on the kk theory and the e theory of amalgamated free products of c algebras abstract we establish six terms exact sequences relating the kk theory groups and the e theory groups of an amalgamated free product c algebra a 1 b a 2 to the respective groups of the three constituents a1 a2 and b in the kk theory case we assume that b is finite dimensional but in the e theory case only that b is nuclear or is the range of a conditional expectation in both a1 and a2 in particular we obtain the conjectured six terms exact sequence for the k theory of an amalgamated free product in this generality

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

dynamical systems associated to separated graphs graph algebras and paradoxical decompositions abstract we attach to each finite bipartite separated graph e c a partial dynamical system e c f where e c is a zero dimensional metrizable compact space f is a finitely generated free group and is a continuous partial action of f on e c the full crossed product c algebra o e c c e c f is shown to be a canonical quotient of the graph c algebra c e c of the separated graph e c similarly we prove that for any field k the algebraic crossed product l k ab e c c k e c alg f is a canonical quotient of the leavitt path algebra l k e c of e c the monoid v l k ab e c of isomorphism classes of finitely generated projective modules over l k ab e c is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs using this we are able to construct an action of a finitely generated free group f on a zero dimensional metrizable compact space z such that the type semigroup s z f k is not almost unperforated where k denotes the algebra of clopen subsets of z finally we obtain a characterization of the separated graphs e c such that the canonical partial action of f on e c is topologically free

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

c algebras generated by elements of a unitary matrix abstract let unnc be the unital c algebra generated by the elements uij 1 i j n satisfying the relations which make uij a unitary matrix unnc is isomorphic to the relative commutant of mn in the amalgamated product m n c c t define un rednc to be the relative commutant of mn in the reduced amalgamated product of m n c c t with respect to the usual traces we show that unnc has no nontrivial projections and that un rednc is a simple c algebra with no nontrivial projections we also compute the k groups of unnc as well as those of the noncommutative grassmanian gnnc

time ordered operators i foundations for an alternative view of reality abstracr the purpose of this paper is to present the proper framework for the mathematical foundations of time ordered operators we introduce a new mathematical process which we call the chronological process this process generalizes the notion of a limit and allows us to recapture many properties lost in time ordering we then construct time ordered integrals and evolution operators we show that under reasonable assumptions the time ordered sum of two generators of contraction semigroups is a generator this result resolves a question that has been debated in physics for forty years

branching systems and representations of cohn leavitt path algebras of separated graphs abstract for each separated graph e c we construct a family of branching systems over a set x and show how each branching system induces a representation of the cohn leavitt path algebra associated with e c as homomorphisms over the module of functions in x we also prove that the abelianized cohn leavitt path algebra of a separated graph with no loops can be written as an amalgamated free product of abelianized cohn leavitt algebras that can be faithfully represented via branching systems

foundations for relativistic quantum theory i feynman s operator calculus and the dyson conjectures in this paper we provide a representation theory for the feynman operator calculus this allows us to solve the general initial value problem and construct the dyson series we show that the series is asymptotic thus proving dyson s second conjecture for quantum electrodynamics in addition we show that the expansion may be considered exact to any finite order by producing the remainder term this implies that every nonperturbative solution has a perturbative expansion using a physical analysis of information from experiment versus that implied by our models we reformulate our theory as a sum over paths this allows us to relate our theory to feynman s path integral and to prove dyson s first conjecture that the divergences are in part due to a violation of heisenberg s uncertainly relations

purely infinite simple reduced c algebras of one relator separated graphs abstract given a separated graph e c there are two different c algebras associated to it the full graph c algebra c e c and the reduced one c red e c for a large class of separated graphs e c we prove that c red e c either is purely infinite simple or admits a faithful tracial state the main tool we use to show pure infiniteness of reduced graph c algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to dykema

convex subshifts separated bratteli diagrams and ideal structure of tame separated graph algebras abstract we introduce a new class of partial actions of free groups on totally disconnected compact hausdorff spaces which we call convex subshifts these serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph we define the notion of a finite type convex subshift and show that any such subshift is kakutani equivalent to the partial action associated with a finite bipartite separated graph we then study the ideal structure of both the full and the reduced tame graph c algebras o e c and o r e c of a separated graph e c and of the abelianized leavitt path algebra l k ab e c as well these algebras are the reduced crossed products with respect to the above mentioned partial actions and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary d saturated subsets of a certain infinite separated graph f d built from e c called the separated bratteli diagram of e c we finally use these tools to study simplicity and primeness of the tame separated graph algebras

derivations of leavitt path algebras abstract in this paper we describe the k module h h 1 l k of outer derivations of the leavitt path algebra l k of a row finite graph with coefficients in an associative commutative ring k with unit we explicitly describe a set of generators of h h 1 l k and relations among them we also describe a lie algebra structure of outer derivation algebra of the toeplitz algebra we prove that every derivation of a leavitt path algebra can be extended to a derivation of the corresponding c algebra

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

k theory for the tame c algebra of a separated graph abstract a separated graph is a pair e c consisting of a directed graph e and a set c v e 0 c v where each c v is a partition of the set of edges whose terminal vertex is v given a separated graph e c such that all the sets x c are finite the k theory of the graph c algebra c e c is known to be determined by the kernel and the cokernel of a certain map denoted by 1 c a e c from z c to z e 0 in this paper we compute the k theory of the tame graph c algebra o e c associated to e c which has been recently introduced by the authors letting denote the natural surjective homomorphism from c e c onto o e c we show that k 1 is a group isomorphism and that k 0 is a split monomorphism whose cokernel is a torsion free abelian group we also prove that this cokernel is a free abelian group when the graph e is finite and determine its generators in terms of a sequence of separated graphs e n c n n 1 naturally attached to e c on the way to showing our main results we obtain an explicit description of a connecting map arising in a six term exact sequence computing the k theory of an amalgamated free product and we also exhibit an explicit isomorphism between ker 1 c a e c and k 1 c e c

time ordered operators and feynman dyson algebras an approach to time ordered operators based upon von neumann s infinite tensor product hilbert spaces is used to define feynman dyson algebras this theory is used to show that a one to one correspondence exists between path integrals and semigroups which are integral operators defined by a kernel the reproducing property of the kernel being a consequence of the semigroup property for path integrals constructed from two semigroups the results are more general than those obtained by the use of the trotter kato formula perturbation series for the feynman dyson operator calculus for time evolution and scattering operators are discussed and it is pointed out that they are asymptotic in the sense of poincare as defined in the theory of semigroups thereby giving a precise formulation to a well known conjecture of dyson stated many years ago in the context of quantum electrodynamics moreover the series converge when these operators possess suitable holomorphy properties

faithfulness of free product states abstract it is proved that the free product state in the reduced free product of c algebras is faithful if the initial states are faithful

certain free products of graph operator algebras abstract we develop a notion of a generalized cuntz krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection we associate to these generalized cuntz krieger families a directed graph with a coloring function on the edge set we call such a directed graph an edge colored directed graph we then study the c algebras and the non selfadjoint operator algebras associated to edge colored directed graphs these algebras arise as free products of directed graph algebras with amalgamation we then determine the c envelopes for a large class of the non selfadjoint algebras finally we relate properties of the edge colored directed graphs to properties of the associated c algebra including simplicity and nuclearity using the free product description of these algebras we investigate the k theory of these algebras

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

time ordered operators ii apres avoir construit les fondements mathematiques generaux pour des equations d evolution lineaires en temps ordonne on applique les resultats pour montrer qu a la fois le developpement en perturbation et la methode des diagrammes de feymann sont mathematiquement sains

c algebras of separated graphs the construction of the c algebra associated to a directed graph e is extended to incorporate a family c consisting of partitions of the sets of edges emanating from the vertices of e these c algebras c e c are analyzed in terms of their ideal theory and k theory mainly in the case of partitions by finite sets the groups k0 c e c and k1 c e c are completely described via a map built from an adjacency matrix associated to e c one application determines the k theory of the c algebras um nnc confirming a conjecture of mcclanahan a reduced c algebra cred e c is also introduced and studied a key tool in its construction is the existence of canonical faithful conditional expectations from the c algebra of any row finite graph to the c subalgebra generated by its vertices differences between cred e c and c e c such as simplicity versus non simplicity are exhibited in various examples related to some algebras studied by mcclanahan