stability in the cuntz semigroup of a commutative c algebra let a be ac algebra the cuntz semigroup w a is an analogue for positive elements of the semigroup v a of murray von neumann equivalence classes of projections in matrices over a we prove stability theorems for the cuntz semigroup of a commutative c algebra which are analogues of classical stability theorems for topological vector bundles over compact hausdorff spaces let sdg denote the class of simple unital and infinite dimensional ah algebras with slow dimension growth and let a be an element of sdg we apply our stability theorems to obtain the following i a has strict comparison of positive elements ii w a is recovered functorially from the elliott invariant of a iii the lower semicontinuous dimension functions on a are weak dense in the dimension functions on a iv the dimension functions on a form a choquet simplex statement ii confirms a conjecture of perera and the author while statements iii and iv confirm for sdg conjectures of blackadar and handelman from the early 1980s

commutators and linear spans of projections in certain finite c algebras assume that a is a unital separable simple c algebra with real rank zero stable rank one strict comparison of projections and that its tracial simplex t a has a finite number of extremal points we prove that every self adjoint element a in a with a 0 for all t a is the sum of two commutators in a and that every positive element of a is a linear combination of projections with positive coefficients assume that a is as above but unital and not necessarily unital then an element resp a positive element a of a is a linear combination resp a linear combination with positive coefficients of projections if and only if ra for every t a where denotes the extension of to a tracial weight on a and ra a denotes the range projection of a assume that a is unital and as above but t a has infinitely many extremal points then a is not the linear span of its projections this result settles two open problems of marcoux in 32

on tracial approximation let c be a class of unital c algebras the class tac of c algebras which can be tracially approximated in the egorov like sense first considered by lin by the c algebras in c is studied lin considered the case that c consists of finite dimensional c algebras or the tensor products of such with c 0 1 in particular the question is considered whether for any simple separable a tac there is a c algebra b which is a simple inductive limit of certain basic homogeneous c algebras together with c algebras in c such that the elliott invariant of a is isomorphic to the elliott invariant of b an interesting case of this question is answered in the final part of the paper the question is also considered which properties of c algebras are inherited by tracial approximation results of this kind are obtained which are used in the proof of the main theorem of the paper and also in the proof of the classification theorem of the second author given in z niu a classification of tracially approximately splitting tree algebra in preparation and z niu a classification of certain tracially approximately subhomogeneous c algebras phd thesis university of toronto 2005 which also uses the main result of the present paper

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

classification of a class of crossed product c algebras associated with residually finite groups a residually finite group acts on a profinite completion by left translation we consider the corresponding crossed product c algebra for discrete countable groups that are central extensions of finitely generated abelian groups by finitely generated abelian groups these are automatically residually finite we prove that all such crossed products are classifiable by k theoretic invariants using techniques from the classification theory for nuclear c algebras

projection decomposition in multiplier algebras in this paper we present new structural information about the multiplier algebra mathcal m mathcal a of a unital purely infinite simple c algebra mathcal a by characterizing the positive elements a in mathcal m mathcal a that are strict sums of projections belonging to mathcal a if a not in mathcal a and a itself is not a projection then the necessary and sufficient condition for a to be a strict sum of projections belonging to mathcal a is that a 1 and that the essential norm a ess geq 1 based on a generalization of the perera rordam weak divisibility of separable simple c algebras of real rank zero to all unital simple c algebras of real rank zero we show that every positive element of mathcal a with norm 1 can be approximated by finite sums of projections based on block tri diagonal approximations we decompose any positive element a in mathcal m mathcal a with a 1 and a ess geq 1 into a strictly converging sum of positive elements in mathcal a with norm 1

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

a class of simple c algebras with stable rank one abstract we study the limits of inductive sequences a i i where each a i is a direct sum of full matrix algebras over compact metric spaces and each partial map of i is diagonal we give a new characterisation of simplicity for such algebras and apply it to prove that the said algebras have stable rank one whenever they are simple and unital significantly our results do not require any dimension growth assumption

