bad normed spaces convexity properties separated sets abstract some years ago a parameter denoted by a 1 x was defined in real banach spaces in the same setting several years before a notion called q convexity had been defined studying these two notions seems to be rather awkward and up until now this has not been done in deep here we indicate some properties and connections between these two parameters and some other related ones in infinite dimensional banach spaces we also consider another notion a natural extension of q convexity and we discuss the case when a 1 x attains its maximum value the spaces where this happens can be considered as bad since they cannot have several properties which are usually considered as nice like uniform non squareness or p convexity

norming c algebras by c subalgebras we introduce and investigate the notion of a norming c subalgebra of c algebra we characterize when von neumann algebras norm b h and construct various examples of abelian and hyperfinite norming subalgebras we then apply these results to von neumann algebra cohomology theory and the bounded projection problem for von neumann algebras

existence positivity and contractivity for quantum stochastic flows with infinite dimensional noise quantum stochastic differential equations of the form govern stochastic flows on a c algebra we analyse this class of equation in which the matrix of fundamental quantum stochastic integrators is infinite dimensional and the coefficient matrix consists of bounded linear operators on weak and strong forms of solution are distinguished and a range of regularity conditions on the mapping matrix are considered for investigating existence and uniqueness of solutions necessary and sufficient conditions on are determined for any sufficiently regular weak solution k to be completely positive the further conditions on for k to also be a contraction process are found and when is a von neumann algebra and the components of are normal these in turn imply sufficient regularity for the equation to have a strong solution weakly multiplicative and homomorphic solutions and their generators are also investigated we then consider the right and left hudson parthasarathy equations in which f is a matrix of bounded hilbert space operators their solutions are interchanged by a time reversal operation on processes the analysis of quantum stochastic flows is applied to obtain characterisations of the generators f of contraction isometry and coisometry processes in particular weak solutions that are contraction processes are shown to have bounded generators and to be necessarily strong solutions

topology recognition with advice in topology recognition each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph with all ports correctly marked this task is usually unfeasible without any a priori information such information can be provided to nodes as advice an oracle knowing the network can give a possibly different string of bits to each node and all nodes must reconstruct the network using this advice after a given number of rounds of communication during each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations the time of completing topology recognition is the number of rounds it takes and the size of advice is the maximum length of a string given to nodes we investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes we provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time in the class of all graphs of size n and diameter d n for any constant 1 in most cases our bounds are asymptotically tight

a note on models for graph representations this paper is intended more to ask questions than give answers specifically we consider models for labeling schemes and discuss issues regarding the number of labels consulted vs the sizes of the labels recently quite a few papers studied methods for representing network properties by assigning informative labels to the vertices of a network consider a graph function f on pairs of vertices for example f can be the distance function in an f labeling scheme the labels are constructed in such a way so that given the labels of any two vertices u and v one can compute the function f u v e g the graph distance between u and v just by looking at these two labels some very involved lower bounds for the sizes of the labels were proven also some highly sophisticated labeling schemes were developed to ensure short labels in this paper we demonstrate that such lower bounds are very sensitive to the number of vertices consulted that is we show several constructions of such labeling schemes that beat the lower bounds by large margins moreover as opposed to the strong technical skills that were needed to develop the traditional labeling schemes most of our schemes are almost trivial the catch is that in our model one needs to consult the labels of three vertices instead of two that is a query about vertices u and v can access also the label of some third vertex w w is determined by the labels of u and v more generally we address the model in which a query about vertices u and v can access also the labels of c other vertices we term our generalized model labeling schemes with queries the main importance of this model is theoretical specifically this paper may serve as a first step towards investigating different tradeoffs between the amount of labels consulted and the amount of information stored at each vertex as we show if all vertices can be consulted then the problem almost reduces to the corresponding sequential problem on the other hand consulting just the labels of u and v or even just the label of u reduces the problem to a purely distributed one therefore in a sense our model spans a range of intermediate notions between the sequential and the distributed settings in addition to the theoretical interest we also show cases that schemes constructed for our model can be translated to the traditional model or to the sequential model thus simplifying the construction for those models as well for implementing query labeling schemes in a distributed environment directly we point at a potential usage for some new paradigms that became common recently such as p2p and overlay networks

