pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

prefix free regular languages and pattern matching we explore the regular expression matching problem with respect to prefix freeness of the pattern we prove that a prefix free regular expression gives only a linear number of matching substrings in the size of a given text based on this observation we propose an efficient algorithm for the prefix free regular expression matching problem furthermore we suggest an algorithm to determine whether or not a given regular language is prefix free

non simple purely infinite c algebras the hausdorff case abstract local and global definitions of pure infiniteness for a c algebra a are compared and equivalence between them is obtained if the primitive ideal space of a is hausdorff and of finite dimension if a has real rank zero or if a is approximately divisible sufficient criteria are given for local pure infiniteness of tensor products they yield that exact simple tensorially non prime c algebras are purely infinite if they have no semi finite lower semi continuous trace one obtains that a is isomorphic to a o if a is 1 purely infinite separable stable nuclear and prim a is a hausdorff space not necessarily of finite dimension

ultraproducts of von neumann algebras abstract we study several notions of ultraproducts of von neumann algebras from a unified viewpoint in particular we show that for a sigma finite von neumann algebra m the ultraproduct m introduced by ocneanu is a corner of the ultraproduct m introduced by groh and raynaud using this connection we show that the ultraproduct action of the modular automorphism group of a normal faithful state of m on the ocneanu ultraproduct is the modular automorphism group of the ultrapower state t t applying these results we obtain several properties of the ocneanu ultraproduct of type iii factors which are not present in the tracial ultraproducts for instance it turns out that the ultrapower m of a type iii0 factor is never a factor moreover we settle in the affirmative a recent problem by ueda about the connection between the relative commutant of m in m and connes asymptotic centralizer algebra m

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

finite difference solution of a two dimensional mathematical model of the cochlea a current linear two dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the model equations the finite difference method is used to discretize laplace s equation over a rectangular region with specified boundary conditions the resulting matrix equation for fluid pressure is solved by using a gaussian block elimination technique numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes the finite difference method is a direct versatile and reasonably efficient means of solving the two dimensional cochlear model

the size of higman haines sets we show that for the family of church rosser languages the higman haines sets which are the sets of all scattered subwords of a given language and the sets of all words that contain some word of a given language as a scattered subword cannot be effectively constructed although both these sets are regular for any language this nicely contrasts the result on the effectiveness of the higman haines sets for the family of context free languages the non effectiveness is based on a non recursive trade off result between the language description mechanism of church rosser languages and the corresponding higman haines sets which in turn is also valid for all supersets of the language family under consideration and in particular for the family of recursively enumerable languages finally for the family of regular languages we prove an upper and a matching lower bound on the size of the higman haines sets in terms of nondeterministic finite automata

on kirchberg s embedding problem kirchberg s embedding problem kep asks whether every separable c algebra embeds into an ultrapower of the cuntz algebra o 2 o 2 in this paper we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses in order to prove this result we study general properties of existentially closed c algebras along the way we establish a connection between existentially closed c algebras the weak expectation property of lance and the local lifting property of kirchberg the paper concludes with a discussion of the model theory of o 2 o 2 several results in this last section are proven using some technical results concerning tubular embeddings a notion first introduced by jung for studying embeddings of tracial von neumann algebras into the ultrapower of the hyperfinite ii 1 factor

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

decidability of involution hypercodes given a finite set x of strings x is a hypercode if a string in x is not a subsequence of any other string in x we consider hypercodes for involution codes which are useful for dna strand design and define an involution hypercode we then tackle the involution hypercode decidability problem that is to determine whether or not a given language is an involution hypercode based on the hypercode properties we design a polynomial runtime algorithm for regular languages we also prove that it is decidable whether or not a context free language is an involution hypercode note that it is undecidable for some other involution codes such as involution prefix codes suffix codes and k intercodes

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

noncommutative l p spaces without the completely bounded approximation property for any 1 leq p leq infty different from 2 we give examples of non commutative lp spaces without the completely bounded approximation property let f be a non archimedian local field if p 4 or p 4 3 and r geq 3 these examples are the non commutative lp spaces of the von neumann algebra of lattices in sl r f or in sl r r for other values of p the examples are the non commutative lp spaces of the von neumann algebra of lattices in sl r f for r large enough depending on p we also prove that if r geq 3 lattices in sl r f or sl r r do not have the approximation property of haagerup and kraus this provides examples of exact c algebras without the operator space approximation property

crossed products and entropy of automorphisms abstract let a be an exact c algebra let g be a locally compact group and let a g be a c dynamical system each automorphism g induces a spatial automorphism ad g on the reduced crossed product a g in this paper we examine the question first raised by e stormer of when the topological entropies of g and ad g coincide this had been answered by n brown for the particular case of discrete abelian groups using different methods we extend his result to preservation of entropy for g when the subgroup of aut g generated by the corresponding inner automorphism ad g has compact closure this property is satisfied by all elements of a wide class of groups called locally fia this class includes all abelian groups both discrete and continuous as well as all compact groups

the blackadar handelman theorem for non unital c algebras abstract a well known theorem of blackadar and handelman states that every unital stably finite c algebra has a bounded quasitrace rather strong generalizations of stable finiteness to the non unital case can be obtained by either requiring the multiplier algebra to be stably finite or alternatively requiring it to be at least stably not properly infinite this paper deals with the question whether the blackadar handelman result can be extended to the non unital case with respect to these generalizations of stable finiteness two examples of c algebras are constructed the first one is a non unital stably commutative c algebra a that contradicts the weakest possible generalization of the blackadar handelman theorem the multiplier algebra of a is stably finite while a has no bounded quasitrace the second example is a simple c algebra b with no bounded quasitrace such that the multiplier algebras of all matrix algebras over b are not properly infinite

a simple separable exact c algebra not anti isomorphic to itself we give an example of an exact stably finite simple separable c algebra d which is not isomorphic to its opposite algebra moreover d has the following additional properties it is stably finite approximately divisible has real rank zero and stable rank one has a unique tracial state and the order on projections over d is determined by traces it also absorbs the jiang su algebra z and in fact absorbs the 3 uhf algebra we can also explicitly compute the k theory of d namely k 0 d cong mathbb z tfrac 1 3 with the standard order and k1 d 0 as well as the cuntz semigroup of d namely w d cong mathbb z tfrac 1 3 sqcup 0 infty

dynamical entropy for automorphisms of exact c algebras abstract some estimations of dynamical entropies are given by applying the entropy ht for an automorphism of a c algebra a with respect to invariant state this ht coincides with the connes narnhofer thirring cnt entropy h when a is abelian let a be the reduced free product of ai i i i if ai i i are finite dimensional then h ht 0 for all aut a concerning the free product 1 2 of i aut ai i we have always that ht 1 2 1 2 max ht 1 1 ht 2 2 and the cnt entropy satisfies this formula for all automorphisms in the ergodic theory

operator algebraic superridigity for sl n ℤ n 3 for n 3 let sln we prove the following superridigity result for in the context of operator algebras let l be the von neumann algebra generated by the left regular representation of let m be a finite factor and let u m be its unitary group let u m be a group homomorphism such that m then either

coding properties of dna languages the computation language of a dna based system consists of all the words dna strands that can appear in any computation step of the system in this work we define properties of languages which ensure that the words of such languages will not form undesirable bonds when used in dna computations we give several characterizations of the desired properties and provide methods for obtaining languages with such properties the decidability of these properties is addressed as well as an application we consider splicing systems whose computation language is free of certain undesirable bonds and is generated by nearly optimal comma free codes

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

hnn extensions of von neumann algebras abstract reduced hnn extensions of von neumann algebras as well as c algebras will be introduced and their modular theory factoriality and ultraproducts will be discussed in several concrete settings detailed analysis on them will be also carried out

tsirelson s problem and asymptotically commuting unitary matrices in this paper we consider quantum correlations of bipartite systems having a slight interaction and reinterpret tsirelson s problem and hence kirchberg s and connes s conjectures in terms of finite dimensional asymptotically commuting positive operator valued measures we also consider the systems of asymptotically commuting unitary matrices and formulate the stronger kirchberg conjecture

