matrix regular operator spaces the concept of the regular or riesz norm on ordered real banach spaces is generalized to matrix ordered complex operator spaces in a way that respects the matricial structure of the operator space a norm on an ordered real banach spaceeis regular if 1 x y ximplies that y x and 2 y 1 implies the existence ofx esuch that x 1 and x y x a matrix ordered operator space is called matrix regular if at each matrix level the restriction of the norm to the self adjoint elements is a regular norm in such a space elements at each matrix level can be written as linear combinations of four positive elements the concept of the matrix ordered operator space is made specific in such a way as to be a natural generalization of ordered real and complex banach spaces for the case wherevis a matrix ordered operator space a natural cone is defined on the operator spacex hv hx with hindicating the haagerup tensor product so as to make it a matrix ordered operator space exploiting the advantages gained by takingxto be the column hilbert spacehc an equivalence is established between the matrix regularity of a space and that of its operator dual this concept of matrix regularity also provides for more accessible proofs of the christensen sinclair multilinear representation and multilinear decomposition theorems

multipliers and dual operator algebras abstract in a previous paper we showed how the main theorems characterizing operator algebras and operator modules fit neatly into the framework of the noncommutative shilov boundary and more particularly via the left multiplier operator algebra of an operator space as well as giving new characterization theorems the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated recent progress of the author with effros and zarikian now enables weak versions of these characterization theorems for example we prove a result analogous to sakai s famous characterization of von neumann algebras as the c algebras with predual namely that the weakly closed unital not necessarily selfadjoint subalgebras of b h for a hilbert space h are exactly the unital operator algebras which possess an operator space predual this removes one of the hypotheses from an earlier characterization due to le merdy we also show that the multiplier operator algebras of dual operator spaces are dual operator algebras using this we refine several known results characterizing dual operator modules

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

operator spaces and residually finite dimensional c algebras abstract for every operator space x the c algebra containing it in a universal way is residually finite dimensional that is has a separating family of finite dimensional representations in particular the free c algebra on any normed space is residually finite dimensional this is an extension of an earlier result by goodearl and menal and our short proof is based on a criterion due to exel and loring

every completely polynomially bounded operator is similar to a contraction it is proved that a bounded operator on a hilbert space is similar to a contraction if and only if it is completely polynomially bounded this gives a partial answer to problem 6 of halmos bull amer math soc 76 1970 877 933 the set of completely bounded maps between c algebras is studied to obtain some structure representation and extension theorems for this class of maps these allow a characterization of the completely bounded representations on a hilbert space of any subalgebra of a c algebra to be obtained the result in the title follows by applying this characterization to the disk algebra

a morita theorem for dual operator algebras abstract we prove that two dual operator algebras are weak morita equivalent in the sense of d p blecher u kashyap morita equivalence of dual operator algebras j pure appl algebra 212 2008 2401 2412 if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak continuous on appropriate morphism spaces moreover in a fashion similar to the operator algebra case we characterize such functors as the module normal haagerup tensor product with an appropriate weak morita equivalence bimodule we also develop the theory of the w dilation which connects the non selfadjoint dual operator algebra with the w algebraic framework in the case of weak morita equivalence this w dilation is a w module over a von neumann algebra generated by the non selfadjoint dual operator algebra the theory of the w dilation is a key part of the proof of our main theorem

c algebras generated by stable relations abstract presentations of certain c algebras such as c0 0 1 mn by generators and relations are proven to be stable by the weakest definition of stability a set p1 pn of star polynomials is stable if operators a1 an which almost satisfy these relations meaning pi a1 an can be perturbed within c a1 an to satisfy these relations exactly homotopy and perturbation results for these c algebras are corollaries in particular we prove some new results about the dimension drop c algebras a n c 0 1 m n 0 1 c i we show that an is semiprojective answering a question of effros and kaminker also we show that whenever an b is a homomorphism approximately contained in a sub c algebra b0 b there is a nearby homomorphism of an into b0 this answers a question raised by elliott we prove technical results regarding mn c0 0 1 which should be useful elsewhere first we show this is projective meaning a star homomorphism mn c0 0 1 b j always lifts to a star homomorphism mn c0 0 1 b secondly we show that for a finitely generated m n a m n c 0 0 1 c 0 0 1 a if the amalgamation is done properly