the hochschild cohomology problem for von neumann algebras in 1967 when kadison and ringrose began the development of continuous cohomology theory for operator algebras they conjectured that the cohomology groups hn m m n 1 for a von neumann algebra m should all be zero this conjecture which has important structural implications for von neumann algebras has been solved affirmatively in the type i ii and iii cases leaving open only the type ii1 case in this paper we describe a positive solution when m is type ii1 and has a cartan subalgebra and a separable predual

grothendieck s theorem past and present probably the most famous of grothendieck s contributions to banach space theory is the result that he himself described as the fundamental theorem in the metric theory of tensor products that is now commonly referred to as grothendieck s theorem gt in short or sometimes as grothendieck s inequality this had a major impact first in banach space theory roughly after 1968 then later on in c algebra theory roughly after 1978 more recently in this millennium a new version of gt has been successfully developed in the framework of operator spaces or non commutative banach spaces in addition gt independently surfaced in several quite unrelated fields in connection with bell s inequality in quantum mechanics in graph theory where the grothendieck constant of a graph has been introduced and in computer science where the grothendieck inequality is invoked to replace certain np hard problems by others that can be treated by semidefinite programming and hence solved in polynomial time in this expository paper we present a review of all these topics starting from the original gt we concentrate on the more recent developments and merely outline those of the first banach space period since detailed accounts of that are already available for instance the author s 1986 cbms notes

controller and estimator for dynamic networks afek awerbuch plotkin and saks identified an important fundamental problem inherent to distributed networks which they called the resource controller problem consider first the problem in which one node called the root is required to estimate the number of events that occurred all over the network this counting problem can be viewed as a useful variant of the heavily studied and used task of topology update that deals with collecting all remote information the resource controller problem generalizes the counting problem such remote events are considered as requests and the counting node i e the root also issues permits for the requests that way the number of requests granted can be controlled bounded an efficient resource controller was constructed in the paper by afek et al and it can operate on a dynamic network assuming that the network is spanned by a tree that may only grow and only by allowing leaves to join the tree in contrast the resource controller presented here can operate under a more general dynamic model allowing the spanning tree of the network to undergo both insertions and deletions of both leaves and internal nodes despite the more dynamic network model we allow the message complexity of our controller is never more than the message complexity of the more restricted controller all the applications for the controller of afek et al can be used also with our controller moreover with the same message complexity our controller can handle these applications under the more general dynamic model mentioned above in particular the new controller can be transformed into an efficient size estimation protocol i e a protocol allowing all nodes to maintain a constant factor estimation of the number of nodes in the dynamically changing network informally the resulting new size estimation protocol uses o log 2n amortized message complexity per topological change assuming that the number of changes in the network size is not too small where n is the current number of nodes in the network an application of the size estimation of afek et al was to solve agreement in the case of initial faults fischer lynch and paterson and leader election under initial faults bar yehuda and kutten hence the controllers in this paper can be useful for these applications too

the structure of decoherence functionals for von neumann quantum histories gell mann and hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role isham linden schreckenberg have given a penetrating analysis of decoherence functionals for l h where h is finite dimensional dimension greater than 2 in this note their conjecture on the significance of boundedness is verified in particular it is shown that when d is a bounded decoherence functional associated with a von neumann algebra a then provided a has no direct summand of type i2 d can be represented as the difference between semi innerproducts on a

harmonizability v boundedness 2 p boundedness of stochastic processes some new classes of discrete time non stationary processes related to the harmonizable andv bounded classes are introduced a few characterizations are obtained which in turn unify thev bounded theory our main results depend on a special form of grothendieck s inequality

compact labeling scheme for ancestor queries we consider the following problem given a rooted tree t label the nodes of t in the most compact way such that given the labels of two nodes u and v one can determine in constant time by looking only at the labels whether u is ancestor of v the best known labeling scheme is rather straightforward and uses labels of length at most 2 log 2 n bits each where n is the number of nodes in the tree our main result in this paper is a labeling scheme with maximum label length log 2 n oh sqrt log n our motivation for studying this problem is enhancing the performance of web search engines in the context of this application each indexed document is a tree and the labels of all trees are maintained in main memory therefore even small improvements in the maximum label length are important

universal gaps for xor games from estimates on tensor norm ratios we define and study xor games in the framework of general probabilistic theories which encompasses all physical models whose predictive power obeys minimal requirements the bias of an xor game under local or global strategies is shown to be given by a certain injective or projective tensor norm respectively the intrinsic i e model independent advantage of global over local strategies is thus connected to a universal function r n m called projective injective ratio this is defined as the minimal constant rho such that vert cdot vert x otimes pi y leqslant rho vert cdot vert x otimes varepsilon y holds for all banach spaces of dimensions dim x n and dim y m where x otimes pi y and x otimes varepsilon y are the projective and injective tensor products by requiring that x y one obtains a symmetrised version of the above ratio denoted by r s n we prove that r n m geqslant 19 18 for all n m geqslant 2 implying that injective and projective tensor products are never isometric we then study the asymptotic behaviour of r n m and r s n showing that up to log factors r s n is of the order sqrt n which is sharp r n n is at least of the order n 1 6 and r n m grows at least as min n m 1 8 these results constitute our main contribution to the theory of tensor norms in our proof a crucial role is played by an ell 1 ell 2 ell infty trichotomy theorem based on ideas by pisier rudelson szarek and tomczak jaegermann the main operational consequence we draw is that there is a universal gap between local and global strategies in general xor games and that this grows as a power of the minimal local dimension in the quantum case we are able to determine this gap up to universal constants as a corollary we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements

solution of the similarity problem for cyclic representations of c algebras in a paper from 1955 13 kadison considered the following problem is any bounded non selfadjoint representation 77 of a c algebra a on a hilbert space h similar to a representation i e does there exist an invertible operator t e b h such that tht t is a representation of a this problem is a natural counterpart to a problem considered by dixmier in 1950 9 is any bounded representation of a group g on a hilbert space similar to a unitary representation the group question was solved in the negative by kunze and stein in 1960 14 which in turn was based on previous work of ehrenpreis and mautner 21 the similarity problem for representations of c algebras is still open although a number of partial results have been obtained using the deep results of connes on infective von neumann algebras 7 and the characterization of nuclear c algebras given in 3 one gets the result fairly easily that every bounded representation of a nuclear c algebra is similar to a representation cf 2 th 3 5 and 5 th 4 1 the similarity problem for bounded cyclic representations of arbitrary c algebras was first considered by barnes in 1 he proved that when 77 is a bounded representation of a c algebra a on a hilbert space h such that 7v a and v a have a common cyclic vector i then there exists a closed infective operator t on h such that t and t are densely defined and such that x tht x t is a representation of a bunce sharpened this in 2 by proving that t can be chosen bounded but not necessarily with bounded inverse he also removed the condition that is cyclic for 7v a finally christensen proved in 5 that every irreducible bounded representation of a c algebra is similar to a representation these two results of bunce and christensen rely on an inequality due to pisier 16 cor 2 3 stating that when s is a bounded linear map from a c algebra a into a c algebra b then for all n e n and a an e a

on compact representations of all pairs shortest path distance matrices let g be an unweighted and undirected graph of n nodes and let d be the nxn matrix storing the all pairs shortest path distances in g since d contains integers in n its plain storage takes n 2log n 1 bits however a simple counting argument shows that n 2 2 bits are necessary to store d in this paper we investigate the question of finding a succinct representation of d that requires o n 2 bits of storage and still supports constant time access to each of its entries this is asymptotically optimal in the worst case and far from the information theoretic lower bound by a multiplicative factor log 23 1 585 as a result o 1 bits per pairs of nodes in g are enough to retain constant time access to their shortest path distance we achieve this result by reducing the storage of d to the succinct storage of labeled trees and ternary sequences for which we properly adapt and orchestrate the use of known compressed data structures this approach can be easily and optimally extended to graphs whose edge weights are positive integers bounded by a constant value

