crossed products by endomorphisms of c_0 x algebras abstract in the first part of the paper we develop a theory of crossed products of a c algebra a by an arbitrary not necessarily extendible endomorphism a a we consider relative crossed products c a j where j is an ideal in a and describe up to morita rieffel equivalence all gauge invariant ideals in c a j and give six term exact sequences determining their k theory we also obtain certain criteria implying that all ideals in c a j are gauge invariant and that c a j is purely infinite in the second part we consider a situation where a is a c 0 x algebra and is such that f a f a a a f c 0 x where is an endomorphism of c 0 x pictorially speaking is a mixture of a topological dynamical system x dual to c 0 x and a continuous field of homomorphisms x between the fibers a x x x of the corresponding c bundle for systems described above we establish efficient conditions for the uniqueness property gauge invariance of all ideals and pure infiniteness of c a j we apply these results to the case when x prim a is a hausdorff space in particular if the associated c bundle is trivial we obtain formulas for k groups of all ideals in c a j in this way we constitute a large class of crossed products whose ideal structure and k theory is completely described in terms of x x x x y where y is a closed subset of x

plethystic algebras and vector symmetric functions an isomorphism is established between the plethystic hopf algebra pleth super l and the algebra of vector symmetric functions the hall inner product of symmetric function theory is extended to the hopf algebra pleth super l

bilinear covariants and spinor fields duality in quantum clifford algebras classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum clifford algebras on minkowski spacetime once the bilinear covariants are expressed in terms of algebraic spinor fields the duality between spinor and quantum spinor fields can be discussed thus by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric quantum algebraic spinor fields and deformed bilinear covariants can be constructed they are thus compared to the classical non quantum ones classes of quantum spinor fields classes are introduced and compared with lounesto s spinor field classification a physical interpretation of the deformed parts and the underlying z grading is proposed the existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity s w hawking the unpredictability of quantum gravity commun

dixmier traces and extrapolation description of noncommutative lorentz spaces we study the connections between dixmier traces functions and traces of heat semigroups beyond the dual of the macaev ideal and in the general context of semifinite von neumann algebras we show that the correct framework for this investigation is that of operator lorentz spaces possessing an extrapolation description we demonstrate the applicability of our results to hormander weyl pseudo differential calculus on rn in this context we prove that the dixmier trace of a pseudo differential operator coincides with the dixmier integral of its symbol

the noncommutative geometry of graph c algebras i the index theorem we investigate conditions on a graph c algebra for the existence of a faithful semifinite trace using such a trace and the natural gauge action of the circle on the graph algebra we construct a smooth 1 summable semi finite spectral triple the local index theorem allows us to compute the pairing with k theory this produces invariants in the k theory of the fixed point algebra and these are invariants for a finer structure than the isomorphism class of c e

dixmier measurability in marcinkiewicz spaces and applications abstract we present new results concerning the description of dixmier measurable elements in the sense of a connes in marcinkiewicz spaces and demonstrate some applications of our results in particular we complement famous connes trace theorem by describing the asymptotics of the singular function of order n classical pseudo differential operators on an n dimensional compact riemannian manifold

group c algebras as decreasing intersection of nuclear c algebras we prove that for every exact discrete group gamma there is an intermediate rm c ast algebra between the reduced group rm c ast algebra and the intersection of the group von neumann algebra and the uniform roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the cuntz algebra cal o 2 in particular when gamma has the ap approximation property the reduced group rm c ast algebra is realized in this way we also study extensions of the reduced free group rm c ast algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear rm c ast algebra is realized in this way

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

twisted cyclic theory equivariant kk theory and kms states given a c algebra a with a kms weight for a circle action we construct and compute a secondary invariant on the equivariant k theory of the mapping cone of a t a both in terms of equivariant kk theory and in terms of a seminite spectral ow this in particular puts the previously considered examples of cuntz algebras 9 and suq 2 13 in a general framework as a new example we consider the araki woods iii representations of the fermion algebra