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

decision support with preference constraints abstract one approach to human centered processing is to take into account preferences of users within the context of multiple criteria optimization the preference model of a problem encloses all the information needed to evaluate the quality of solutions in this article we propose a new soft constraint called preference constraint based on the decision theory concept of binary preference relations preference based constraint systems are defined and a generic algorithm searching for best quality solutions is then described finally global constraints based on a customizable level consistency g verfaillie d martinez c bessiere a generic customizable framework for inverse local consistency in proc of the 16th national conf on artificial intelligence and 11th conf on innovative applications of artificial intelligence orlando florida are proposed for solving the preference constraint associated to the pareto aggregation rule this new model offers greater flexibility to represent and make complex decisions on computers

relative yoneda cohomology for operator spaces abstract we provide two alternate presentations of the completely bounded hochschild cohomology one as a relative yoneda cohomology i e as equivalence classes of n resolutions which are relatively split and the second as a derived functor the first presentation makes clear the importance of certain relative notions of injectivity projectivity and amenability which we introduce and study we prove that every von neumann algebra is relatively injective as a bimodule over itself and consequently h n cb m m 0 for any von neumann algebra m a result obtained earlier by christensen and sinclair we prove that the relatively amenable c algebras are precisely the nuclear c algebras and hence exactly those which are amenable as banach algebras in a similar vein we prove that the only relatively projective c algebras are finite dimensional this result implies that the only c algebras that are projective as banach algebras are finite dimensional a result first obtained by selivanov and helemskii in a slightly different direction we prove that b h viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module this result is in some sense a generalization of the sz nagy foias commutant lifting theorem or of the hypoprojectivity introduced by douglas and the author

isomorphisms of function modules and generalized approximation in modulus for a function algebra a we investigate relations between the following three topics isomorphisms of singly generated a modules morita equivalence bimodules and real harmonic functions with respect to a we also consider certain groups which are naturally associated with a uniform algebra a we illustrate the notions considered with several examples

compactness properties of operator multipliers abstract we continue the study of multidimensional operator multipliers initiated in k juschenko i g todorov l turowska multidimensional operator multipliers trans amer math soc in press we introduce the notion of the symbol of an operator multiplier we characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers we give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two c algebras is automatically compact we give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the c algebra of compact operators in terms of tensor products generalising results of saar h saar kompakte vollstandig beschrankte abbildungen mit werten in einer nuklearen c algebra diplomarbeit universitat des saarlandes saarbrucken 1982

noncommutative fourier transforms of bounded bilinear forms and completely bounded multilinear operators abstract let g1 gn be locally compact groups and h a hilbert space any bounded n linear operator c g 1 c g n l h is known to have a unique separately weakly continuous extension gf w g 1 w g n l h the fourier transform of is the function gf g 1 g n l h defined by gf s 1 s n gf 1 s 1 n s n where i g i w g i l h i is the universal representation in the case n 2 l h c characterizations based on the grothendieck pisier haagerup inequality are given for such fourier transforms in terms of weakly harmonizable and hemihomogeneous random fields and continuous unitary jordan representations in the case of arbitrary n attention is confined to completely bounded n linear l h valued operators their fourier transforms are characterized and a convolution operation for them is defined in the case of l h c the completely bounded n linear forms on c g 1 c g n form a commutative banach algebra whose maximal ideal space contains the image of b g1 b gn under a separately continuous injection with a continuous left inverse where b gi is the maximal ideal space of the fourier stieltjes algebra of gi

modules over operator algebras and the maximal c dilation abstract we continue our study of the general theory of possibly nonselfadjoint algebras of operators on a hilbert space and modules over such algebras developing a little more technology to connect nonselfadjoint operator algebra with the c algebraic framework more particularly we make use of the universal or maximal c algebra generated by an operator algebra and c dilations this technology is quite general however it was developed to solve some problems arising in the theory of morita equivalence of operator algebras and as a result most of the applications given here and in a companion paper are to that subject other applications given here are to extension problems for module maps and characterizations of c algebras

operator space tensor products of c algebras for c algebras a and b the identity map from a widehat otimes b into a otimes b is shown to be injective next we deduce that the center of the completion of the tensor product a b of two c algebras a and b with centers z a and z b under operator space projective norm is equal to z a widehat otimes z b a characterization of isometric automorphisms of a widehat otimes b and a otimes h b is also obtained