on ram priority queues priority queues are some of the most fundamental data structures for example they are used directly for task scheduling in operating systems moreover they are essential to greedy algorithms we study the complexity of integer priority queue operations on a ram with arbitrary word size modeling the possibilities in standard imperative programming languages such as c we present exponential improvements over previous bounds and we show tight relations to sorting our first result is a ram priority queue supporting find min in constant time and insert and delete min in time o log log n where n is the current number of keys in the queue this is an exponential improvement over the o sqrt log n bound of fredman and willard proceedings of the 22nd acm symposium on the theory of computing baltimore md pp 1 7 plugging this priority queue into dijkstra s algorithm gives an o mlog log m algorithm for the single source shortest path problem on a graph with m edges as compared with the previous o m sqrt log m bound based on fredman and willard s priority queue the above bounds assume o n 2 varepsilon w space where w is the word length and varepsilon 0 they can however be achieved in linear space using randomized hashing our second result is a general equivalence between sorting and priority queues a priority queue is monotone if the minimum is nondecreasing over time as in many greedy algorithms we show that on a ram the amortized operation cost of a monotone priority queue is equivalent to the per key cost of sorting for example the equivalence implies that the single source shortest paths problem on a graph with m edges is no harder than that of sorting m keys with the current ram sorting this gives an o m log log m time bound as above but the relation holds regardless of the future developments in ram sorting from the equivalence result for any fixed varepsilon 0 we derive a randomized monotone o sqrt log n 1 varepsilon priority queue with expected constant time decrease key plugging this into dijkstra s algorithm gives an o n sqrt log n 1 varepsilon m algorithm for the single source shortest path problem on a graph with n nodes and m edges complementing the above o mlog log m algorithm if m gg n this improves the o nlog n log log n m bound by fredman and willard proceedings of the 31st ieee symposium on the foundations of computer science st louis mo 1990 pp 719 725 based on their o log n log log n priority queue with constant decrease key

distributed computing of efficient routing schemes in generalized chordal graphs efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks two of them are of special interest low logarithmic diameter and high clustering coefficient high clustering coefficient implies the existence of few large induced cycles considering this fact we propose here a routing scheme that computes short routes in the class of k chordal graphs i e graphs with no induced cycles of length more than k in the class of k chordal graphs our routing scheme achieves an additive stretch of at most k 1 i e for all pairs of nodes the length of the route never exceeds their distance plus k 1 in order to compute the routing tables of any n node graph with diameter d we propose a distributed algorithm which uses o logn bit messages and takes o d time the corresponding routing scheme achieves the stretch of k 1 on k chordal graphs we then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs notice that chordal graphs are 3 chordal graphs in this case distributed computation of routing tables takes o min dd n time where d is the maximum degree of the graph our routing schemes use addresses of size logn bits and local memory of size 2 d 1 logn bits per node of degree d

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

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

the fixed point property via dual space properties abstract a banach space has the weak fixed point property if its dual space has a weak sequentially compact unit ball and the dual space satisfies the weak uniform kadec klee property and it has the fixed point property if there exists e 0 such that for every infinite subset a of the unit sphere of the dual space a a fails to be 2 e separated in particular e convex banach spaces a class of spaces that includes the uniformly nonsquare spaces have the fixed point property

dynamic reachability in planar digraphs with one source and one sink abstract in this paper we investigate the reachability problem on spherical st graphs which are planar acyclic digraphs with exactly one source and exactly one sink these digraphs find specific applications to visibility representations planarity testing planar graph embedding graph drawing and motion planning let n be the number of vertices first we show that reachability queries in a spherical st graph can be answered in o 1 time using an o n space data structure that can be constructed in o n time next we present a dynamic data structure which uses o n space and supports reachability queries and updates in o log n time finally we show how to extend these results to planar digraphs that have one source and one sink and may have directed cycles

perturbations of nuclear c algebras kadison and kastler introduced a natural metric on the collection of all c subalgebras of the bounded operators on a separable hilbert space they conjectured that sufficiently close algebras are unitarily conjugate we establish this conjecture when one algebra is separable and nuclear we also consider one sided versions of these notions and we obtain embeddings from certain near inclusions involving separable nuclear c algebras at the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme

search space size in contraction hierarchies contraction hierarchies are a speed up technique to improve the performance of shortest path computations which works very well in practice despite convincing practical results there is still a lack of theoretical explanation for this behavior in this paper we develop a theoretical framework for studying search space sizes in contraction hierarchies we prove the first bounds on the size of search spaces that depend solely on structural parameters of the input graph that is they are independent of the edge lengths to achieve this we establish a connection with the well studied elimination game our bounds apply to graphs with treewidth k and to any minor closed class of graphs that admits small separators for trees we show that the maximum search space size can be minimized efficiently and the average size can be approximated efficiently within a factor of 2 we show that under a worst case assumption on the edge lengths our bounds are comparable to those in the recent paper vc dimension and shortest path algorithms of abraham et al 1 whose analysis depends also on the edge lengths as a side result we link their notion of highway dimension a parameter that is conjectured to be small but is unknown for all practical instances with the notion of pathwidth this is the first relation of highway dimension with a well known graph parameter

grothendieck type inequalities in combinatorial optimization we survey connections of the grothendieck inequality and its variants to combinatorial optimization and computational complexity 2011 wiley periodicals inc

succinct encoding of arbitrary graphs we consider the problem of encoding graphs with n vertices and m edges compactly supporting adjacency neighborhood and degree queries in constant time in the q logn bit word ram model the adjacency query asks whether there is an edge between two vertices the neighborhood query reports the neighbors of a given vertex in constant time per neighbor and the degree query reports the number of incident edges to a given vertex we study the problem in the context of succinctness where the goal is to achieve the optimal space requirement as a function of n and m to within lower order terms we prove a lower bound in the cell probe model indicating it is impossible to achieve the information theory lower bound up to lower order terms unless the graph is either too sparse namely m o n d for any constant d 0 or too dense namely m w n 2 d for any constant d 0 furthermore we present a succinct encoding of graphs supporting aforementioned queries in constant time the space requirement of the encoding is within a multiplicative 1 e factor of the information theory lower bound for any arbitrarily small constant e 0 this is the best achievable space bound according to our lower bound where it applies the space requirement of the representation achieves the information theory lower bound tightly within lower order terms where the graph is very sparse m o n d for any constant d 0 or very dense m n 2 lg 1 dn for an arbitrarily small constant d 0

dilations and rigid factorisations on noncommutative lp spaces abstract we study some factorisation and dilation properties of completely positive maps on noncommutative l p spaces we show that akcoglu s dilation theorem for positive contractions on classical commutative l p spaces has no reasonable analog in the noncommutative setting our study relies on nonsymmetric analogs of pisier s operator space valued noncommutative l p spaces that we investigate in the first part of the paper

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

on coding labeled trees we consider the problem of coding labeled trees by means of strings of node labels different codes have been introduced in the literature by prufer neville and deo and micikevicius for all of them we show that both coding and decoding can be reduced to integer radix sorting closing several open problems within a unified framework that can be applied both in a sequential and in a parallel setting our sequential coding and decoding schemes require optimal o n time when applied to n node trees yielding the first linear time decoding algorithm for a code presented by neville these schemes can be parallelized on the erew pram model so as to work in o logn time with cost o n or o nlogn depending on the code and on the operation in all cases they either match or improve the performances of the best ad hoc approaches known so far

a stochastic stinespring theorem completely positive markovian cocycles on a von neumann algebra adapted to a fock filtration are realised as conjugations of homomorphic markovian cocycles the conjugating processes are affiliated to the algebra and are governed by quantum stochastic differential equations whose coefficients evolve according to the homomorphic process some perturbation theory for quantum stochastic flows is developed in order to achieve the above stinespring decomposition

