevich, whichstates that the Quillen cohomology of a based C_k-algebra (inthe category of chain complexes) is equivalent to a shift ofits Hochschild cohomology, as well as prove that the operadC_*C_k is Koszul-dual to itself up to a shift in the derived category.This gives one a natural notion of (derived) Koszul dual C_*C_k-algebras.I show that the cochain complex of X and the chain complex of^k X are Koszul dual to each other as C_*C_k-algebras, and thatthe chain complex of Maps(S^k, X)^S(X) is naturally equivalentto their (equivalent) Hochschild cohomology in the categoryof C_* C_k-algebras. 2000 Mathematics Subject Classification 55P48(primary), 16E40, 55N45, 18D50 (secondary).     

Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation is considered on a unital C^*-algebra, with separablenoise dimension space. Necessary conditions on the matrix ofbounded linear maps for the existence of a completely positivecontractive solution are shown to be sufficient. It is knownthat for completely positive contraction processes, k satisfiessuch an equation if and only if k is a regular Markovian cocycle.‘Feller’ refers to an invariance condition analogousto probabilistic terminology if the algebra is thought of asa non-commutative topological space. 2000 Mathematics SubjectClassification 81S25, 46L07, 46L53, 47D06.     

Approximately Local Derivations

Certain linear operators from a Banach algebra A into a BanachA-bimodule X, which are called approximately local derivations,are studied. It is shown that when A is a C^*-algebra, a Banachalgebra generated by idempotents, a semisimple annihilator Banachalgebra, or the group algebra of a SIN or a totally disconnectedgroup, bounded approximately local derivations from A into Xare derivations. This, in particular, extends a result of B.E. Johnson that ‘local derivations on C^*-algebras arederivations’ and provides an alternative proof of it.     

On {sigma}-Normal C*-Algebras

A C*-algebra A is said to be monotone (respectively monotone-) complete if every increasing net (respectively increasingsequence) of elements in the ordered space A_h of all hermitianelements of A has a supremum in A_h. It is straightforward toverify that every monotone complete C*-algebra is an AW*-algebra.For type I AW*-algebras, the converse is known to be true. However,for general AW*-algebras, this question is still open, althoughan impressive attack on the problem was made by E. Christensenand G. K. Pedersen, who showed that properly infinite AW*-algebrasare monotone -complete [4]. 1991 Mathematics Subject Classification46L05, 46L06.     

Strict Completely Positive Maps between Locally C*-Algebras and Representations on Hilbert Modules

The structure of the continuous strict completely positive linearmaps between locally C^*-algebras is described.     

Generating Examples of Shock Propagation by Three Principles

In a medium characterized by a scalar speed C(x), a shock arrivesat the point x, after time T(x), with its magnitude decreasedby A(x). Symmetric C, T, and A in two dimensions can be convertedto cylindrically symmetric results in three dimensions by applyinga dimension-increasing principle: "Let C(x, y), T(x, y), andA(x, y) be even functions of y. They can be extended into threedimensions by using the formulas C(x, y)C(x, r), T(x, y)T(x,r), and A(x,y)A(x,r) [r^–1 cos(x, r)]^?, where r = (x²+2²)^?and is an auxiliary function." When C(x) is a function of asingle variable, the auxiliary function is given by cos(x,y) = T_y(x, y). In two dimensions, there is a conformal mappingprinciple: "Under the conformal mapping x+iy = f(x^*+iy^*), thefunctions T(x, y) and A(x, y) go into functions associated witha medium having speed C^*,y^*) = C(Re[f), Im[f]/f¹(x^*+iy."Thereis also an unchanged wavefronts principle: "If g is a smoothfunction with g(0) = 0 and g'(0)>0 then T^*(x) = g(T(x) andA^*(x) = A(x)[g'(x)/g'^1/2 are associated with a medium havingspeed C^*(x) = C(x)/g'(T(x))." in two dimensions, alternatingthe application of the last two principles generates a sequenceof media with their associated T(x, y) and A(x, y). Some ofthese can be extended into three dimensions by applying thefirst principle.     

The Alternative Dunford-Pettis Property, Conjugations and Real Forms of C*-Algebras

Let be a conjugation, alias a conjugate linear isometry oforder 2, on a complex Banach space X and let X be the real formof X of -fixed points. In contrast to the Dunford–Pettisproperty, the alternative Dunford–Pettis property neednot lift from X to X. If X is a C^*-algebra it is shown thatX has the alternative Dunford–Pettis property if and onlyif X does and an analogous result is shown when X is the dualspace of a C^*-algebra. One consequence is that both Dunford–Pettisproperties coincide on all real forms of C^*-algebras.     

Optimal policy for a general repair replacement model: average reward case

For a general repair replacement model, we study two types ofreplacement policy.Replacement policy T replaces the systemat time T since the installation or last replacement, whilereplacement policy N replaces the system at the time of Nthfailure. Let T^* and N^* be the optimal among all policies T andN respectively. Under the expected average reward criterion,then we show that the optimal policy N^* is at least as goodas the optimal policy T^*. Furthermore, for a monotone processmodel, we determine the optimal policy N^* explicitly throughtwo different approaches.     

Right Cancellation in the LUC-Compactification of a Locally Compact Group

Write G^* = G^LUC\G where G^LUC is the largest semigroup compactificationof the locally compact group G. Then the set of points of G^*which are right cancellable in G^* = G^LUC is large; in fact ithas an interior in G^* which is dense in G^*. Corollaries aregiven about the number of left ideals in G^* = G^LUC and the sizeof right ideals in the algebra LUC(G)^*.     

Dickson-Mui Invariants

Let V be a finite-dimensional vector space over a finite field.The general and special linear groups, GL(V) and SL(V), acton the exterior algebras ^*V and ^*V^* of V and its dual V^*, andon the symmetric algebra S^*V. The subring of SL(V)-invariantsof ^*VS^*V was determined by Dickson and Mui. This paper describesthe equivalent, but simpler, calculation of the invariant subringof ^*VS^*V as a representation of GL(V)/SL(V). 2000 MathematicsSubject Classification 13A50.     

Real Rank of Crossed products by Connected Compact Groups

We study general properties of the real rank under the operationsof taking extensions and crossed products. For example, consideringthe C^*-exact sequence 0JAA/J0, we prove RR(A) max{RR(M(J)),RR(A/J)}, where M(J) is the multiplier algebra of J. If A isa C^*-algebra with a continuous trace and J = C_o(X) k, we obtainRR(A) = max{RR(J),RR(A/J)}. Finally, we present the correctestimate for the real rank of C^*-crossed products by connectedcompact groups.     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

Full Crossed Products by Coactions, C0(X)-Algebras and C*-Bundles

Let be a coaction of a locally compact group G on a C*-algebraA. We show that if I is a -invariant ideal in A, then for full crossed products, as Landstadet al. have done for spatial crossed products by coactions.We prove that for suitable coactions, the crossed products ofC₀(X)-algebras are again C₀(X)-algebras, and the crossed productsof continuous C*-bundles by a locally compact group are againcontinuous C*-bundles. 1991 Mathematics Subject Classification46L55.