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

colp spaces the local structure of non commutative lp spaces abstract developing the theory of col p spaces a variation of the non commutative analogue of l p spaces we provide new tools to investigate the local structure of non commutative l p spaces under mild assumptions on the underlying von neumann algebras non commutative l p spaces with grothendieck s approximation property behave locally like the space of matrices equipped with the p norm of the sequences of their singular values as applications we obtain a basis for non commutative l p spaces associated with hyperfinite von neumann algebras with separable predual von neumann algebras generated by free groups and obtain a basis for separable nuclear c algebras

the effros marechal topology in the space of von neumann algebras ii we continue our study 9 of the effros marechal topology on the space vn h of all von neumann algebras acting on a fixed separable hilbert space h in particular we prove that factors of each of the types ii1 ii and for fixed 0 1 iii form a dense subset of vn h moreover the density of type i factors or equivalently of injective factors in vn h is proved to be equivalent to famous conjectures by a connes and e kirchberg

a non nuclear c algebra with the weak expectation property and the local lifting property we construct the first example of a c algebra a with the properties in the title this gives a new example of non nuclear a for which there is a unique c norm on a otimes a op this example is of particular interest in connection with the connes kirchberg problem which is equivalent to the question whether c mathbb f 2 which is known to have the llp also has the wep our c algebra a has the same collection of finite dimensional operator subspaces as c mathbb f 2 or c mathbb f infty in addition our example can be made to be quasidiagonal and of similarity degree or length 3 in the second part of the paper we reformulate our construction in the more general framework of a c algebra that can be described as the limit both inductive and projective for a sequence of c algebras c n when each c n is a subquotient of c n 1 we use this to show that for certain local properties of injective non surjective homomorphisms there are c algebras for which the identity map has the same properties as the homomorphisms

herrero s approximation problem for quasidiagonal operators let t be a quasidiagonal operator on a separable hilbert space it is shown that there exists a sequence of operators tn such that dim c tn and t tn 0 if and only if c t is exact

three dimensional numerical modeling for global cochlear dynamics a hybrid analytical numerical model using galerkin approximation to variational equations has been developed for predicting global cochlear responses the formulation provides a flexible framework capable of incorporating morphologically based mechanical models of the cochlear partition and realistic geometry the framework is applied for a simplified model with an emphasis on application of hybrid methods for three dimensional modeling the resulting formulation is modular where matrices representing fluid and cochlear partition are constructed independently computational cost is reduced using two methods a modal finite element method and a boundary element finite element method the first uses a cross mode expansion of fluid pressure 2 5d model and the second uses a waveguide green s function based boundary element method bem a novel wave number approach to the boundary element formulation for interior problem results in efficient computation of the finite element matrix for the two methods a c

the mode coupling liouville green approximation for a two dimensional cochlear model the liouville green or wentzel kramers brillouin wkb approximation for the two dimensional cochlear mechanics problem disagrees with the finite difference solution in the region after the response peak this disagreement has left doubts about the validity of the liouville green approximation and has never been satisfactorily explained in this paper it is shown that the liouville green approximation fails to satisfy laplace s equation a new solution is proposed called the mode coupling liouville green approximation in which energy is coupled into a second wave mode so as to obey laplace s equation the new approximation gives excellent quantitative agreement with the finite difference solution furthermore it may provide an explanation for a second vibration mode observed in biological cochleas also proposed is a high order formulation of the stapes displacement term which is necessary to obtain good agreement between the liouville green approximation and finite difference solutions at low frequencies

topological entropy and the af core of a graph c algebra abstract let c e be the c algebra associated with a locally finite directed graph e and a e be the af core of c e for the topological entropy ht e in the sense of brown voiculescu of the canonical completely positive map e on the graph c algebra it is known that if e is finite ht e ht e a e h b e h l e where h b e respectively h l e is the block respectively the loop entropy of e in case e is irreducible and infinite h l e ht e a e h b e t is known recently where e t is the graph e with the edges directed reversely then by monotonicity of entropy h l e ht e is clear in this paper we show that ht e h b e t holds for locally finite infinite graphs e the af core a e is known to be stably isomorphic to the graph c algebra c e c z of certain skew product e c z and we also show that ht e c z ht e a e examples e p p 1 of irreducible graphs with ht e p log p are discussed

central sequence c algebras and tensorial absorption of the jiang su algebra we study properties of the central sequence algebra a a 0 of a c algebra a and we present an alternative approach to a recent result of matui and sato they prove that every unital separable simple nuclear c algebra whose trace simplex is nite dimensional tensorially absorbs the jiang su algebra if and only if it has the strict comparison property we extend their result to the case where the extreme boundary of the trace simplex is closed and of nite topological dimension we are also able to relax the assumption on the c algebra of having the strict com parison property to a weaker property that we call local weak comparison namely we prove that a unital separable simple nuclear c algebra whose trace simplex has nite dimensional closed extreme boundary tensorially absorbs the jiang su algebra if and only if it has the local weak comparison property we can also eliminate the nuclearity assumption if instead we assume the si property of matui and sato and moreover that each ii1 factor representation of the c algebra is a mcdu factor