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

c algebras generated by stable relations abstract presentations of certain c algebras such as c0 0 1 mn by generators and relations are proven to be stable by the weakest definition of stability a set p1 pn of star polynomials is stable if operators a1 an which almost satisfy these relations meaning pi a1 an can be perturbed within c a1 an to satisfy these relations exactly homotopy and perturbation results for these c algebras are corollaries in particular we prove some new results about the dimension drop c algebras a n c 0 1 m n 0 1 c i we show that an is semiprojective answering a question of effros and kaminker also we show that whenever an b is a homomorphism approximately contained in a sub c algebra b0 b there is a nearby homomorphism of an into b0 this answers a question raised by elliott we prove technical results regarding mn c0 0 1 which should be useful elsewhere first we show this is projective meaning a star homomorphism mn c0 0 1 b j always lifts to a star homomorphism mn c0 0 1 b secondly we show that for a finitely generated m n a m n c 0 0 1 c 0 0 1 a if the amalgamation is done properly

mean dimension and ah algebras with diagonal maps mean dimension for ah algebras with diagonal maps is introduced it is shown that if a simple unital ah algebra with diagonal maps has mean dimension zero then it has strict comparison on positive elements in particular the strict order on projections is determined by traces moreover a lower bound of the mean dimension is given in term of toms comparison radius using classification results if a simple unital ah algebra with diagonal maps has mean dimension zero it must be an ah algebra without dimension growth two classes of ah algebras with diagonal maps are shown to have mean dimension zero the class of ah algebras with at most countably many extremal traces and the class of ah algebras with numbers of extreme traces which induce same state on the k0 group being uniformly bounded in particular this class includes ah algebras with real rank zero

theory of strongly irreducible operators and its applications in the matrix theory of finite dimensional space the famous jordan canonical form theorem sufficiently reveals the internal structure of matrices in the survey we will give similar jordan canonical form theorems of operator theory in a complex separable infinite dimensional hilbert space by the way we also show their applications in other mathematical branches

classification of simple c algebras and higher dimensional noncommutative tori we show that unital simple c algebras with tracial topological rank zero which are locally approximated by subhomogeneous c algebras can be classified by their ordered k theory we apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered k theory in particular irrational higher dimensional noncommutative tori of the form c k are in fact inductive limits of circle algebras

k group and similarity classification of operators abstract let h be a complex separable hilbert space and l h denote the collection of bounded linear operators on h an operator a in l h is said to be a cowen douglas operator if there exist a connected open subset of complex plane c and n a positive integer such that a a z c a z is not invertible b ran a z y a z x y x h h for z in c z ker a z h and d dim ker a z n for z in in the paper we give a similarity classification of cowen douglas operators by using the ordered k group of the commutant algebra as an invariant and characterize the maximal ideals of the commutant algebras of cowen douglas operators the theorem greatly generalizes the main result in canada j math 156 4 2004 742 by simply removing the restriction of strong irreducibility of the operators the research is also partially inspired by the recent classification theory of simple ah algebras of elliott gong in documenta math 7 2002 255 on the classification of simple inductive limit c algebras ii the isomorphism theorem preprint also see j funct anal 1998 1 ann math 144 1996 497 amer j math 1996 187

classification of homomorphisms and dynamical systems let a be a unital simple c algebra with tracial rank zero and let x be a compact metric space suppose that h 1 h 2 c x a are two unital monomorphisms we show that h 1 and h 2 are approximately unitarily equivalent if and only if h 1 h 2 in kl c x a and o h 1 f o h 2 f for every f c x and every trace r of a inspired by a theorem of tomiyama we introduce a notion of approximate conjugacy for minimal dynamical systems let x be a compact metric space and let x x be two minimal homeomorphisms using the above mentioned result we show that two dynamical systems are approximately conjugate in that sense if and only if a k theoretical condition is satisfied in the case that x is the cantor set this notion coincides with the strong orbit equivalence of giordano putnam and skau and the k theoretical condition is equivalent to saying that the associate crossed product c algebras are isomorphic another application of the above mentioned result is given for c dynamical systems related to a problem of kishimoto let a be a unital simple ah algebra with no dimension growth and with real rank zero and let a aut a we prove that if r fixes a large subgroup of k 0 a and has the tracial rokhlin property then ax z is again a unital simple ah algebra with no dimension growth and with real rank zero

optimal and near optimal advection diffusion finite difference schemes ii unsteadiness and non uniform grid it is shown how time dependent coefficients and non uniform unsteady grid spacing modifies the optimal weights in a 3 spatial points 2 time step implicit approximation for the numerical solution of the equation t c c u x c k 2 x c q x f 2 x g at the grid points the extremely high order of approximation for the numerical solutions is such that if loss of accuracy is to be avoided then interpolation must use values extending beyond the local 3 by 2 computational module illustrative examples show that reasonable accuracy is possible with extremely long time steps on sparse non uniform moving grids

