strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

the cone of lower semicontinuous traces on a c algebra the cone of lower semicontinuous traces is studied with a view to its use as an invariant its properties include compactness hausdorffness and continuity with respect to inductive limits a suitable notion of dual cone is given the cone of lower semicontinuous 2 quasitraces on a non exact c algebra is considered as well these results are applied to the study of the cuntz semigroup it is shown that if a c algebra absorbs the jiang su algebra then the subsemigroup of its cuntz semigroup consisting of the purely non compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2 quasitraces this yields a computation of the cuntz semigroup for the following two classes of c algebras c algebras that absorb the jiang su algebra and have no non zero simple subquotients and simple c algebras that absorb the jiang su algebra 1 introduction the most standard invariants in the classification of nu clear c algebras are their k groups and their traces the traces at least for simple c algebras are assumed to be bounded in the unital case and lower semicontinuous and densely finite in the non unital case if one has in mind the classification of non simple c algebras it is clear that these two kinds of traces will not suffice and a broader class should be considered in this paper we study the properties of the cone of all lower semicontinuous traces on a c algebra with the purpose of applying our results to questions in the classification of non simple c algebras one sign that this is natural is that this cone in particular determines the lattice of closed two sided ideals an important invariant in its own right see 3 4 we shall also consider lower semicontinuous 2 quasitraces since these appear naturally as functionals on the cuntz semigroup of the algebra if the

towards a theory of classification abstract the well known difficulties arising in a classification which is not set theoretically trivial involving what is sometimes called a non smooth quotient have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor the very existence of which is already a surprise here the notion of such a functor is developed abstractly and a number of examples are considered including those which have arisen for various classes of operator algebras

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

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

infinite non simple c algebras absorbing the cuntz algebra o abstract the first named author has given a classification of all separable nuclear c algebras a that absorb the cuntz algebra o we say that a absorbs o if a is isomorphic to a o motivated by this classification we investigate here if one can give an intrinsic characterization of c algebras that absorb o this investigation leads us to three different notions of pure infiniteness of a c algebra all given in terms of local algebraic conditions on the c algebra the strongest of the three properties strongly purely infinite is shown to be equivalent to absorbing o for separable nuclear c algebras that either are stable or have an approximate unit consisting of projections in a previous paper 2000 amer j math 122 637 666 we studied an intermediate and perhaps more natural condition purely infinite that extends a well known property for simple c algebras the weakest condition of the three weakly purely infinite is shown to be equivalent to the absence of quasitraces in an ultrapower of the c algebra the three conditions may be equivalent for all c algebras and we prove this to be the case for c algebras that are either simple of real rank zero or approximately divisible

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

the krull dimension of power series rings over almost dedekind domains abstract let d be an almost dedekind domain that is not dedekind m be a non invertible maximal ideal of d x be an indeterminate over d and d x be the power series ring over d we first construct an example of 1 sets and we then use this 1 set and m to give a simple proof of dim d x 2 1 we show that ht m x m d x 2 1 when d has only countably many non invertible maximal ideals or when m is countably generated we finally construct a simple example of almost dedekind domains that are not dedekind with given cardinal number of non invertible maximal ideals

classification of certain infinite simple c algebras the class of c algebras that arise as the crossed product of a stable simple af algebra with a z action determined by an automorphism which maps a projection in the algebra onto a proper subprojection is proved to consist of simple purely infinite c algebras and a specific subclass of it is proved to be classified by k theory this subclass is large enough to exhaust all possible k groups if g0 and g1 are countable abelian groups with g1 torsion free as it must be then there is a c algebras a in the classified subclass with k0 a g0 and k 1 a g1 the subclass contains the cuntz algebras o with n even and the cuntz krieger algebras o a with k0 o a of finite old order and it is closed under forming inductive limits the c algebras in the classified subclass can be viewed as classifiable models in a strong sense of general simple purely infinite c algebras with the same k theory

krull dimension of power series rings over non sft domains abstract a ring d is called an sft ring if for each ideal i of d there exist a finitely generated ideal j of d with j i and a positive integer k such that a k j for all a i for a cardinal number and a ring d we say that dim d if d has a chain of prime ideals with length arnold showed that if d is a non sft ring then dim d x 0 let c be the class of non sft domains the class c includes the class of finite dimensional nondiscrete valuation domains the class of non noetherian almost dedekind domains the class of completely integrally closed domains that are not krull domains the class of integral domains with non noetherian prime spectrum and the class of integral domains with a nonzero proper idempotent ideal the ring of algebraic integers the ring of integer valued polynomials on z and the ring of entire functions are also members of the class c in this paper we prove that dim d x 2 1 for every d c and that under the continuum hypothesis 2 1 is the greatest lower bound of dim d x for d c on the other hand there exists a finite dimensional sft domain d such that dim d x 2 1

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

stability for integer actions on uhf c algebras on montre qu il est possibile d obtenir une version norme continue de la solution de connes a la 1 cohomologie qui est adaptable a la situation des c algebres on exhibe diverses actions modeles stables

a universal multicoefficient theorem for the kasparov groups that holds in the same generality as the universal coefficient theorem of rosenberg and schochet there are advantages in some circumstances to using hom k a k b in place of kk a b these advantages derive from the fact that k a can be equipped with order and scale structures similar to those on k0 a with this additional structure the module k a becomes a powerful invariant of c algebras we show that it is a complete invariant for the class of real rank zero ad algebras the ad algebras are a certain kind of approximately subhomogeneous c algebras which may have torsion in k1 ell in addition to classifying these algebras we calculate their automorphism groups up to approximately innerautomorphisms

krull dimension of a power series ring over a valuation domain abstract let v be a one dimensional nondiscrete valuation domain and let v v 0 we prove that krull dim v x v 2 1 which is an analogue of the fact that krull dim e 2 1 where e is the ring of entire functions the lower bound 2 1 is sharp in fact if v is countable then krull dim v x v 2 1 under the continuum hypothesis we construct a chain of prime ideals in v x with length 2 1 such that each prime ideal in the chain has height 2 1 and contracts to 0 in v we also show that for a finite dimensional valuation domain w either krull dim w x or krull dim w x 2 1

equivariant homomorphisms rokhlin constraints and equivariant uhf absorption abstract we classify equivariant homomorphisms between c dynamical systems associated to actions of finite groups with the rokhlin property in addition the given actions are classified an obstruction is obtained for the cuntz semigroup of a c algebra allowing such an action we also obtain an equivariant uhf absorption result

pullbacks c x algebras and their cuntz semigroup abstract in this paper we analyse the structure of the cuntz semigroup of certain c x algebras for compact spaces of low dimension that have no k 1 obstruction in their fibres in a strong sense the techniques developed yield computations of the cuntz semigroup of some surjective pullbacks of c algebras as a consequence this allows us to give a complete description in terms of semigroup valued lower semicontinuous functions of the cuntz semigroup of c x a where a is a not necessarily simple c algebra of stable rank one and vanishing k 1 for each closed two sided ideal we apply our results to study a variety of examples

constructing chains of primes in power series rings for an integral domain d of dimension n the dimension of the polynomial ring d x is known to be bounded by n 1 and 2n 1 while n 1 is a lower bound for the dimension of the power series ring d x it often happens that d x has infinite chains of primes for example such chains exist if d is either an almost dedekind domain that is not dedekind or a rank one nondiscrete valuation domain one concern here is developing schemes by which such chains can be constructed in d x when d is an almost dedekind domain a consequence of these constructions is that there are chains of primes similar to the set of 1 transfinite sequences of 0 s and 1 s ordered lexicographically