spectral synthesis for banach algebras ii this paper continues the study of spectral synthesis and the topologies tau infty and tau r on the ideal space of a banach algebra concentrating particularly on the class of haagerup tensor products of c algebras for this class it is shown that spectral synthesis is equivalent to the hausdorffness of tau infty under a weak extra condition spectral synthesis is shown to be equivalent to the hausdorffness of tau r

grothendieck s theorem for operator spaces we prove several versions of grothendieck s theorem for completely bounded linear maps t e f when e and f are operator spaces we prove that if e f are c algebras of which at least one is exact then every completely bounded t e f can be factorized through the direct sum of the row and column hilbert operator spaces equivalently t can be decomposed as t t r t c where t r resp t c factors completely boundedly through a row resp column hilbert operator space this settles positively at least partially some earlier conjectures of effros ruan and blecher on the factorization of completely bounded bilinear forms on c algebras moreover our result holds more generally for any pair e f of exact operator spaces this yields a characterization of the completely bounded maps from a c algebra or from an exact operator space to the operator hilbert space oh as a corollary we prove that up to a complete isomorphism the row and column hilbert operator spaces and their direct sums are the only operator spaces e such that both e and its dual e are exact we also characterize the schur multipliers which are completely bounded from the space of compact operators to the trace class

factorization of completely bounded bilinear operators and injectivity a completely bounded bilinear operator m m m on a von neumann algebra m is said to have a factorization in m if there exist completely bounded linear operators j j m m such that x y sum j j x j y x y m where convergence of the sum is made precise below the main result of the paper is that all completely bounded bilinear operators m m m have factorizations in m if and only if m is injective

on the peterson hit problem let pk f2 x1 x2 xk pk f2 x1 x2 xk be the polynomial algebra over the prime field of two elements f2f2 in k variables x1 x2 xkx1 x2 xk each of degree 1 we study the hit problem set up by f peterson of finding a minimal set of generators for pkpk as a module over the mod 2 steenrod algebra aa in this paper we study a minimal set of generators for aa module pkpk in some so called generic degrees and apply these results to explicitly determine the hit problem for k 4k 4

a topology for operator modules overw algebras abstract given a von neumann algebraron a hilbert space h the so calledr topology is introduced into b h which is weaker than the norm and stronger than the ultrastrong operator topology a rightr submodulexof b h is closed in ther topology if and only if for eachb b h the right ideal consisting of alla rsuch thatba x is weak closed inr equivalently xis closed in ther topology if and only if for eachb b h and each orthogonal family of projectionseiinrwith the sum 1 the conditionbei xfor alliimplies thatb x

an optimal k consistency algorithm abstract this paper generalizes the arc consistency algorithm of mohr and henderson 4 and the path consistency algorithm of han and lee 2 to a k consistency algorithm arc consistency and path consistency being 2 consistency and 3 consistency respectively the algorithm is a development of freuder s synthesis algorithm 1 it simultaneously establishes i consistency for each 1 i k it has worst case time and space complexity which is optimal when k is a constant and almost optimal for all other values of k in the case that all order i constraints exist for all 1 i n this algorithm is a solution to the consistent labeling problem with almost optimal worst case time and space complexity

the grothendieck inequality for bilinear forms on c algebras abstract the following generalization of grothendieck s inequality is proved for any bounded bilinear form v on a pair of c algebras a b there exist two states 1 2 on a and two states 1 2 on b such that v x y v 1 x x 2 xx 2 1 2 1 y y 2 yy 2 1 2 for all x a and all y b an inequality of this type was proved a few years ago by pisier in the case where one of the c algebras has the bounded approximation property it follows from the above inequality that any bounded linear map t of a c algebra into the dual of a c algebra has a factorization t r s through a hilbert space such that r s 2 t

on stably decomposing products of classifying spaces john martino1 stewart priddy2 jason douma3 1 department of mathematics and statistics western michigan university kalamazoo mi 49008 usa e mail martino math stat wmich edu 2 department of mathematics northwestern university evanston il 60208 usa e mail priddy math nwu edu 3 department of mathematics university of sioux falls sioux falls sd 57105 usa e mail jason douma thecoo edu