classification of unital extensions of at algebras abstract we prove that the six term exact sequence in k theory together with the elliott invariants of the ideal and quotient is a complete invariant for certain unital extensions of a t algebras up to stable isomorphism we also give some necessary and sufficient conditions for such extensions being a t algebras or quasidiagonal extensions and prove the above invariant with the equivalence class of unit is a complete invariant for unital quasidiagonal extensions of a t algebras up to isomorphism

the classification of simple separable unital z stable locally ash algebras let a be a unital simple separable locally approximately subhomogeneous c algebra locally ash algebra it is shown that a qa q can be tracially approximated by unital elliott thomsen algebras with trivial k1k1 group where q is the universal uhf algebra in particular it follows that a is classifiable by the elliott invariant if a is jiang su stable

distance between unitary orbits of normal elements in simple c algebras of real rank zero let x y x y be two normal elements in a unital simple c c algebra a we introduce a function d c x y d c x y and show that in a unital simple af algebra there is a constant 1 c 0 1 c 0 such that c d c x y dist u x u y d c x y c d c x y dist u x u y d c x y where u x u x and u y u y are the closures of the unitary orbits of x and of y respectively we also generalize this to unital simple c c algebras with real rank zero stable rank one and weakly unperforated k 0 k 0 group more complicated estimates are given in the presence of non trivial k 1 k 1 information

optimal and near optimal advection diffusion finite difference schemes i constant coefficient in one dimension an implicit finite difference scheme is derived which achieves exact results for the area centroid variance skewness and kurtosis flatness of solutions to the unforced advection diffusion equation t c u c u x c s 2x c 0 sufficient conditions for computational stability are that the grid spacing j x and time step j t be small enough that u j x s 121 2 u j t j x 1 21 2 at a loss of exactness in the kurtosis flatness the upper bound on j x can be removed and the upper bound on j t relaxed to u j t j x 1 for forcing terms the spatial moments are very accurate rather than exact

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

k theoretic rigidity and slow dimension growth let a be an approximately subhomogeneous ash c algebra with slow dimension growth we prove that if a is unital and simple then the cuntz semigroup of a agrees with that of its tensor product with the jiang su algebra mathcal z in tandem with a result of w winter this yields the equivalence of mathcal z stability and slow dimension growth for unital simple ash algebras this equivalence has several consequences including the following classification theorem unital ash algebras which are simple have slow dimension growth and in which projections separate traces are determined up to isomorphism by their graded ordered k theory and none of the latter three conditions can be relaxed in general

almost commuting unitaries and classification of purely infinite simplec algebras we show that for any irrational rotational algebraa a o2 o2 this is proved by combining recently established results forc algebras of real rank zero with the following result for anye 0 there is 0 such that for any pair of unitariesu vin any purely infinite simplec algebraawithk1 a 0 satisfies uv vu and u v 0 there is a pair of commuting unitariesu v awith finite spectrum such that u u eand v v e more general results concerning almost commuting unitaries are also obtained

classification of simple c algebras with unique traces we classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following let a and b be two unital separable simple c algebras of real rank zero stable rank one with weakly unperforated k 0 groups and with unique normalized quasi traces suppose that both a and b are locally ah and k a k a 1 a k b k b 1 b then a is isomorphic to b

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

universal coefficient theorems for the stable ext groups let a be a unital separable nuclear c algebra and let b be a stable c algebra using k theory and kk theory we establish universal coefficient theorems for the stable ext groups of unital extensions of a by b when a and b have certain properties which generalize a result of l brown and m dadarlat for the strong ext groups the class of extensions being studied are also enlarged

simple nuclear c algebras of tracial topological rank one we give a classification theorem for unital separable nuclear c algebras with tracial rank no more than one let a and b be two unital separable simple nuclear c algebras with tr a tr b 1 which satisfy the universal coefficient theorem we show that a b if and only if there is an order and unit preserving isomorphism 0 1 2 k0 a k0 a 1a k1 a t a k0 b k0 b 1b k1 b t b where 2 1 x 0 x for each x k0 a and t b

asymptotically unitary equivalence and asymptotically inner automorphisms let c be a unital ah algebra and let a be a unital separable simple c algebra with tracial rank zero suppose that phi 1 phi 2 colon c to a are two unital monomorphisms we show that there is a continuous path of unitaries u t colon t in 0 infty of a such that lim t to infty u t phi 1 a u t phi 2 a quad hbox rm for all quad a in c if and only if phi 1 phi 2 in kk c a tau circ phi 1 tau circ phi 2 for all tau in t a and the rotation map tilde eta phi 1 phi 2 associated with phi 1 and phi 2 is zero in particular an automorphism alpha on a unital separable simple c algebra a in cal n with tracial rank zero is asymptotically inner if and only if alpha rm id a quad rm in quad kk a a and the rotation map tilde eta phi 1 phi 2 is zero let a be a unital ah algebra not necessarily simple and let alpha in aut a be an automorphism as an application we show that the associated crossed product a rtimes alpha bbb z can be embedded into a unital simple af algebra if and only if a admits a strictly positive alpha invariant tracial state

dimension growth for c algebras we introduce the growth rank of a c algebra a valued invariant whose minimal instance is equivalent to the condition that an algebra absorbs the jiang su algebra tensorially and prove that its range is exhausted by simple nuclear c algebras as consequences we obtain a well developed theory of dimension growth for approximately homogeneous ah c algebras establish the existence of simple nuclear and non stable c algebras which are not tensorially prime and show the assumption of stability to be particularly natural when seeking classification results for nuclear c algebras via k theory the properties of the growth rank lead us to propose a universal property which can be considered inside any class of unital and nuclear c algebras we prove that satisfies this universal property inside a large class of locally subhomogeneous algebras representing the first uniqueness theorem for which does not depend on the classification theory of nuclear c algebras

mathbb z actions on ah algebras and mathbb z 2 actions on af algebras we consider mathbb z actions single automorphisms on a unital simple ah algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the rohlin property under some technical assumptions moreover two mathbb z actions with the rohlin property on such a c algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent we also prove that locally approximately inner and uniformly outer mathbb z 2 actions on a unital simple af algebra with a unique trace have the rohlin property and classify them up to cocycle conjugacy employing the orderext group as classification invariants

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

similarity classification of holomorphic curves abstract let h be a complex separable hilbert space for an open connected subset of c we shall say that a map f gr n h is a holomorphic curve if there exist n holomorphic h valued functions 1 2 n on such that f 1 n where gr n h denotes the grassmann manifold the set of all n dimensional subspaces of h in the paper we give a similarity classification of some holomorphic curves by using the k group of its commutant algebra as an invariant

the classification of certain non simple c algebras of tracial rank zero abstract it is proved that the z 2 graded ordered k theory group with order unit k a k a 1 a is a complete invariant for the class of unital c algebras which are inductive limits of finite direct sums of unital amenable separable simple c algebras with tracial rank zero which satisfy the uct

crossed products by minimal homeomorphisms let x be an infinite compact metric space with finite covering dimension and let h x x be a minimal homeomorphism we show that the associated crossed product c algebra a 1 4 c z x h has tracial rank zero whenever the image of k0 a in a t a is dense as a consequence we show that these crossed product c algebras are in fact simple ah algebras with real rank zero when x is connected and h is further assumed to be uniquely ergodic then the above happens if and only if the rotation number associated to h has irrational values by applying the classification result for nuclear simple c algebras with tracial rank zero we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered k theory

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

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

optimal and near optimal advection diffusion finite difference schemes vii radionuclide chain transport optimally accurate numerical schemes are derived for the concentrations of successive isotopes in a radionuclide chain as they transmute are sorbed into the rock matrix and are transported and spread out with the groundwater flow a key step is a change of dependent variables based on classical work by bateman that non local change of dependent variables can be performed numerically to the requisite high accuracy using the corresponding steady numerical scheme an exact point release solution derived in an appendix is used to demonstrate the accuracy of the numerical scheme with large time steps and low spatial resolution

finite group actions on c algebras with the rohlin property ii abstract we classify finite group actions on some classes of c algebras with the rohlin property in terms of the k groups a group cohomological obstruction is obtained for the k groups of a c algebra allowing such an action for example the tate cohomology groups with the k groups as the coefficient modules must vanish model actions with the rohlin property are constructed for given k theoretical invariants quasi free actions of finite cyclic groups on the cuntz algebras and their dual actions are investigated from the view point of the rohlin property