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

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

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

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

universal lattices and property tau we prove that the universal lattices the groups g sl d r where r x 1 x k have property for d 3 this provides the first example of linear groups with which do not come from arithmetic groups we also give a lower bound for the constant with respect to the natural generating set of g our methods are based on bounded elementary generation of the finite congruence images of g a generalization of a result by dennis and stein on k 2 of some finite commutative rings and a relative property t of mathrm sl 2 r ltimes r 2 r 2

quotients exactness and nuclearity in the operator system category abstract we continue our study of tensor products in the operator system category we define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products one of our main goals is to relate these refinements of nuclearity to the kirchberg conjecture in particular we prove that the kirchberg conjecture is equivalent to the statement that every operator system that is min er nuclear is also el c nuclear we show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient we prove that an operator system is exact for the min tensor product if and only if it is min el nuclear we give many characterizations of operator systems that are min er nuclear el c nuclear min el nuclear and el max nuclear these characterizations involve operator system analogues of various properties from the theory of c algebras and operator spaces including the wep and llp

a characterisation of large finitely presented groups in this paper we will consider finitely presented groups that have a finite index subgroup which admits a surjective homomorphism onto a non abelian free group gromov called these groups large 4 large groups have particularly nice properties for example super exponential subgroup growth they also play an important role in lowdimensional topology it is a major conjecture that the fundamental group of any closed hyperbolic 3 manifold is large our main theorem is a characterisation of these groups in terms of the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

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

universal lattices and unbounded rank expanders we study the representations of non commutative universal lattices and use them to compute lower bounds of the constant for the commutative universal lattices g d k sl d x 1 x k for d 3 with respect to several generating sets as an application we show that the cayley graphs of the finite groups text sl 3k mathbb f p can be made expanders with a suitable choice of generators this provides the first example of expander families of groups of lie type where the rank is not bounded and provides counter examples to two conjectures of a lubotzky and b weiss

prime and almost prime integral points on principal homogeneous spaces we develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space in particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors in many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers when the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number and thus establish the zariski density of matrices all of whose entries are prime numbers among the key tools used are explicit approximations to the generalized ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms

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 amenability of affine algebras abstract we introduce the notion of amenability for affine algebras we characterize amenability by folner sequences paradoxicality and the existence of finitely invariant dimension measures then we extend the results of rowen on ranks from affine algebras of subexponential growth to amenable affine algebras

the product replacement algorithm and kazhdan s property t a problem of great importance in computational group theory is to generate nearly uniformly distributed random elements in a finite groupg a good example of such an algorithm should start at any given set of generators use no prior knowledge of the structure of g and in a polynomial number of group operations return the desired random group elements see bb2 then these random elements can be used further to determine the structure of g in a pioneer paper bb1 babai found such an algorithm which provably generates nearly uniformly distributed random elements in o log g group multiplications too many for practical applications a different heuristic the product replacement algorithm was designed by leedham green and soicher lg and studied by celler et al in clmno in spite of the fact that very little theoretical justification was known practical experiments showed excellent performance so it quickly became the most popular practical algorithm to generate random group elements and was included in the two most frequently used group algebra packages gap sc and magma bcp a systematic and quantitative approach was carried out by diaconis and saloffcoste ds1 ds2 see also bb2 cg but their results did not reveal the mystery of the truly outstanding performance of the algorithm the aim of this paper is to propose a conceptual explanation based on kazhdan s property t from representation theory of lie groups and to improve some of the previous estimates on the running time the product replacement algorithm works as follows clmno given a finite group g let k g be the set of k tuples g g1 gk of elements of g such that g1 gk g we call elements of k g the generating k tuples given a generating k tuple g1 gk define a move on it in the following way choose uniformly a pair i j such that 1 i 6 j k then apply one of the following four operations with equal probability

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

walks on generating sets of groups we study a markov chain on generating n tuples of a fixed group which arises in algorithms for manipulating finite groups the main tools are comparison of two markov chains on different but related state spaces and combinatorics of random paths the results involve group theoretical parameters such as the size of minimal generating sets the number of distinct generating k tuples for different k s and the maximal diameter of the group

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

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

new comparison results for impulsive integro differential equations and applications we prove some new maximum principles for ordinary integro differential equations this allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first order impulsive integro differential equation

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

generic case complexity decision problems in group theory and random walks abstract we give a precise definition of generic case complexity and show that for a very large class of finitely generated groups the classical decision problems of group theory the word conjugacy and membership problems all have linear time generic case complexity we prove such theorems by using the theory of random walks on regular graphs

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