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

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

an optimal ancestry labeling scheme with applications to xml trees and universal posets in this article we solve the ancestry labeling scheme problem which aims at assigning the shortest possible labels bit strings to nodes of rooted trees so ancestry queries between any two nodes can be answered by inspecting their assigned labels only this problem was introduced more than 20 years ago by kannan et al 1988 and is among the most well studied problems in the field of informative labeling schemes we construct an ancestry labeling scheme for n node trees with label size log 2n o log log n bits thus matching the log 2n log log n bits lower bound given by alstrup et al 2003 our scheme is based on a simplified ancestry scheme that operates extremely well on a restricted set of trees in particular for the set of n node trees with a depth of at most d the simplified ancestry scheme enjoys label size of log 2n 2log2d o 1 bits since the depth of most xml trees is at most some small constant such an ancestry scheme may be of practical use in addition we also obtain an adjacency labeling scheme that labels n node trees of depth d with labels of size log 2n 3log 2d o 1 bits all our schemes assign the labels in linear time and guarantee that any query can be answered in constant time finally our ancestry scheme finds applications to the construction of small universal partially ordered sets posets specifically for any fixed integer k it enables the construction of a universal poset of size o nk for the family of n element posets with a tree dimension of at most k up to lower order terms this bound is tight thanks to a lower bound of nk o 1 by to alon and scheinerman 1988

succinct representations of permutations and functions we investigate the problem of succinctly representing an arbitrary permutation p on 0 n 1 so that p k i can be computed quickly for any i and any positive or negative integer power k a representation taking 1 e nlgn o 1 bits suffices to compute arbitrary powers in constant time for any positive constant e 1 a representation taking the optimal lgn o n bits can be used to compute arbitrary powers in o lgn lglgn time we then consider the more general problem of succinctly representing an arbitrary function f n n so that f k i can be computed quickly for any i and any integer power k we give a representation that takes 1 e nlgn o 1 bits for any positive constant e 1 and computes arbitrary positive powers in constant time it can also be used to compute f k i for any negative integer k in optimal o 1 f k i time we place emphasis on the redundancy or the space beyond the information theoretic lower bound that the data structure uses in order to support operations efficiently a number of lower bounds have recently been shown on the redundancy of data structures these lower bounds confirm the space time optimality of some of our solutions furthermore the redundancy of one of our structures surpasses a recent lower bound by golynski golynski soda 2009 thus demonstrating the limitations of this lower bound

compact oracles for reachability and approximate distances in planar digraphs it is shown that a planar digraph can be preprocessed in near linear time producing a near linear space oracle that can answer reachability queries in constant time the oracle can be distributed as an o log n space label for each vertex and then we can determine if one vertex can reach another considering their two labels only the approach generalizes to give a near linear space approximate distances oracle for a weighted planar digraph with weights drawn from 0 n it approximates distances within a factor 1 e in o log log nn 1 e time our scheme can be extended to find and route along correspondingly short dipaths

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

space efficient path reporting approximate distance oracles we consider approximate path reporting distance oracles distance labeling and labeled routing with extremely low space requirements for general undirected graphs for distance oracles we show how to break the nlog n space bound of thorup and zwick if approximate paths rather than distances need to be reported for approximate distance labeling and labeled routing we break the previously best known space bound of o log n words per vertex the cost for such space efficiency is an increased stretch

hochschild cohomology for von neumann algebras with cartan subalgebras the main result of this paper is that h n m m 0 n 1 for von neumann algebras m with cartan subalgebras and separable preduals

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

relative dixmier property and complete boundedness of modular maps in this paper we are concerned with the connection between the relative dixmier property and the complete boundedness of bounded modular maps for an inclusion of ii1 factors with the relative dixmier property we will show that is properly infinite where is the gns representation associated with a singular state on we show that a bounded multimodular bilinear map is completely bounded and that a bounded bimodule map is completely bounded finally we prove the row boundedness of a normal bimodule map where is hyperfinite without the relative dixmier property