self duality for the haagerup tensor product and hilbert space factorizations d blecher and v paulsen showed that the haagerup tensor product v h w for operator spaces v and w preserves inclusions it is proved to also preserve complete quotient maps and to be self dual in the sense that it induces the haagerup norm on the algebraic tensor product v w the full operator dual space v h w is computed it coincides with the natural operator space gg2 v w of maps v w which have completely bounded factorizations through hilbert spaces with vectors identified with row matrices more generally one has the natural complete isometry gg2 v h w x gg2 v gg2 w x given hilbert spaces h and k with vectors regarded as column matrices it is shown that one may identify the operator spaces b h k and cb h k

the shilov boundary of an operator space and the characterization theorems abstract we study operator spaces operator algebras and operator modules from the point of view of the noncommutative shilov boundary in this attempt to utilize some noncommutative choquet theory we find that hilbert c modules and their properties which we studied earlier in the operator space framework replace certain topological tools we introduce certain multiplier operator algebras and c algebras of an operator space which generalize the algebras of adjointable operators on a c module and the imprimitivity c algebra it also generalizes a classical banach space notion this multiplier algebra plays a key role here as applications of this perspective we unify and strengthen several theorems characterizing operator algebras and modules we also include some general notes on the commutative case of some of the topics we discuss coming in part from joint work with christian le merdy about function modules

completely bounded module maps and the haagerup tensor product abstract we obtain a representation theorem for module maps defined on the algebra k h of compact operators this is used to show that the haagerup tensor product of two operator algebras enjoys strong versions of tomita s commutant theorem and the slice map property we also give a general theorem concerning the automatic complete boundedness of module maps

spectral flow and the unbounded kasparov product abstract we present a fairly general construction of unbounded representatives for the interior kasparov product as a main tool we develop a theory of c 1 connections on operator modules we do not require any smoothness assumptions our unitality assumptions are minimal furthermore we use work of kucerovsky and our recent local global principle for regular operators in hilbert c modules as an application we show that the spectral flow theorem and more generally the index theory of dirac schrodinger operators can be nicely explained in terms of the interior kasparov product

equivalence of norms on operator space tensor products of algebras the haagerup norm 11 ilh on the tensor product a x b of two c algebras a and b is shown to be banach space equivalent to eith er the banach space projective norm 11 11 or the operator space projective norm 11 iia if and only if either a or b is finite dimensional or a and b are infinite dimensional and subhomogeneous the banach space projective norm and the operator space projective nortn are equivalent on a x b if and only if a or b is subhomogeneous

tensor products of operator systems abstract the purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems after giving an axiomatic definition of tensor products in this category we examine in detail several particular examples of tensor products including a minimal maximal maximal commuting maximal injective and some asymmetric tensor products we characterize these tensor products in terms of their universal properties and give descriptions of their positive cones we also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system we examine notions of nuclearity for our tensor products which on the category of c algebras reduce to the classical notion we exhibit an operator system s which is not completely order isomorphic to a c algebra yet has the property that for every c algebra a the minimal and maximal tensor product of s and a are equal

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

the hit problem for the modular invariants of linear groups abstract let the mod 2 steenrod algebra a and the general linear group glk gl k f 2 act on pk f 2 x1 xk with deg xi 1 in the usual manner we prove that for a family of some rather small subgroups g of glk every element of positive degree in the invariant algebra pkg is hit by a in pk in other words pkg a pk where pkg and a denote respectively the submodules of pkg and a consisting of all elements of positive degree this family contains most of the parabolic subgroups of glk it should be noted that the smaller the group g is the harder the problem turns out to be remarkably when g is the smallest group of the family the invariant algebra pkg is a polynomial algebra in k variables whose degrees are 8 and fixed while k increases it has been shown by hu ng trans amer math soc 349 1997 3893 3910 that for g glk the inclusion pkglk a pk is equivalent to a weak algebraic version of the long standing conjecture stating that the only spherical classes in q0s0 are the elements of hopf invariant 1 and those of kervaire invariant 1