a problem of cayley from 1857 and how he could have solved it abstract in a paper of 1857 cayley suggests the problem of determining all the symmetric functions of the common solutions of a linear cubic homogeneous system of ternary equations as functions of the coefficients of the equations we show how cayley could have solved the problem himself if he had taken another look at it all the facts necessary for the solution were known to cayley including a crucial formula known as cayley s identity we study the problem by mixing techniques from vector symmetric function theory with techniques from the study of determinants and permanents we develop some new formulas for determinants of matrices which arise by the hadamard or pointwise product from other matrices

spectral flow and dixmier traces abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi finite von neumann algebra our results have several applications we deduce a formula for the chern character of an odd l 1 summable breuer fredholm module in terms of a hochschild 1 cycle we explain how to derive a wodzicki residue for pseudo differential operators along the orbits of an ergodic r n action on a compact space x finally we give a short proof of an index theorem of lesch for generalised toeplitz operators

amenable minimal cantor systems of free groups arising from diagonal actions we study amenable minimal cantor systems of free groups arising from the diagonal actions of the boundary actions and certain cantor systems it is shown that every virtually free group admits continuously many amenable minimal cantor systems whose crossed products are mutually non isomorphic kirchberg algebras in the uct class with explicitly determined k theory the technique developed in our study also enables us to compute the k theory of certain amenable minimal cantor systems we apply it to the diagonal actions of the boundary actions and the products of the odometer transformations and determine their k theory then we classify them in terms of the topological full groups continuous orbit equivalence strong orbit equivalence and the crossed products

measures from dixmier traces and zeta functions for l functions on a closed compact riemannian manifold the noncommutative residue and the dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the lebesgue integral the identifications are shown to continue to and be sharp at l2 functions for functions strictly in lp 1 p 2 symmetrised noncommutative residue and dixmier trace formulas must be introduced for which the identification is shown to continue for the noncommutative residue however a failure is shown for the dixmier trace formulation at l1 functions it is shown the noncommutative residue remains finite and recovers the lebesgue integral for any integrable function while the dixmier trace expression can diverge the results show that a claim in the monograph j m gracia bondia j c varilly h figueroa elements of noncommutative geometry birkhauser adv texts birkhauser boston 2001 that the equality on c functions between the lebesgue integral and an operator theoretic expression involving a dixmier trace obtained from connes trace theorem can be extended to any integrable function is false the results of this paper include a general presentation for finitely generated von neumann algebras of commuting bounded operators including a bounded borel or l functional calculus version of c results in iv 2 of a connes noncommutative geometry academic press new york 1994

type iii representations and modular spectral triples for the noncommutative torus abstract it is well known that for any irrational rotation number the noncommutative torus a must have representations such that the generated von neumann algebra a is of type iii therefore it could be of interest to exhibit and investigate such kind of representations together with the associated spectral triples whose twist of the dirac operator and the corresponding derivation arises from the tomita modular operator in the present paper we show that this program can be carried out at least when is a liouville number satisfying a faster approximation property by rationals in this case we exhibit several type i i and ii i 0 1 factor representations and modular spectral triples the method developed in the present paper can be generalised to ccr algebras based on a locally compact abelian group equipped with a symplectic form

the dixmier trace and asymptotics of zeta functions abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup we prove our results in a general semi finite von neumann algebra we find for p 1 that the asymptotics of the zeta function determines an ideal strictly larger than l p on which the dixmier trace may be defined we also establish stronger versions of other results on dixmier traces and zeta functions

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

spectral triples and wavelets for higher rank graphs abstract in this paper we present a new way to associate a finitely summable spectral triple to a higher rank graph via the infinite path space of moreover we prove that this spectral triple has a close connection to the wavelet decomposition of which was introduced by farsi gillaspy kang and packer in 2015 we first introduce the concept of stationary k bratteli diagrams in order to associate a family of ultrametric cantor sets and their associated pearson bellissard spectral triples to a finite strongly connected higher rank graph we then study the zeta function abscissa of convergence and dixmier trace associated to the pearson bellissard spectral triples of these cantor sets and show these spectral triples are regular in the sense of pearson and bellissard we obtain an integral formula for the dixmier trace given by integration against a measure and show that is a rescaled version of the measure m on which was introduced by an huef laca raeburn and sims finally we investigate the eigenspaces of a family of laplace beltrami operators associated to the dirichlet forms of the spectral triples we show that these eigenspaces refine the wavelet decomposition of l 2 m which was constructed by farsi et al