an algebraic approach to the radius of comparison the radius of comparison is an invariant for unital c algebras which extends the theory of covering dimension to noncommutative spaces we extend its definition to general c algebras and give an algebraic as opposed to functional theoretic reformulation this yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces we then give several applications of these results new examples of c algebras with finite radius of comparison are given and the question of when the cuntz classes of finitely generated hilbert modules form a hereditary subset of the cuntz semigroup is addressed most interestingly perhaps we treat the question of when a full hereditary subalgebra b of a stable c algebra a is itself stable giving a characterization in terms of the radius of comparison we also use the radius of comparison to quantify the least n for which a c algebra d without bounded 2 quasitraces or unital quotients has the property that mn d is stable

isomorphism of hilbert modules over stably finite c algebras abstract it is shown that if a is a stably finite c algebra and e is a countably generated hilbert a module then e gives rise to a compact element of the cuntz semigroup if and only if e is algebraically finitely generated and projective it follows that if e and f are equivalent in the sense of coward elliott and ivanescu cei and e is algebraically finitely generated and projective then e and f are isomorphic in contrast to this we exhibit two cei equivalent hilbert modules over a stably finite c algebra that are not isomorphic

asymptotic behaviour of output feedback for a class of non deterministic discrete event systems a class of non deterministic discrete event systems des where the point probability distributions are known is considered the non deterministic des models are introduced by using minimax algebra to formalize the treatment of time sequences and by the usual classical algebra to introduce a markovian structure that assigns the transition probabilities to events based on these representations the closed non deterministic des models are derived by output feedback control the asymptotic behaviour of the system is investigated to predict the long range performance of discrete event processes the results are illustrated by some examples of flexible manufacturing systems

krull dimension of power series rings abstract an ideal i of a commutative ring r with identity is called an sft strong finite type ideal if there exist a finitely generated ideal j of r with j i and a positive integer k such that a k j for each a i a ring r is called an sft ring if every ideal of r is an sft ideal arnold showed that if r is a non sft ring then the krull dimension of the power series ring r x is infinite let c be the class of non sft rings in this paper we show that dim r x 2 1 for every r c in other words if r is a non sft ring then there exists a chain of prime ideals in r x with length at least 2 1 we also prove that under the continuum hypothesis 2 1 is the greatest lower bound of dim r x for r c if m is a non sft maximal ideal of a ring r such that m is the radical of a countably generated ideal then we construct a chain p of prime ideals in r x with length at least 2 1 lying between m r x and m x i e m r x p m x for each the same result holds when m is the non sft maximal ideal of a zero dimensional quasi local ring or a one dimensional quasi local domain r

discrete event systems with stochastic processing times discrete event dynamic systems are studied in which the underlying algebra is the max algebra and the coefficients in the system referring to processing times in practice are stochastic the processing times and or the transportation times within a network show stochastic fluctuations the restrictions are that the stochastic processing times of the nodes in the network are independent and identically distributed the asymptotic behavior of the system is investigated and the average duration of one cycle of the process is calculated a specific example of the theory is considered the state space is two dimensional and the probability distributions are exponential it is shown that the process approaches a stationary limit as time proceeds the case when the probability distributions are discrete is also treated several examples are given two dimensional systems and more generally finite dimensional systems are considered

finite group actions on certain stably projectionless c algebras with the rohlin property we introduce the rohlin property and approximate representability for finite group actions on stably projectionless c algebras and study their basic properties we give some examples of finite group actions on the razakjacelon algebra w and show some classification results of these actions this study is based on the work of izumi robert s classification theorem and kirchberg s central sequence c algebras

brief paper infinitesimal and finite perturbation analysis for queueing networks the sample path perturbation analysis technique is extended to include finite and possibly large perturbations typically introduced by changes in queue sizes or other parameters it is shown that there is a natural hierarchy of perturbation analysis which takes care of increasingly large perturbations experiments with zero infinitesimal and first order finite perturbation analysis show that significant accuracy improvement can be obtained with small increase in computational effort

sur la classification des automorphismes periodiques des c algebres uhf abstract we obtain a complete classification up to conjugacy and up to outer conjugacy of finite tensor product type automorphisms of uhf c algebras which are periodic of period n n prime