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

optimal induced universal graphs and adjacency labeling for trees in this article we show that there exists a graph g with o n nodes such that any forest of n nodes is an induced subgraph of g furthermore for constant arboricity k the result implies the existence of a graph with o nk nodes that contains all n node graphs of arboricity k as node induced subgraphs matching a nk lower bound of alstrup and rauhe our upper bounds are obtained through a log2n o 1 labeling scheme for adjacency queries in forests we hereby solve an open problem being raised repeatedly over decades by authors such as kannan et al chung and fraigniaud and korman

weakly compact operators and the strong topology for a banach space the strong topology s x of a banach space x is defined as the locally convex topology generated by the seminorms x 7 ksxk for bounded linear maps s from x into hilbert spaces the w right topology for x x is a stronger locally convex topology which may be analogously characterised by taking reflexive banach spaces in place of hilbert spaces for any banach space y a linear map t x y is known to be weakly compact precisely when t is continuous from the w right topology to the norm topology of y the main results deal with conditions for and consequences of the coincidence of these two topologies on norm bounded sets a large class of banach spaces including all c algebras and more generally all jb triples exhibit this behaviour

amenability and virtual diagonals for von neumann algebras abstract it is shown that a certain universal bimodule over a von neumann algebra is automatically normal this leads to a simple direct proof of haagerup s result that a von neumann algebra is amenable or equivalently injective if and only if it has a normal virtual diagonal haagerup has used this fact to prove that nuclear c algebras are amenable

labeling schemes for flow and connectivity this paper studies labeling schemes for flow and connectivity functions a flow labeling scheme using o log n cdot log hat omega log 2 n bit labels is presented for general n vertex graphs with maximum integral capacity hat omega this is shown to be asymptotically optimal for edge connectivity this yields a tight bound of theta log 2 n bits a k vertex connectivity labeling scheme is then given for general n vertex graphs using at most 3 log n bits for k 2 5 log n bits for k 3 and 2k log n bits for k 3 finally a lower bound of omega k log n is established for k vertex connectivity on n vertex graphs where k is polylogarithmic in n

operator space projective tensor product of c algebras for a finite dimensional c algebra a and any c algebra b we determine a constant of equivalence of operator space projective norm cdot wedge and the banach space projective norm cdot gamma on a otimes b we also discuss the banach algebra a widehat otimes b

the little grothendieck theorem and khintchine inequalities for symmetric spaces of measurable operators abstract we prove the little grothendieck theorem for any 2 convex noncommutative symmetric space let m be a von neumann algebra equipped with a normal faithful semifinite trace and let e be an r i space on 0 let e m be the associated symmetric space of measurable operators then to any bounded linear map t from e m into a hilbert space h corresponds a positive norm one functional f e 2 m such that x e m t x 2 k 2 t 2 f x x x x where e 2 denotes the 2 concavification of e and k is a universal constant as a consequence we obtain the noncommutative khintchine inequalities for e m when e is either 2 concave or 2 convex and q concave for some q we apply these results to the study of schur multipliers from a 2 convex unitary ideal into a 2 concave one

the effros ruan conjecture for bilinear forms on c algebras in 1991 effros and ruan conjectured that a certain grothendieck type inequality for a bilinear form on c algebras holds if and only if the bilinear form is jointly completely bounded in 2002 pisier and shlyakhtenko proved that this inequality holds in the more general setting of operator spaces provided that the operator spaces in question are exact moreover they proved that the conjecture of effros and ruan holds for pairs of c algebras of which at least one is exact in this paper we prove that the effros ruan conjecture holds for general c algebras with constant one more precisely we show that for every jointly completely bounded for short j c b bilinear form on a pair of c algebras a and b there exist states f 1 f 2 on a and g 1 g 2 on b such that for all a a and b b u a b leq u mathrm jcb big f 1 aa 1 2 g 1 b b 1 2 f 2 a a 1 2 g 2 bb 1 2 big while the approach by pisier and shlyakhtenko relies on free probability techniques our proof uses more classical operator algebra theory namely tomita takesaki theory and special properties of the powers factors of type iii 0 1