ring c algebras we associate reduced and full c algebras to arbitrary rings and study the inner structure of these ring c algebras as a result we obtain conditions for them to be purely infinite and simple we also discuss several examples originally our motivation comes from algebraic number theory

the local index formula in semifinite von neumann algebras i spectral flow abstract we generalise the local index formula of connes and moscovici to the case of spectral triples for a subalgebra a of a general semifinite von neumann algebra in this setting it gives a formula for spectral flow along a path joining an unbounded self adjoint breuer fredholm operator affiliated to the von neumann algebra to a unitarily equivalent operator our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is almost a b b cocycle in the cyclic cohomology of a

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

a new approach to recent constructions of c algebras from modular index theory abstract we present a new approach to c algebras recently constructed in the context of modular index theory by carey phillips putnam and rennie it turns out that their constructions can be identified with full corners of ideals in semigroup c algebras this new point of view leads to a systematic analysis of these algebras and their canonical maximal abelian subalgebras

k theoretic duality for hyperbolic dynamical systems the k theoretic analog of spanier whitehead duality for noncommutative c algebras is shown to hold for the ruelle algebras associated to irreducible smale spaces this had previously been proved only for shifts of finite type implications of this result as well as relations to the baum connes conjecture and other topics are also considered hn i x h i s n 1 x given by slant product this was generalized by spanier and whitehead to allow s n 1 x to be replaced by a space of the homotopy type of a finite complex dn x for which the analog of the above relations hold it is in this form that these notions extend naturally to the noncommutative setting in noncommutative topology the roles of homology and cohomology are played by k theory and k homology these have been combined into a bivariant theory by kasparov for which one has k theory k a kk c a and k homology k a kk a c kasparov s theory comes equipped with a product that is the analog of the slant product used above of course noncommutative c algebras play the role of algebras of continuous functions on ordinary spaces with this in hand the generalization of spanier whitehead duality to this setting is easily accomplished let a be a c algebra we will consider indices in k theory to be modulo 2 a c algebra dn a is a spanier whitehead n dual of a or simply a dual when the context

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

simple purely infinite c algebras and n filling actions let n be a positive integer we introduce a concept which we call the n filling property for an action of a group on a separable unital c algebra a if a c is a commutative unital c algebra and the action is induced by a group of homeomorphisms of then the n filling property reduces to a weak version of hyperbolicity the n filling property is used to prove that certain crossed product c algebras are purely infinite and simple a variety of group actions on boundaries of symmetric spaces and buildings have the n filling property an explicit example is the action of sln z on the projective n space

measurable operators and the asymptotics of heat kernels and zeta functions abstract in this note we answer some questions inspired by the introduction in connes 1988 1994 6 7 of the notion of measurable operators using dixmier traces these questions concern the relationship of measurability to the asymptotics of functions and heat kernels the answers have remained elusive for some 15 years

plethystic vertex operators and boson fermion correspondences we study the algebraic properties of plethystic vertex operators introduced in 2010 j phys a math theor 43 405202 underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group defined to stabilize tensors of young symmetry type characterized by a partition of arbitrary shape here we establish an extension of the well known boson fermion correspondence involving schur functions and their associated bernstein vertex operators for each the modes generated by the plethystic vertex operators and their suitably constructed duals satisfy the anticommutation relations of a complex clifford algebra the combinatorial manipulations underlying the results involve exchange identities exploiting the hopf algebraic structure of certain symmetric function series and their plethysms

k theory of certain purely infinite crossed products it was shown by rordam and the second named author that a countable group g admits an action on a compact space such that the crossed product is a kirchberg algebra if and only if g is exact and non amenable this construction allows a certain amount of choice we show that different choices can lead to different algebras at least with the free group

minimal ambient nuclear c algebras abstract we provide examples of ambient nuclear c algebras of non nuclear c algebras with no proper intermediate c algebras in particular this gives the first examples of minimal ambient nuclear c algebras of non nuclear c algebras for this purpose we study generic cantor systems of infinite free product groups

on lineability of sets of continuous functions we study the existence of vector spaces of dimension at least two of continuous functions on subsets of r every non zero element of which admits one and only one absolute maximum

integration on locally compact noncommutative spaces abstract we present an ab initio approach to integration theory for nonunital spectral triples this is done without reference to local units and in the full generality of semifinite noncommutative geometry the main result is an equality between the dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators we also examine definitions for integrable bounded elements of a spectral triple based on zeta function heat kernel and dixmier trace techniques we show that zeta functions and heat kernels yield equivalent notions of integrability which imply dixmier traceability

dixmier traces as singular symmetric functionals and applications to measurable operators abstract we unify various constructions and contribute to the theory of singular symmetric functionals on marcinkiewicz function operator spaces this affords a new approach to the non normal dixmier and connes dixmier traces introduced by dixmier and adapted to non commutative geometry by connes living on a general marcinkiewicz space associated with an arbitrary semifinite von neumann algebra the corollaries to our approach stated in terms of the operator ideal l 1 which is a special example of an operator marcinkiewicz space are i a new characterization of the set of all positive measurable operators from l 1 i e those on which an arbitrary connes dixmier trace yields the same value in the special case when the operator ideal l 1 is considered on a type i infinite factor a bounded operator x belongs to l 1 if and only if the sequence of singular numbers s n x n 1 in the descending order and counting the multiplicities satisfies x 1 sup n 1 1 log 1 n n 1 n s n x in this case our characterization amounts to saying that a positive element x l 1 is measurable if and only if lim n 1 log n n 1 n s n x exists ii the set of dixmier traces and the set of connes dixmier traces are norming sets up to equivalence for the space l 1 l 0 1 where the space l 0 1 is the closure of all finite rank operators in l 1 in the norm 1

a class of c algebras generalizing both graph algebras and homeomorphism c algebras iv pure infiniteness this is the final one in the series of papers where we introduce and study the c algebras associated with topological graphs in this paper we get a sufficient condition on topological graphs so that the associated c algebras are simple and purely infinite using this result we give one method to construct all kirchberg algebras as c algebras associated with topological graphs

boundary quotients and ideals of toeplitz c algebras of artin groups abstract we study the quotients of the toeplitz c algebra of a quasi lattice ordered group g p which we view as crossed products by a partial actions of g on closed invariant subsets of a totally disconnected compact hausdorff space the nica spectrum of g p our original motivation and our main examples are drawn from right angled artin groups but many of our results are valid for more general quasi lattice ordered groups we show that the nica spectrum has a unique minimal closed invariant subset which we call the boundary spectrum and we define the boundary quotient to be the crossed product of the corresponding restricted partial action the main technical tools used are the results of exel laca and quigg on simplicity and ideal structure of partial crossed products which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets when there exists a generalised length function or controlled map defined on g and taking values in an amenable group we prove that the partial action is amenable on arbitrary closed invariant subsets the topological freeness of the boundary action depends on topological freeness of the restriction to a certain lattice subgroup of g the core of g p which often turns out to be trivial our main results are obtained for right angled artin groups with trivial centre that is those with no cyclic direct factor they include a presentation of the boundary quotient in terms of generators and relations that generalises cuntz s presentation of o n a proof that the boundary quotient is purely infinite and simple and a parametrisation of the ideals of the toeplitz c algebra in terms of subsets of the standard generators of the artin group

invariant banach limits and applications let l be the space of all bounded sequences x x1 x2 with the norm x l supn xn and let l l be the set of all bounded linear operators on l we present a set of easily verifiable sufficient conditions on an operator h l l guaranteeing the existence of a banach limit b on l such that b bh we apply our results to the classical cesaro operator c on l and give necessary and sufficient condition for an element x l to have fixed value bx for all cesaro invariant banach limits b finally we apply the preceding description to obtain a characterization of measurable elements from the dixmier macaev sargent ideal of compact operators with respect to an important subclass of dixmier traces generated by all cesaro invariant banach limits it is shown that this class is strictly larger than the class of all measurable elements with respect to the class of all dixmier traces

hopf algebras distributive laplace pairings and hash products a unified approach to tensor product decompositions of group characters we show for bicommutative graded connected hopf algebras that a certain distributive laplace subgroup of the convolution monoid of 2 cochains parameterizes certain well behaved hopf algebra deformations using the laplace group or its frobenius subgroup we define higher derived hash products and develop a general theory to study their main properties applying our results to the universal bicommutative graded connected hopf algebra of symmetric functions we show that classical tensor product and character decompositions such as those for the general linear group mixed co and contravariant or rational characters orthogonal and symplectic group characters thibon and reduced symmetric group characters are special cases of higher derived hash products in the appendix we discuss a relation to formal group laws

generalized limits with additional invariance properties and their applications to noncommutative geometry abstract let n be the set of all natural numbers and l l n resp l l 0 be the banach space of all bounded sequences x x 1 x 2 resp classes of lebesgue measurable bounded functions on 0 with the uniform norm we work with subsets of the set of all normalized positive functionals on l and l which are used in noncommutative geometry to define various classes of dixmier traces singular positive traces on the ideal m 1 of compact operators on an infinite dimensional hilbert space h with logarithmic divergence of the partial sums of their singular values these classes of traces are of importance in noncommutative geometry and with each such a class say b of traces a subset m 1 b m 1 on which all traces from b coincide is linked the main objective of the present article is to give new characterizations of m 1 b for commonly occurring classes of dixmier traces we also answer a question from benameur and fack 2006 4 concerning conditions on a dixmier trace under which lidskii type formulae from that article hold

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

the structure ofc algebras associated with hyperbolic dynamical systems we consider the stable unstablec algebras and the ruelle algebras associated to a mixing smale space in the case of a shift of finite type these are the af algebras studied by w krieger and the stabilized cuntz krieger algebras in the general case we show that the stable and unstable algebras are simple and amenable we also show that the ruelle algebras are simple amenable and purely infinite

families of type iii kms states on a class of c algebras containing on and qn abstract we construct a family of purely infinite c algebras q for 0 1 that are classified by their k groups there is an action of the circle t with a unique kms state on each q for 1 n q 1 n o n with its usual t action and kms state for p q rational in lowest terms q o n n q p 1 with uhf fixed point algebra of type p q for any n 1 q o n for infinitely many with distinct kms states and uhf fixed point algebras for any 0 1 q o for irrational the fixed point algebras are not af and the q are usually not cuntz algebras for transcendental k 1 q k 0 q z so that q is cuntz q n cuntz 2008 16 if and 1 are both algebraic integers the only o n which appear are those for which n 3 mod 4 for each the representation of q defined by the kms state generates a type iii factor these algebras fit into the framework of modular index theory twisted cyclic theory of carey et al 2010 8 carey et al 2009 12 carey et al in press 5

dixmier traces are weak dense in the set of all fully symmetric traces abstract we extend dixmier s construction of singular traces see 2 to arbitrary fully symmetric operator ideals in fact we show that the set of dixmier traces is weak dense in the set of all fully symmetric traces that is those traces which respect hardy littlewood submajorization our results complement and extend earlier work of wodzicki 22

banach limits and traces on l1 abstract we introduce a new approach to traces on the principal ideal l 1 generated by any positive compact operator whose singular value sequence is the harmonic sequence distinct from the well known construction of j dixmier the new approach provides the explicit construction of every trace of every operator in l 1 in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues the approach is based on a remarkable bijection between the set of all traces on l 1 and the set of all translation invariant functionals on l this bijection allows us to identify all known and commonly used subsets of traces dixmier traces connes dixmier traces etc in terms of invariance properties of linear functionals on l and definitively classify the measurability of operators in l 1 in terms of qualified convergence of sums of eigenvalues this classification has led us to a resolution of several open problems for the class l 1 from 7 as an application we extend connes classical trace theorem to positive normalised traces

infinite dimensional holomorphic non extendability and algebraic genericity abstract in this note the linear structure of the family h e g of holomorphic functions in a domain g of a complex banach space that are not holomorphically continuable beyond the boundary of g is analyzed more particularly we prove that h e g contains except for zero a closed and a dense vector space having maximal dimension as well as a maximally generated free algebra the results obtained complete a number of previous ones by several authors

large linear manifolds of noncontinuable boundary regular holomorphic functions we prove in this paper that if g is a domain in the complex plane satisfying appropriate topological or geometrical conditions then there exists a large dense or closed infinite dimensional linear submanifold of boundary regular holomorphic functions on g all of whose nonzero members are not continuable across any boundary point of g

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given