unification and extension of monolithic state space and iterative cochlear models time domain cochlear models have primarily followed a method introduced by allen and sondhi j acoust soc am 66 123 132 1979 recently the state space formalism proposed by elliott et al j acoust soc am 122 2759 2771 2007 has been used to simulate a wide range of nonlinear cochlear models it used a one dimensional approach that is extended to two dimensions in this paper using the finite element method the recently developed state space formalism in fact shares a close relationship to the earlier approach working from diependaal et al j acoust soc am 82 1655 1666 1987 the two approaches are compared and the relationship formalized understanding this relationship allows models to be converted from one to the other in order to utilize each of their strengths a second method to derive the state space matrices required for the state space formalism is also presented this method offers improved numerical properties because it uses the information available about the mod

c algebras associated with groups with kazhdan s property t has exhibited block diagonal operators on separable hilbert spaces which are not norm limits of generators of singly generated finite dimensional c algebras it turns out that voiculescu s construction which is based upon a certain class of groups with property t can be put into a more general context involving c algebras which arise very naturally from such groups it is these c algebras which are the subject of this paper as well as giving a natural setting in which to view voiculescu s result these c algebras have almost fortuitously a number of other important properties which are very easily established like the full c algebras of free groups these c algebras are not

an approximate universal coefficient theorem an approximate universal coefficient theorem auct for certain c algebras is established we present a proof that kirchberg phillips s classification theorem for separable nuclear purely infinite simple c algebras is valid for c algebras satisfying the auct instead of the uct it is proved that two versions of auct are in fact the same we also show that c algebras that are locally approximated by c algebras satisfying the auct satisfy the auct as an application we prove that certain simple c algebras which are locally type i are in fact isomorphic to simple ah algebras as another application we show that a sequence of residually finite dimensional c algebras which are asymptotically nuclear and which asymptotically satisfies the auct can be embedded into the same simple af 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

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

maurey s factorization theory for operator spaces we prove an operator space version of maurey s theorem which claims that every absolutely p 1 summing map on c k is automatically absolutely q summing for q p our results imply in particular that every completely bounded map from b h with values in pisier s operator space oh is completely p summing for p 2 this fails for p 2 as applications we obtain eigenvalue estimates for translation invariant maps defined on the von neumann algebra vn g associated with a discrete group g we also develop a notion of cotype which is compatible with factorization results on noncommutative lp spaces

three dimensional viscous finite element formulation for acoustic fluid structure interaction abstract a three dimensional viscous finite element model is presented in this paper for the analysis of the acoustic fluid structure interaction systems including but not limited to the cochlear based transducers the model consists of a three dimensional viscous acoustic fluid medium interacting with a two dimensional flat structure domain the fluid field is governed by the linearized navier stokes equation with the fluid displacements and the pressure chosen as independent variables the mixed displacement pressure based formulation is used in the fluid field in order to alleviate the locking in the nearly incompressible fluid the structure is modeled as a mindlin plate with or without residual stress the hinton huang s 9 noded lagrangian plate element is chosen in order to be compatible with 27 4 u p fluid elements the results from the full 3d fem model are in good agreement with experimental results and other fem results including beltman s thin film viscoacoustic element w m beltman p j m van der hoogt r m e j spiering h tijdeman implementation and experimental validation of a new viscothermal acoustic finite element for acousto elastic problems j sound vib 216 1 1998 159 185 and two and half dimensional inviscid elements a a parthasarathi k grosh a l nuttall three dimensional numerical modeling for global cochlear dynamics j acoust soc am 107 2000 474 485 although it is computationally expensive it provides a benchmark solution for other numerical models or approximations to compare to besides experiments and it is capable of modeling any irregular geometries and material properties while other numerical models may not be applicable

a generalization of expander graphs and local reflexivity of uniform roe algebras abstract we introduce a generalization of expander graphs which is called a weak expander sequence it is proved that a uniform roe algebra of a weak expander sequence is not locally reflexive it follows that uniform roe algebras of expander graphs are not exact we introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework key tools for the proof are amenable traces and measured groupoids associated with generalized box spaces

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