labeling schemes for weighted dynamic trees a distance labeling scheme is a type of localized network representation in which short labels are assigned to the vertices allowing one to infer the distance between any two vertices directly from their labels without using any additional information sources as most applications for network representations in general and distance labeling schemes in particular concern large and dynamically changing networks it is of interest to focus on distributed dynamic labeling schemes the paper considers dynamic weighted trees where the vertices of the trees are fixed but the positive integral weights of the edges may change the two models considered are the edge dynamic model where from time to time some edge changes its weight by a fixed quanta and the increasing dynamic model in which edge weights can only grow the paper presents distributed approximate distance labeling schemes for the two dynamic models which are efficient in terms of the required label size and communication complexity involved in updating the labels following the weight changes

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

cones and norms in the tensor product of matrix spaces abstract this paper is of expository nature in the tensor product of two matrix spaces m m and m n we consider three natural cones the first is the cone p 0 of positive semi definite matrices the second is the cone p generated by tensor products of positive semi definite matrices in the factor spaces and the third cone p is defined as the dual cone of p correspondingly in the tensor space identified with m mn besides the spectral and the trace norms we consider several tensor norms generated by sepectral and or trace norms in the factor spaces we will study relations among those norms as well as relations among the three kinds of order intervals in the tensor product

physical symmetries of quantum histories gell mann and hartle have proposed a significant generalisation of quantum theory in which decoherence functionals play a key role physical symmetries of quantum history systems are shown under mild physical assumptions to correspond to isometric jordan automorphisms of the underlying von neumann algebra unitary representations of groups of symmetries of quadratic forms over operator algebras are discussed and related to symmetries of decoherence functionals

convexity conditions and normal structure of banach spaces we prove that f convexity the property dual to p convexity of kottman implies uniform normal structure moreover in the presence of the worth property normal structure follows from a weaker convexity condition than f convexity the latter result improves the known fact that every uniformly nonsquare space with the worth property has normal structure

some geometric properties on the fourier and fourier stieltjes algebras of locally compact groups arens regularity and related problems let g be a locally compact topological group and a g b g be respectively the fourier and fourier stieltjes algebras of g it is one of the purposes of this paper to investigate the rnp radon nikodym property and some other geometric properties such as weak rnp the dunford pettis property and the schur property on the algebras a g and b g and to relate these properties to the properties of the multiplication operator on the group c algebra c g we also investigate the problem of arens regularity of the projective tensor products c g a when b g c g has the rnp and a is any c algebra some related problems on the measure algebra the group algebra and the algebras a p g pf p g pm p g 1 p are also discussed

p convexity of orlicz bochner function spaces endowed with the orlicz norm abstract in this paper criteria for p convexity of orlicz bochner function spaces equipped with orlicz norm are given although criteria for p convexity of orlicz function spaces equipped with the orlicz norm were known we can easily deduce them from our main results

hochschild cohomology of factors with property gamma the main result of this paper is that the k th continuous hochschild cohomology groups h k m m and h k m b h of a von neumann factor m b h o f type ii 1 with property are zero for all positive integers k the method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of grothendieck type for multilinear maps we prove joint continuity in the 2 norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology which is already solved

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

fast algorithms for shortest paths in planar graphs with applications graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems improved

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

undirected single source shortest paths with positive integer weights in linear time the single source shortest paths problem sssp is one of the classic problems in algorithmic graph theory given a positively weighted graph g with a source vertex s find the shortest path from s to all other vertices in the graph since 1959 all theoretical developments in sssp for general directed and undirected graphs have been based on dijkstra s algorithm visiting the vertices in order of increasing distance from s thus any implementation of dijkstra s algorithm sorts the vertices according to their distances from s however we do not know how to sort in linear time here a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights the algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure identifying vertex pairs that may be visited in any order

little grothendieck s theorem for real jb triples we prove that given a real jb triple e and a real hilbert space h then the set of those bounded linear operators t from e to h such that there exists a norm one functional varphi in e and corresponding pre hilbertian semi norm varphi on e such that