representations of completely bounded multilinear operators a deli ion of a completely bounded multilinear operator from one c algebra into another is introduced each completely bounded multilinear operator from a algebra into the algebra of bounded linear operators on a hilbert space is shown to be representable in terms of representations of the c algebra and interlacing operators this result extends wittstock s theorem that decomposes a completely bounded linear operator from a c algebra into an injective c algebra into completely positive linear operators 0 1987 academic press inc stinespring s theorem gives a useful representation for a completely positive linear operator from a c algebra into the algebra bl h for continuous linear operators on a hilbert space h 12 11 using this representation with wittstock s theorem that decomposes a completely bounded linear operator as a finite linear combination of completely positive linear operators 14 lo one obtains a representation of a completely bounded linear operator from a c algebra into bl h our main result is a representation theorem for completely bounded multilinear operators from a c algebra into bl h which generalizes this representation of completely bounded linear operators corollaries give multilinear generalizations of several results known for completely bounded linear operators we shall briefly describe the type of representations to be 151

ideals of a c algebra generated by an operator algebra in this paper we consider ideals of a c algebra c b generated by an operator algebra b a closed ideal j subset of c b is called a k boundary ideal if the restriction of the quotient map on b has a completely bounded inverse with cb norm equal to k 1 for k 1 one gets the notion of boundary ideals introduced by arveson we study properties of the k boundary ideals and characterize them in the case when operator algebra lambda norms itself several reformulations of the kadison similarity problem are given in particular the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from c b onto b which is a projection on b is completely bounded moreover we prove that kadison s similarity problem is decided on one particular c algebra which is a completion of the double of m 2 c

the relative burnside module and the stable maps between classifying spaces of compact lie groups tom dieck s burnside ring of compact lie groups is generalized to the relative case for any g d n a compact lie group and its normal subgroup a g n is defined to be an appropriate set of the equivalence classes of compact g enr s with free n action in such a way that i a g d n 7g n so b n g where b n g is the classifying space of principal n g bundle under the product situation i e g f x k n k a f x k d k is also denoted by a f k as it turns out to be the usual a f k when both f and k are finite then a couple of applications are given to the study of stable maps between classifying spaces of compact lie groups a conceptual proof of feshbach s double coset formula and a density theorem on the map ap a l h a a bl bh p for any compact lie groups l k when p is odd some restriction is applied to l when p 2 this latter result may be regarded as the pushout of feshbach s density theorem and the theorem of may snaith zelewski over the celebrated carlsson solution of segal s burnside ring conjecture 0 introduction even before the arrival of carlsson s celebrated paper c the segal conjecture for finite groups was known to imply the complete calculation of the stable maps between classifying spaces of finite groups agm lmm more recently snaith s2 got interested in the stable maps between classifying spaces of compact lie groups while studying his explicit brauer induction problem see s2 which is a nice reference on this topic snaith s research led to the further generalization of the segal conjecture by msz theorem a msz let f and j be finite groups and let k be a compact lie group then a induces the isomorphism a a f x j k a j fxj bfj bfk f where a f x j k is the grothendieck a f x j module of principal f x j k bundles over finite f x j sets and aj fxj is the completion of received by the editors june 23 1993 1991 mathematics subject classification primary 55p42 55r35 57s15 secondary 55q91 55r9 1

automorphisms of p groups given as cyclic by elementary abelian central extensions abstract we determine the structure of aut g and its relation to symplectic groups when g is a non abelian p group given by a central extension of the form 1 z p g z p z p 1

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

logmodularity and isometries of operator algebras we generalize some facts about function algebras to operator algebras using the noncommutative shilov boundary or c envelope first considered by arveson in the first part we study and characterize complete isometries between operator algebras in the second part we introduce and study a notion of logmodularity for operator algebras we also give a result on conditional expectations many miscellaneous applications are provided

the similarity degree of an operator algebra ii for every integer d ge 1 there is a unital closed subalgebra a d subset b h with similarity degree equal precisely to d in the sense of our previous paper this means that for any unital homomorphism u colon a d to b h we have u cb le k u d with k 0 independent of u and the exponent d in this estimate cannot be improved the proof that the degree is larger than d 1 crucially uses an upper bound for the norms of certain gaussian random matrices due to haagerup and thorbjornsen we also include several complements to our previous publications on the same subject

