krieger factors isomorphic to their tensor square and pure point spectrum flows a factor m isomorphic to its tensor square whose sakai flip aut m m is approximately inner has a flow of weights with pure point spectrum

spectral triples for the sierpinski gasket abstract we construct a family of spectral triples for the sierpinski gasket k for suitable values of the parameters we determine the dimensional spectrum and recover the hausdorff measure of k in terms of the residue of the volume functional a tr a d s at its abscissa of convergence d d which coincides with the hausdorff dimension d h of the fractal we determine the associated connes distance showing that it is bi lipschitz equivalent to the distance on k induced by the euclidean metric of the plane and show that the pairing of the associated fredholm module with odd k theory is non trivial when the parameters belong to a suitable range the abscissa of convergence d of the energy functional a tr d s 2 d a 2 d s 2 takes the value d e log 12 5 log 2 which we call energy dimension and the corresponding residue gives the standard dirichlet form on k

symplectic tori in rational elliptic surfaces let e 1 p denote the rational elliptic surface with a single multiple fiber fp of multiplicity p we construct an infinite family of homologous non isotopic symplectic tori representing the primitive 2 dimensional homology class fp in e 1 p when p 1 as a consequence we get infinitely many non isotopic symplectic tori in the fiber class of the rational elliptic surface open image in new window we also show how these tori can be non isotopically embedded as homologous symplectic submanifolds in other symplectic 4 manifolds

non isotopic symplectic tori in the same homology class for any pair of integers n 1 and q 2 we construct an infinite family of mutually non isotopic symplectic tori representing the homology class q f of an elliptic surface e n where f is the homology class of the fiber we also show how such families can be non isotopically and symplectically embedded into a more general class of symplectic 4 manifolds

exotic smooth structures on small 4 manifolds let m be either mathbb cp 2 3 overline mathbb cp 2 or 3 mathbb cp 2 5 overline mathbb cp 2 we construct the first example of a simply connected irreducible symplectic 4 manifold that is homeomorphic but not diffeomorphic to m

ζ function and heat kernel formulae abstract we present a systematic study of asymptotic behaviour of generalised functions and heat kernels used in noncommutative geometry and clarify their connections with dixmier traces we strengthen and complete a number of results from the recent literature and answer in the affirmative the question raised by m benameur and t fack 2006 1

milnor fibers and symplectic fillings of quotient surface singularities abstract we determine a one to one correspondence between milnor fibers and minimal symplectic fillings of a quotient surface singularity up to diffeomorphism type by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3 folds and pinkham s negative weight smoothing as by products we show that milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions for this we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow downs from a special resolution so called the maximal resolution of the singularity which is an analogue of the one to one correspondence between the irreducible components of the reduced versal deformation space and the so called p resolutions of a quotient surface singularity

combinatorial cobordism maps in hat heegaard floer theory in a previous article sarkar and wang 15 gave a combinatorial description of the hat version of heegaard floer homology for three manifolds given a cobordism between two connected three manifolds there is an induced map between their heegaard floer homologies assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism modulo torsion under this assumption we present a procedure for finding the rank of the induced heegaard floer map combinatorially in the hat version

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

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

a construction of horikawa surface via q gorenstein smoothings we prove that the fintushel stern construction of a horikawa surface which is obtained from elliptic surface via a rational blow down surgery in the smooth category can be performed in the complex category the main technique involved is mathbb q gorenstein smoothings

the geography problem for irreducible spin four manifolds we study the geography problem for smooth irreducible simplyconnected spin four manifolds for a large class of hoinotopy types we exhibit both symplectic and non symplectic representatives we also compute the seiberg witten invariants of all the four manifolds we construct

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

lagrangian tori in homotopy elliptic surfaces let e 1 k denote the symplectic four manifold homotopy equivalent to the rational elliptic surface corresponding to a fibred knot k in s 3 constructed by r fintushel and r j stern in 1998 we construct a family of nullhomologous lagrangian tori in e 1 k and prove that infinitely many of these tori have complements with mutually non isomorphic fundamental groups if the alexander polynomial of k has some irreducible factor which does not divide t n 1 for any positive integer n we also show how these tori can be non isotopically embedded as nullhomologous lagrangian submanifolds in other symplectic 4 manifolds

on the diffeomorphism types of certain algebraic surfaces i l invariant de donaldson la geometrie de certaines surfaces rationnelles surfaces dolgachev generalites sur des formes de type 1 n la classe canonique d une chambre et un coin le cone de kahler d une bonne surface generique une etude de p cellules les super p cellules

the normalizer group of an ergodic automorphism of type iii and the commutant of an ergodic flow abstract it was proved by w krieger that for an ergodic automorphism t of type iii there is an ergodic flow f t t r whose isomorphism class uniquely determines the weak equivalence class of t it will be shown that if n t is the normalizer group of the full group t and t is the closure of t then the quotient group n t t is topologically isomorphic to the commutant c f t t r of the flow f t t r for various examples of flows f t t r the commutant c f t t r is studied

spectral triples for noncommutative solenoidal spaces from self coverings abstract examples of noncommutative self coverings are described and spectral triples on the base space are extended to spectral triples on the inductive family of coverings in such a way that the covering projections are locally isometric such triples are shown to converge in a suitable sense to a semifinite spectral triple on the direct limit of the tower of coverings which we call noncommutative solenoidal space some of the self coverings described here are given by the inclusion of the fixed point algebra in a c algebra acted upon by a finite abelian group in all the examples treated here the noncommutative solenoidal spaces have the same metric dimension and volume as on the base space but are not quantum compact metric spaces namely the pseudo metric induced by the spectral triple does not produce the weak topology on the state space

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

the best constants for operator lipschitz functions on schatten classes suppose that f is a lipschitz function on r with kfklip 1 let a be a bounded self adjoint operator on a hilbert space h let p 1 and suppose that x b h is an operator such that the commutator a x is contained in the schatten class sp it is proved by the last two authors that then also f a x sp and there exists a constant cp independent of x and f such that k f a x kp cpk a x kp the main result of this paper is to give a sharp estimate for cp in terms of p namely we show that cp p 2 p 1 in particular this gives the best estimates for operator lipschitz inequalities we treat this result in a more general setting this involves commutators of n self adjoint operators a1 an for which we prove the analogous result the case described here in the abstract follows as a special case

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

small exotic rational surfaces without 1 and 3 handles we give new rational blowdown constructions of exotic cp 2 ncp 2 5 n 9 without using elliptic fibrations we also show that our 4 manifolds admit handle decompositions without 1 and 3 handles for 7 n 9 a strategy for rational blowdown constructions of exotic cp 2 ncp 2 1 n 4 is also proposed

symplectic rational blowdowns we prove that the rational blowdown a surgery on smooth 4 manifolds introduced by fintushel and stern can be performed in the symplectic category as a consequence interesting families of smooth 4 manifolds including the exotic k3 surfaces of gompf and mrowka admit symplectic structures a basic problem in symplectic topology is to understand what smooth manifolds admit a symplectic structure a closed non degenerate 2 form in this paper we focus on this question in dimension 4 currently the primary methods for constructing smooth irreducible 4 manifolds in such a way that one can distinguish them by donaldson or seiberg witten invariants are surgery constructions that use complex manifolds as building blocks these surgery methods are smooth logarithmic transforms rational blowdowns and connect sums along surfaces it is interesting to see when these surgeries can be performed in the symplectic category in this paper we prove that performing a rational blowdown of a symplectic manifold along symplectic surfaces yields a symplectic manifold this result establishes that certain exotic 4 manifolds including the exotic k3 surfaces of gompf and mrowka 9 are symplectic in any even dimension two symplectic manifolds can be summed along codimension 2 symplectic submanifolds to yield a symplectic manifold we refer to this symplectic operation which was proposed by gromov 11 as the symplectic sum gompf 8 used the symplectic sum to construct a plethora of interesting symplectic manifolds including the first examples of simply connected symplectic 4 manifolds that are not homotopic to any complex surface and some exotic k3 surfaces more recently fintushel and stern 6 have used the connect sum along smoothly embedded tori to produce a rich class of exotic 4 manifolds some homeomorphic to a k3 surface many of which cannot admit a symplectic structure the author is grateful for the support of an nsf post doctoral fellowship dsm9627749

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

computations of instanton invariants using donaldson floer theory we compute the donaldson su 2 invariants of the double cover of 2 branched over a smooth algebraic curve of degree eight from this we deduce a formula for the relative invariants of the blow up of the gompf nucleusn2 and we show how this gives a blow up formula for a class of 4 manifolds which includes essentially all the simply connected 4 manifolds known to have big diffeomorphism group we apply the result on the nucleus also to prove a formula for the invariants of minimal simply connected elliptic surfaces which reduces the computation to the case of geometric genus one in particular we compute all the donaldson invariants of minimal simply connected elliptic surfaces without multiple fibers our main tool is donaldson floer theory

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

spectral metric spaces for gibbs measures abstract we construct spectral metric spaces for gibbs measures on a one sided topologically exact subshift of finite type that is for a given gibbs measure we construct a spectral triple and show that connes corresponding pseudo metric is a metric and that its metric topology agrees with the weak topology on the state space over the set of continuous functions defined on the subshift moreover we show that each gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant gibbs measure

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

weighted homogeneous singularities and rational homology disk smoothings we classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings the nonexistence of rational homology disk smoothings is shown by symplectic geometric methods while the existence is verified via smoothings of negative weight

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

the blowup formula for donaldson invariants k k which is calculated in 4 below this formula has been the target of much recent work the abstract fact that there exists such a formula which is independent of x was first proved by c taubes using techniques of t j bryan b and p ozsvath o have independently calculated the coefficients throughb10 x quite recently j morgan and ozsvath have announced a scheme which can recursively compute all of the bk x the special case of the blowup formula for manifolds of simple type see 5 below was first given by p kronheimer and t mrowka however none of the techniques in these cases approach the simplicity of that offered here before presenting the formula we shall first establish notation for the donaldson in variants of a simply connected 4 manifold x with b 1 and odd the hypothesis of simple connectivity is not necessary but makes the exposition easier an orientation of x together with an orientation of h 2 x r is called a homology orientation of x such a homology orientation determines the su 2 donaldson invariant a linear function d dx a x sym h0 x h2 x r

dixmier traces and residues on weak operator ideals we develop the theory of modulated operators in general principal ideals of compact operators for laplacian modulated operators we establish connes trace formula in its local euclidean model and a global version thereof it expresses dixmier traces in terms of the vector valued wodzicki residue we demonstrate the applicability of our main results in the context of log classical pseudo differential operators studied by lesch and a class of operators naturally appearing in noncommutative geometry c 2020 elsevier inc all rights reserved

singular traces and perturbation formulae of higher order abstract let h v be self adjoint operators such that v belongs to the weak trace class ideal we prove higher order perturbation formula f h v j 0 n 1 1 j d j d t j f h t v t 0 r f n t d m n t where is a trace on the weak trace class ideal and m n is a finite measure that is not necessarily absolutely continuous this result extends the first and second order perturbation formulas of dykema and shripka who generalised the krein and koplienko trace formulas to the weak trace class ideal we also establish the perturbation formulae when the perturbation v belongs to the quasi banach ideal weak l n for any n 1

the optimal range of the calderon operator and its applications abstract we identify the optimal range of the calderon operator and that of the classical hilbert transform in the class of symmetric quasi banach spaces further consequences of our approach concern the optimal range of the triangular truncation operator operator lipschitz functions and commutator estimates in ideals of compact operators

type ii non commutative geometry i dixmier trace in von neumann algebras abstract we define the notion of connes von neumann spectral triple and consider the associated index problem we compute the analytic chern connes character of such a generalized spectral triple and prove the corresponding local formula for its hochschild class this formula involves the dixmier trace for ii von neumann algebras in the case of foliations we identify this dixmier trace with the corresponding measured wodzicki residue

the local index formula in semifinite von neumann algebras ii the even case abstract we generalise the even local index formula of connes and moscovici to the case of spectral triples for a subalgebra a of a general semifinite von neumann algebra the proof is a variant of that for the odd case which appears in part i to allow for algebras with a non trivial centre we have to establish a theory of unbounded fredholm operators in a general semifinite von neumann algebra and in particular prove a generalised mckean singer formula

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

symplectic surgeries along certain singularities and new lefschetz fibrations abstract we define a new 4 dimensional symplectic cut and paste operations arising from the generalized star relations t a 0 t a 1 t a 2 t a 2 g 1 2 g 1 t b 1 t b 2 g t b 3 also known as the trident relations in the mapping class group g 3 of an orientable surface of genus g 1 with 3 boundary components we also construct new families of lefschetz fibrations by applying the generalized star relations and the chain relations to the families of words t c 1 t c 2 t c 2 g 1 t c 2 g t c 2 g 1 2 t c 2 g t c 2 g 1 t c 2 t c 1 2 n 1 t c 1 t c 2 t c 2 g t c 2 g 1 2 g 2 n 1 and t c 1 t c 2 t c 2 g 1 t c 2 g 2 2 g 1 n 1 in the mapping class group g of the closed orientable surface of genus g 1 and n 1 furthermore we show that the total spaces of some of these lefschetz fibrations are irreducible exotic symplectic 4 manifolds using the degenerate cases of the generalized star relations we also realize all elliptic lefschetz fibrations and genus two lefschetz fibrations over s 2 with non separating vanishing cycles

double node neighborhoods and families of simply connected 4 manifolds with b 1 journal of the american mathematical society volume 19 number 1 pages 171 180 s 0894 0347 05 00500 x article electronically published on august 15 2005 double node neighborhoods and families of simply connected 4 manifolds with b 1 ronald fintushel and ronald j stern 1 introduction a basic question of 4 manifold topology is whether the complex projective plane cp 2 admits exotic smooth structures thus one is interested in knowing the smallest m for which cp 2 m cp admits an exotic smooth structure in the late 1980s dieter kotschick k proved that the barlow surface which was known to be homeomorphic to cp 2 8cp is not di eomorphic to it in the following years the subject of simply connected smooth 4 manifolds with b 1 languished because of a lack of suitable examples however largely due to a beautiful paper of jongil park p who found the rst examples of exotic simply connected 4 manifolds with b 1 and b 7 the past year has found renewed interest in this subject peter ozsvath and zoltan szabo os proved that park s manifold is minimal and andras stipsicz and szabo ss used a technique similar to park s to construct an exotic manifold with b 1 and b 6 the rational elliptic surface e 1 cp 2 9 cp admits in nitely many distinct smooth structures up to now it has not been known whether cp 2 m cp can have an in nite family of smooth structures when m 9 it is the purpose of this paper to introduce a new technique which we use to show that for m 6 7 8 cp 2 m cp does have an in nite family of smooth structures the construction of these examples to some extent follows the ideas of park p after performing suitable knot surgeries fs3 on e 1 the essential new idea is that after certain knot surgeries on e 1 one is able to nd an immersed sphere of self intersection 1 representing the pseudo section this is accomplished by studying double node neighborhoods as described below shortly after the appearance of a preliminary version of this article park stip sicz and szabo pss used the techniques described herein to give even better examples an in nite family of pairwise nondi eomorphic smooth 4 manifolds all homeomorphic to cp 2 5 cp in the nal section of this paper we give a quick construction of such a family of examples received by the editors january 13 2005 2000 mathematics subject classi cation primary 14j26 53d05 57r55 57r57 key words and phrases 4 manifold seiberg witten invariant knot surgery rational blowdown the rst author was partially supported by nsf grant dms0305818 and the second author by nsf grant dms0204041 c 2005 american mathematical society reverts to public domain 28 years from publication

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

actions of _ whose ii factors and orbit equivalence relations have prescribed fundamental group we show that given any subgroup f of r which is either countable or belongs to a certain large class of uncountable subgroups there exist continuously many free ergodic measure preserving actions i of the free group with infinitely many generators f1 on probability measure spaces xi i such that their associated group measure space ii1 factors mi l 1 xi o i f1 and orbit equivalence relationsri r f1 y xi have fundamental group equal tof and with mi respectivelyri stably non isomorphic moreover these actions can be taken so thatri has no outer automorphisms and any automorphism of mi is unitary conjugate to an automorphism that acts trivially on the subalgebra l 1 xi of mi

derivations and dirichlet forms on fractals abstract we study derivations and fredholm modules on metric spaces with a local regular conservative dirichlet form in particular on finitely ramified fractals we show that there is a non trivial fredholm module if and only if the fractal is not a tree i e not simply connected this result relates fredholm modules and topology refines and improves known results on p c f fractals we also discuss weakly summable fredholm modules and the dixmier trace in the cases of some finitely and infinitely ramified fractals including non self similar fractals if the so called spectral dimension is less than 2 in the finitely ramified self similar case we relate the p summability question with estimates of the lyapunov exponents for harmonic functions and the behavior of the pressure function

on symplectic fillings of lens spaces le st be the contact structure naturally induced on the lens space l p q s 3 z pz by the standard contact structure st on the three sphere s 3 we obtain a complete classification of the symplectic fillings of l p q st up to orientation preserving diffeomorphisms in view of our results we formulate a conjecture on the diffeomorphism types of the smoothings of complex two dimensional cyclic quotient singularities

small lefschetz fibrations and exotic 4 manifolds we explicitly construct genus 2 lefschetz fibrations whose total spaces are minimal symplectic 4 manifolds homeomorphic to complex rational surfaces mathbb cp 2 p overline mathbb cp 2 for p 7 8 9 and to 3 mathbb cp 2 q overline mathbb cp 2 for q 12 ldots 19 complementarily we prove that there are no minimal genus 2 lefschetz fibrations whose total spaces are homeomorphic to any other simply connected 4 manifold with b le 3 with one possible exception when b 3 meanwhile we produce positive dehn twist factorizations for several new genus 2 lefschetz fibrations with small number of critical points including the smallest possible example which follow from a reverse engineering procedure we introduce for this setting we also derive exotic minimal symplectic 4 manifolds in the homeomorphism classes of mathbb cp 2 4 overline mathbb cp 2 and 3 mathbb cp 2 6 overline mathbb cp 2 from small lefschetz fibrations over surfaces of higher genera

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

a simply connected surface of general type with p g 0 and k 2 2 in this paper we construct a simply connected minimal complex surface of general type with pg 0 and k2 2 using a rational blow down surgery and a gorenstein smoothing theory

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p

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

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

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

the geography of spin symplectic 4 manifolds in this paper we construct a family of simply connected spin non complex symplectic 4 manifolds which cover all but finitely many allowed lattice points chi mathbf c lying in the region 0 leq mathbf c leq 8 76 chi furthermore as a corollary we prove that there exist infinitely many exotic smooth structures on 2n 1 s 2 times s 2 for all n large enough

turbulence and araki woods factors abstract using baire category techniques we prove that araki woods factors are not classifiable by countable structures as a result we obtain a far reaching strengthening as well as a new proof of the well known theorem of woods that the isomorphism problem for itpfi factors is not smooth we derive as a consequence that the odometer actions of z that preserve the measure class of a finite non atomic product measure are not classifiable up to orbit equivalence by countable structures

estimating dixmier traces of hankel operators in lorentz ideals abstract in this paper we study dixmier traces of powers of hankel operators in lorentz ideals we extend results of englis zhang to the case of powers p 1 and general lorentz ideals starting from abstract extrapolation results of gayral sukochev in the special case p 2 4 6 we give an exact formula for the dixmier trace for general p we give upper and lower bounds on the dixmier trace we also construct for any p and any lorentz ideal examples of non measurable hankel operators