on the morita equivalence of tensor algebras our objective is two fold first we want to develop a notion of morita equivalence for c correspondences that guarantees that if two c correspondences e and f are morita equivalent then the tensor algebras of e and f taoeu and taofu are strongly morita equivalent in the sense of 8 the toeplitz algebras toeu and tofu are strongly morita equivalent in the sense of rieffel 32 see 33 also and the cuntz pimsner algebras 28 ooeu and oofu are strongly morita equivalent in the same sense we were inspired by earlier work of curto williams and the first author in 15 and combes 14 these papers investigate conditions implying that the crossed product of two c dynamical systems are strongly morita equivalent given that the coefficient algebras are strongly morita equivalent abadie eilers and exel 1 were similarly inspired and developed a theory of morita equivalence for special c correspondences our analysis extends theirs but our arguments are different see example 2 5 and remarks 3 3 and 3 6 for a comparison of their results and ours second we want to investigate the converse implication that is we study the following question if taoeu and taofu are strongly morita equivalent in the sense of 8 when are e and f morita equivalent in the sense we define for this converse direction we focus on the tensor algebras rather than the toeplitz and cuntz pimsner algebras because very little can be said in the self adjoint setting here we were inspired by arveson s paper 2 in which it is proved that two ergodic measure preserving transformations are conjugate if and only if certain non self adjoint crossed products associated with the transformations are algebraically isomorphic see 3 also his analysis was further extended by k saito and the first author to cover non self adjoint crossed products built from automorphisms of general von neumann algebras they showed that non self adjoint crossed products built from properly outer automorphisms of von neumann algebras are isomorphic if and only if the automorphisms are conjugate see 22 and 23 our principal result in the converse direction theorem 7 2 asserts that if e and f are two aperiodic correspondences and if taoeu is strongly morita equivalent to taofu in the sense of 8 then e and f are morita equivalent as we shall see in 5 the notion of aperiodicity is an exact analogue for correspondences of the concept of free action in ergodic theory free action was a central hypothesis in arveson s

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

c algebras generated by operator systems an operator systemx such thatx is ac algebra and such that the canonical embedding ofxinx is a unital complete isometry is called ac system ifais a unitalc algebra andlis a closed left ideal ofa thena l l has a natural operator system structure relative to which it is ac system it is shown that any separablec system is of this form for some separablea and that an arbitrary inseparablec system is an inductive limit of separablec systems associated with any operator systemxis a universalc algebrac u x which containsxas a generating subsystem and is maximal among suchc algebras in the sense that any other suchc algebra is canonically a homomorphic image ofc u x ifxis additionally ac system then thec subalgebrac r x ofx generated byxis minimal amongc algebras containingxas a generating subsystem in the sense thatc r x is canonically a homomorphic image of any other suchc algebra it follows that there is a canonical homomorphism c u x c r x which extends the identity map onx whenxis ac algebra of dimension greater than 1 is never injective one of the main results of the paper is that there exists a separable nuclearc systemxcontainingm2 c for which is an isomorphism this implies that for any embedding ofxas an operator subsystem of ac algebraa thec subalgebra ofagenerated byxis isomorphic toc u x moreoverc u m2 c is notc exact andc u m2 c c u x from which it follows thatc u x and hencea are not exactc algebras thusx though a nuclear operator system cannot be embedded in a nuclearc algebra

the negative answer to kameko s conjecture on the hit problem abstract the goal of this paper is to give a negative answer to kameko s conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra p k considered as a module over the steenrod algebra a is dominated by an explicit quantity depending on the number of the polynomial algebra s variables k the conjecture was shown by kameko himself for k 3 in his phd thesis in the johns hopkins university in 1990 and recently proved by us and kameko for k 4 however we claim that it turns out to be wrong for any k 4 in order to deny kameko s conjecture we study a minimal set of generators for a module p k in some so call generic degrees what we mean by generic degrees is a bit different from that of other authors in the fields such as crabb hubbuck nam repka selick wood we prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables k increases as an immediate consequence of this inductive formula we recognize that kameko s conjecture is no longer true for any k 4

cohomology for operator algebras the mayer vietoris sequence abstract for an inclusion b a of operator algebras we introduce the b nil cohomology groupshn a b which can be easier to compute than the standard cohomology groupshn a we then develop two long exact sequences the first linkinghn a b hn a andhn b the second corresponding to the mayer vietoris sequence of algebraic topology these sequences provide powerful computational tools as illustrated by the examples in the last two sections of the paper

multilinear maps and tensor norms on operator systems abstract we extend work of christensen and sinclair on completely bounded multilinear forms to the case of subspaces of c algebras and obtain a representation theorem and a hahn banach extension theorem for such maps in the second part of the paper the haagerup norms on tensor products are investigated and we obtain new characterizations of these quantities

ein operatorwertiger hahn banach satz abstract we generalize arveson s extension theorem for completely positive mappings 1 to a hahn banach principle for matricial sublinear functionals with values in an injective c algebra or an ideal in b h we characterize injective w algebras by a matricial order condition we illustrate the matricial hahn banach principle by three applications 1 let a b b be unital c algebras b a subalgebra of a and b b injective if a b is a completely bounded self adjoint b bihomomorphism then it can be expressed as the difference of two completely positive b bihomomorphism 2 let m be a w algebra containing 1 h on a hilbert space h if m is finite and hyperfinite there exists an invariant expectation mapping p of b h onto m p is an extension of the center trace 3 combes 7 proved that a lower semicontinuous scalar weight on a c algebra is the upper envelope of bounded positive functionals we generalize this result to unbounded completely positive mappings with values in an injective w algebra

making choices with a binary relation relative choice axioms and transitive closures abstract this article presents an axiomatic analysis of the best choice decision problem from a reflexive crisp binary relation on a finite set a digraph with respect to a transitive digraph optimality and maximality are usually accepted as the best fitted choice axioms to the intuitive notion of best choice however beyond transitivity resp acyclicity optimality and maximality can characterise distinct choice sets resp empty sets accordingly different and rather unsatisfying concepts have appeared such as von neumann morgenstern domination weak transitive closure and kernels here we investigate a new family of eight choice axioms for digraphs relative choice axioms within choice theory these axioms generalise top cycle for tournaments gocha getcha and rational top cycle for complete digraphs we present their main properties such as existence uniqueness idempotence internal structure and cross comparison we then show their strong relationship with optimality and maximality when the latter are not empty otherwise these axioms identify a non empty choice set and underline conflicts between chosen elements in strict preference circuits finally we exploit the close link between this family and transitive closures to compute choice sets in linear time followed by a relevant practical application

the weak conjecture on spherical classes let mathcal a be the mod 2 steenrod algebra we construct a chain level representation of the dual of singer s algebraic transfer which maps singer s invariant theoretic model of the dual of the lambda algebra gamma k wedge to bf f 2 x 1 pm 1 ldots x k pm 1 and is the inclusion of the dickson algebra d k subset gamma k wedge into bf f 2 x 1 ldots x k this chain level representation allows us to confirm the weak conjecture on spherical classes see 9 assuming the truth of 1 either the conjecture that the dickson invariants of at least k 3 variables are homologically zero in tor k mathcal a bf f 2 bf f 2 2 or a conjecture on mathcal a decomposability of the dickson algebra in gamma k wedge we prove the conjecture in item 1 for k 3 and also show a weak form of the conjecture in item 2

the cohomology of the steenrod algebra and representations of the general linear groups let tr k be the algebraic transfer that maps from the coinvariants of certain gl k representations to the cohomology of the steenrod algebra this transfer was defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial more precisely that tr k is an isomorphism for k 1 2 3 and that tr k tr k is a homomorphism of algebras in this paper we first recognize the phenomenon that if we start from any degree d and apply sq repeatedly at most k 2 times then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the gl k representations as a consequence every finite sq 0 family in the coinvariants has at most k 2 nonzero elements two applications are exploited the first main theorem is that tr k is not a n isomorphism for k 5 furthermore for every k 5 there are infinitely many degrees in which tr k is not an isomorphism we also show that if tr detects a nonzero element in certain degrees of ker sq 0 then it is not a monomorphism and further for each k tr k is not a monomorphism in infinitely many degrees the second main theorem is that the elements of any sq 0 family in the cohomology of the steenrod algebra except at most its first k 2 elements are either all detected or all not detected by tr k for every k applications of this study to the cases k 4 and 5 show that tr 4 does not detect the three families g d 3 and p and that tr 5 does not detect the family h n 1g n n 1

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces