minimal dynamical systems on the product of the cantor set and the circle we prove that a crossed product algebra arising from a minimal dynamical system on the product of the cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated cantor minimal system in the case that cocycles take values in the rotation group it is also shown that this condition implies tracial rank zero and in particular the crossed product algebra is isomorphic to a unital simple at algebra of real rank zero under the same assumption we show that two systems are approximately k conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps open image in new window

fundamental group of simple c algebras with unique trace abstract we introduce the fundamental group f a of a unital simple c algebra a with a unique normalized trace we compute fundamental groups f a of several nuclear or non nuclear c algebras a k theoretical obstruction enables us to compute the fundamental group easily our study is essentially based on the computation of picard groups by kodaka

energy decay problems in the design of a point stabilizer for coupled string vibrating systems this paper solves completely the following problem posed by chen coleman and west siam j appl math 47 pp 751 780 when can we achieve the desirable uniform exponential decay property of the vibration energy for two coupled vibrating strings with a stabilizer or damper installed at the coupling point an approach using abstract semigroups is shown the paper proves the following for the cases when the stabilizers are symmetrically placed the system s energy does not decay to zero with respect to time for some initial states for all the other cases if the proportion of the wave speeds on the two vibrating strings c 2 c 1 is a rational number applying the formula of the growth order of a c 0 semigroup on a hilbgrt space derived by huang 2 proves that the semigroup satisfies the spectrum determined growth assumption so the uniform exponential decay property depends completely on the representative form of the ratio c 2 c 1 finally the paper proves that if

extensions of exchange rings we define non unital exchange rings and we prove that ifiis an ideal of a ringr thenris an exchange ring if and only ifiandr iare exchange rings and idempotents can be lifted moduloi we also show that we can replace the condition on liftability of idempotents with the condition that the canonical mapk0 r k0 r i be surjective

an adaptive observer design for n 1 coupled linear hyperbolic pdes based on swapping in this paper we use swapping design filters to bring systems of n 1 partial differential equations of the hyperbolic type to static form standard parameter identification laws can then be applied to estimate unknown parameters in the boundary conditions proof of boundedness of the adaptive laws are offered and the results are demonstrated in simulations

ranks of operators in simple c algebras let a be a unital simple separable c algebra with strict comparison of positive elements we prove that the cuntz semigroup of a is recovered functorially from the murray von neumann semigroup and the tracial state space t a whenever the extreme boundary of t a is compact and of finite covering dimension combined with a result of winter we obtain z a a whenever a moreover has locally finite decomposition rank as a corollary we confirm elliott s classification conjecture under reasonably general hypotheses which notably do not require any inductive limit structure these results all stem from our investigation of a basic question what are the possible ranks of operators in a unital simple c algebra

efficient accurate reanalysis for structural optimization the combined approximations method developed recently is an efficient reanalysis method providing high quality results through the use of this approach the computed terms of a series expansion are used as basis vectors in a reduced basis expression by solving a reduced system of equations first and second order approximations were demonstrated in previous studies for small structures the efficiency and the accuracy of the method are improved and results are illustrated for larger structures by the utilization of a gram schmidt orthogonalization procedure new set of basis vectors is generated and normalized such that the reduced system of equations becomes uncoupled the advantage in using the latter vectors is that all expressions for evaluating the displacements are explicit functions of the design variables consequently additional vectors can be considered without modifying the calculations that have already been carried out in addition the uncoupled system is more well conditioned some considerations related to the efficiency of the solution process and the accuracy of the results are discussed and the effect of various parameters on the accuracy is studied numerical results are demonstrated for several medium and large scale structures it is shown that accurate and efficient approximations are achieved for very large changes in the design

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

boundary observer design for hyperbolic pde ode cascade systems the paper presents an observer design for a class of hyperbolic pde ode cascade systems with a boundary measurement the cascade systems consist of coupled pdes featuring one rightward and one leftward convecting first order transport pdes and a set of odes which enter the pdes through the left boundary of the systems the design which is based on the volterra integral transformation relies only on a single sensor at the right boundary of the system the observer consists of a copy of the plant plus output injection terms both in the pdes and the odes the observer is constructed in a collocated setup which means both sensing and actuation are located at the same boundary the observer gains are computed analytically by solving goursat type pdes in terms of bessel function of the first kind the observer design is tested against a field scale flow loop test experiment in stavanger by statoil oil company the results show that the observer converges to the actual values and that the design can be used as a process monitoring tool in oil well drilling

output regulation for general linear heterodirectional hyperbolic systems with spatially varying coefficients abstract this article presents a backstepping solution to the output regulation problem for general linear heterodirectional hyperbolic systems with spatially varying coefficients the disturbances can act at both boundaries distributed in domain or at the output to be controlled the latter is defined at a boundary distributed or pointwise in domain and has not to be available for measurement by utilizing backstepping coordinates it is shown that all design equations are explicitly solvable this allows a simple determination of a state feedback regulator that is implemented by a reference and a disturbance observer furthermore an easy evaluation of the existence conditions for the resulting output feedback regulator is possible in terms of the plant transfer behaviour in order to facilitate the parameterization of the regulator the resulting closed loop dynamics is directly related to the design parameters the proposed backstepping based design of the output feedback regulator is demonstrated for an unstable heterodirectional 4 4 hyperbolic system

nuclear dimension and mathcal z stability simple separable unital monotracial and nuclear mathrm c algebras are shown to have finite nuclear dimension whenever they absorb the jiang su algebra mathcal z tensorially this completes the proof of the toms winter conjecture in the unique trace case

gromov translation algebras over discrete trees are exchange rings it is shown that the gromov translation ring of a discrete tree over a von neumann regular ring is an exchange ring this provides a new source of exchange rings including for example the algebras g 0 of x matrices over a field of constant bandwidth an extension of these ideas shows that for all real numbers r in the unit interval 0 1 the growth algebras g r introduced by hannah and o meara in 1993 are exchange rings consequently over a countable field countable dimensional exchange algebras can take any prescribed bandwidth dimension r in 0 1

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

new concepts of transferred and potential transferred forces in structures abstract new concepts of transferred force and potential transferred force from the loaded surface to the support surface for a structure are introduced the basic ideas of these new types of forces are explained using a general continuum model for the structure numerical procedures to calculate these forces are presented several example problems are solved and a procedure to depict transferred and potential transferred forces as force transfer diagrams is developed these forces and transfer diagrams show relative importance of various parts of the structure that can be useful in redesign or optimum design of the structure

abelian strict approximation in aw algebras and weyl von neumann type theorems in this paper for a c algebra a with m m a an aw algebra or equivalently for an essential norm closed two sided ideal a of an aw algebra m we investigate the strict approximability of the elements of m from commutative c subalgebras of a in the relevant case of the norm closed linear span a of all finite projections in a semi finite aw algebra m we shall give a complete description of the strict closure in m of any maximal abelian self adjoint subalgebra masa of a we shall see that the situation is completely different for discrete respectively continuous m in the discrete case for any masa c of a the strict closure of c is equal to the relative commutant c m while in the continuous case under certain conditions concerning the center valued quasitrace of the finite reduced algebras of m satisfied by all von neumann algebras c is already strictly closed thus in the continu ous case no elements of m which are not already belonging to a can be strictly approximated from commutative c subalgebras of a in spite of this pathology of the strict topology in the case of the norm closed linear span of all finite projections of a continuous semi finite aw algebra we shall prove that in general situations including also this case any normal y 2 m is equal modulo a to some x 2 m which belongs to an order theoretical closure of an appropriate commutative c subalgebra of a in other words if we replace the strict topology with order theoretical approximation weyl von neumann berg sikonia type theo rems will hold in substantially greater generality

k theoretically simple von neumann regular rings abstract the differences between simplicity of a von neumann regular ring and simplicity of its ordered grothendieck group k0 are investigated in particular we construct examples of stably finite regular rings which are not simple but have simple k0 all the nontrivial factor rings of these examples are directly infinite also we prove that a simple regular ring satisfies weak comparability if and only if its category of finitely generated projective modules is strictly unperforated

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

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

stabilization of a 2 2 system of hyperbolic pdes with recirculation in the unactuated channel abstract a result by guo and jin 2010 considered boundary control of a string with point damping at the exact midpoint an interpretation of the model admits a system of first order hyperbolic transport pdes with actuation at the boundary of one pde the unactuated second pde exhibits recirculation phenomena the recirculation coupling of these two pdes gives rise to a stabilization problem with nonlocal terms that prior has not been considered in this paper we consider a general class of 2 2 hyperbolic pdes where both the unactuated pde and the actuated pde have strict feedback recirculation from the outlet in the upstream direction in addition the state of the unactuated pde feeds in a non local fashion into the domain of the actuated pde we introduce a novel set of transformations through which we arrive at a simple target system with desirable exponential stability characteristics a backstepping observer and output feedback controller design are also given for this 2 2 hyperbolic pde system we then apply our control design to the string and midway antidamper application found in guo and jin and illustrate the result with simulations

simple riesz groups of rank one having wild intervals abstract we prove that every partially ordered simple group of rank one which is not riesz embeds into a simple riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals using this result we construct examples of simple riesz groups of rank one g containing unbounded intervals d n n 1 and d that satisfy a for each n 1 t d n g for every t q n but q n d n g where q n is a sequence of relatively prime integers b for every n 1 n d g we sketch some potential applications of these results in the context of k theory

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

extending states on preordered semigroups and the existence of quasitraces on c algebras abstract it is shown that every state on a subsemigroup of a scaled preordered semigroup extends to the whole semigroup extending a theorem of goodearl and handelman as an application every state on k0 a for a unital c algebra a comes from a quasitrace

brief paper arbitrary decay rate for two connected strings with joint anti damping by boundary output feedback in this paper we are concerned with the boundary stabilization of two connected strings with middle joint anti damping for which all eigenvalues of the control free system are located on the right complex plane we first design an explicit state feedback controller to achieve exponential stability for the closed loop system consequently we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using one boundary stabilizer only the output feedback can make the closed loop system exponentially stable with arbitrary decay rate

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

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

controllability and stabilizability of coupled strings with control applied at the coupled points controllability and stabilizability of a system of coupled strings with control applied at the coupled points is studied by investigating the properties of certain exponential series it is shown that the system is approximate controllable if and only if related systems of uncoupled strings do not share a common eigenvalue a sufficient condition for exact controllability is also obtained in terms of the riesz basis properties of those exponential series

stabilization of a system of n 1 coupled first order hyperbolic linear pdes with a single boundary input we solve the problem of stabilization of a class of linear first order hyperbolic systems featuring n rightward convecting transport pdes and one leftward convecting transport pde we design a controller which requires a single control input applied on the leftward convecting pde s right boundary and an observer which employs a single sensor on the same pde s left boundary we prove exponential stability of the origin of the resulting plant observer controller system in the spatial l2 sense

control of homodirectional and general heterodirectional linear coupled hyperbolic pdes research on stabilization of coupled hyperbolic pdes has been dominated by the focus on pairs of counter convecting heterodirectional transport pdes with distributed local coupling and with controls at one or both boundaries a recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport pdes that convect at different speeds against the direction of the pde whose boundary is actuated in this paper we present a solution to the fully general case in which the number of pdes in either direction is arbitrary and where actuation is applied on only one boundary to all the pdes that convect downstream from that boundary to solve this general problem we solve as a special case the problem of control of coupled homodirectional hyperbolic linear pdes where multiple transport pdes convect in the same direction with arbitrary local coupling our approach is based on pde backstepping and yields solutions to stabilization by both full state and observer based output feedback and trajectory tracking problems

boundary feedback stabilization of a class of coupled hyperbolic equations with nonlocal terms this paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations hyperbolic equations with boundary trace and integral nonlocal terms using the backstepping approach the controller is designed by formulating an integral operator whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation by applying the method of successive approximations the kernel s well posedness is given we prove the exponential stability of the origin of the system in a suitable hilbert space moreover a wave system with nonlocal terms is stabilized by applying the above result

stable range one for rings with many idempotents an associative ring r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit the purpose of this note is to prove the following facts theorem 3 an exchange ring r has stable range 1 if and only if every regular element of r is unit regular theorem 5 if r is a strongly 7r regular ring with the property that all powers of every regular element are regular then r has stable range 1 the latter generalizes a recent result of goodearl and menal 5 let r be an associative ring with identity r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit this definition is left right symmetric by vaserstein 9 theorem 2 furthermore by a theorem of kaplansky all one sided units are two sided in rings having stable range 1 cf vaserstein 10 theorem 2 6 it is well known that a von neumann regular ring r has stable range 1 if and only if r is unit regular see for example goodearl 4 proposition 4 12 call a ring r strongly 7 regular if for every element a e r there exist a number n depending on a and an element x e r such that an an lx this is in fact a two sided condition 3 it is an open question whether all strongly 7r regular rings have stable range 1 goodearl and menal 5 proved that strongly 7r regular rings are unit regular and hence have stable range 1 theorem 5 8 p 278 in this note we first extend the above result for von neumann regular rings to a larger class of rings which includes all strongly 7r regular rings 7r regular rings von neumann regular rings and algebraic algebras as an application of this we prove that a strongly 7r regular ring r has stable range 1 if powers of every regular element are regular the latter is a generalization of the abovementioned result of goodearl and menal for strongly 7r regular regular rings as one can see from our proofs rings in these classes have a large supply of idempotents throughout r stands for an associative ring with identity and j r for the jacobson radical of r modules are unitary right r modules except otherwise specified for other undefined terms readers are referred to 4 let mr be a right r module following crawley and jonsson 2 mr is said to have the exchange property if for every module ar and any two received by the editors january 24 1994 and in revised form may 3 1994 originally communicated to the proceedings of the ams by k a goodearl 1991 mathematics subject classification primary 1 6d70 1 6p70

load paths visualization in plane elasticity using load function method abstract the paper proposes a load function method for load path visualization and load flow calculation in plane elasticity problems the load functions are directly derived from 2d equilibrium equations the stresses are written in terms of the load functions and the load flow is calculated using the load function gradient the load functions are defined using generalized beltrami representation a constructive proof is given for the existence of load functions a set of poisson s equations in terms of load functions and stress components are developed and an efficient numerical procedure to solve them is discussed numerical results show the proposed load function method is able to define the load paths and obtain the load flow using an airy stress function for problems with available closed form solution and also can be easily integrated into the numerical approaches to calculate load paths for complex problems

then metric completion of regular rings we have two main objectives in this paper let r be a unit regular ring the standard reference is g1 r admits a pseudo metric topology and the completion denoted s is called the n completion of r we study the relations between r and s complementary to a recent article of goodearl on n complete regular rings the rings r s are representable as rings of sections of a sheaf like object and we clarify and extend this notion the collection of pseudo rank functions of r terms not defined here will be found in g1 denoted ip r is a compact convex set with extreme boundary denoted either oelp r or e we define n r 0 1 via n r sup p r lpslp r then n induces a pseudo metric topology on r by means of d r s n r s this is a metric precisely when n r 0 implies r 0 r is n torsion free in gh2 p 208 r has a hausdorff family of pseudorank functions we shall consider only such rings chapter 1 deals with properties of the n completion of a general unit regular ring r called s our main result here which takes the bulk of the chapter to establish is that s possesses no new pseudo rank functions i e the natural map ip s ip r is an affine homeomorphism this was claimed in h1 proposition 15 but the injectivity part of the proof has a gap it follows that s is complete in its intrinsic n metric and so the recent results of goodearl i g2 apply to s when the appropriate identifications are made the sup norm completion of the image of ko r in afflp r the affine continuous functions on ip r is just ko s some consequences of these results include i if k is a metrizable choquet simplex there is an n complete regular ring s such that ko s is order isomorphic to aft k ii if g u is an unperforated interpolation group with order unit see ehs then the sup norm closure of its image in its natural representation as a group of affine functions on a simplex is an interpolation group

compensating a string pde in the actuation or sensing path of an unstable ode how to control an unstable linear system with a long pure delay in the actuator path this question was resolved using predictor or finite spectrum assignment designs in the 1970s here we address a more challenging question how to control an unstable linear system with a wave partial differential equation pde in the actuation path physically one can think of this problem as having to stabilize a system to whose input one has access through a string the challenges of overcoming string wave dynamics in the actuation path include their infinite dimension finite propagation speed of the control signal and the fact that all of their infinitely many eigenvalues are on the imaginary axis in this technical note we provide an explicit feedback law that compensates the wave pde dynamics at the input of an linear time invariant ordinary differential equation and stabilizes the overall system in addition we prove robustness of the feedback to the error in a priori knowledge of the propagation speed in the wave pde finally we consider a dual problem where the wave pde is in the sensing path and design an exponentially convergent observer

dye s theorem and gleason s theorem for aw algebras abstract we prove that any map between projection lattices of a w algebras a and b where a has no type i 2 direct summand that preserves orthocomplementation and suprema of arbitrary elements is a restriction of a normal jordan homomorphism between a and b this allows us to generalize dye s theorem from von neumann algebras to a w algebras we show that mackey gleason bunce wright theorem can be extended to homogeneous a w algebras of type i the interplay between dye s theorem and gleason s theorem is shown as an application we prove that jordan homomorphisms are commutatively determined another corollary says that jordan parts of a w algebras can be reconstructed from posets of their abelian subalgebras

products of idempotents in regular rings ii abstract this paper characterizes products of idempotents in von neumann regular rings which are unit regular or right self injective for unit regular rings the minimum number of idempotents needed in such a product is determined thereby generalizing a result of c s ballantine products of idempotent matrices linear algebra appl 19 1978 81 86 in the case of a matrix with entries from a field

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

boundary stabilization of wave equation with velocity recirculation nonlocal terms have been the mainstay of the applications of partial differential equation pde backstepping methods to parabolic pdes the problem of similar nonlocal terms for wave equations is still open for wave equations similar nonlocal terms have not been studied in this paper we open the topic of exploration of control of wave pdes with nonlocal terms this paper is concerned with the wave equation with in domain feedback recirculation of a boundary velocity with a spatially constant coefficient due to this nonlocal term the passivity of the wave equation is destroyed we first design an explicit state feedback controller to achieve exponential stability for the closed loop system then we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using two measurements only the output feedback makes the closed loop system exponentially stable

the corona factorization property and refinement monoids the corona factorization property of a c algebra originally defined to study extensions of c algebras has turned out to say something important about intrinsic struc tural properties of the c algebra we show in this paper that a unital c algebra a of real rank zero has the corona factorization property if and only if its monoid v a of murray von neumann equivalence classes of projections in matrix algebras over a has a certain rather weak comparability property that we call the corona factorization property for monoids we show that a projection in such a c algebra is properly infinite if and only if a multiple of it is properly infinite the latter result is obtained from some more general result we establish about conical refinement monoids we show that the set of order units together with the zero element in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility and if u is an element in a refinement monoid such that nu is properly infinite then u can be written as a sum u s t such that ns and nt are properly infinite

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

continuous quasi states on c algebras a quasi state on d is a positive quasi linear functional p on d that satisfies the normalization condition sup p a ia e d a 0 ia i 1 1 if d is unital with unit 1 this is equivalent to the condition p l 1 in this paper we are concerned with the question of when positive quasilinear functionals on c algebras are linear in view of the condition iii and since positive quasi linear functionals are scalar multiples of quasistates this reduces to the question of additivity of quasi states on the self adjoint elements of a c algebra this question and its various forms depending both on the nature of quasi states and on the structure of the underlying algebra were considered before 1 3 4 6 7 9 13 17 181 substantial progress had been made in a number of cases one of the most remarkable advances was the pioneering work of gleason 6 in which the question was settled in the allirmative for quasistates on b x the algebra of all bounded operators acting on a hilbert space 2 with the property of being completely additive on orthogonal projections provided the dimension of y is different from 2 in the twodimensional case the answer is in general negative one can construct discontinuous quasi states by simple geometrical arguments so that one cannot expect all quase states to be linear on c algebras that admit two dimensional irreducible representations following gleason s result contributions to the problem were made by aarnes l and gunson 7 and in recent years the general problem for von neumann algebras was 111

on rings whose projective modules have the exchange property it is well known that all projective modules over a perfect ring have the exchange property this was shown by yamagata 32 and independently by harada and ishii lo provided that the identity of the ring can be expressed as a sum of orthogonal primitive idempotents the converse is also true 32 that it fails in general was demonstrated by kutami and oshiro ls who constructed a nonartinian boolean ring whose projective modules nevertheless have the exchange property as it turned out this outsider example is representative of an extensive class of rings in fact both perfect and regular rings are subsumed in the hypotheses of the following

minimum time control of heterodirectional linear coupled hyperbolic pdes we solve the problem of stabilizing a general class of linear first order hyperbolic systems considered systems feature an arbitrary number of coupled transport pdes convecting in either direction using the backstepping approach we derive a full state feedback law and a boundary observer enabling stabilization by output feedback unlike previous results finite time convergence to zero is achieved in the theoretical lower bound for control time

local exponential h2 stabilization of 2x2 quasilinear hyperbolic systems using backstepping in this work we consider the problem of boundary stabilization for a quasilinear 2 times2 system of first order hyperbolic pdes we design a new full state feedback control law with actuation on only one end of the domain which achieves h 2 exponential stability of the closed loop system our proof uses a backstepping transformation to find new variables for which a strict lyapunov function can be constructed the kernels of the transformation are found to verify a goursat type 4 times4 system of first order hyperbolic pdes whose well posedness is shown using the method of characteristics and successive approximations once the kernels are computed the stabilizing feedback law can be explicitly constructed from them

metric completions of ordered groups and k0 of exchange rings we give a description of the closure of the natural affine continuous function representation of k0 r for any exchange ring r this goal is achieved by extending the results of goodearl and handelman about metric completions of dimension groups to a more general class of pre ordered groups which includes k0 of exchange rings as a consequence the results about k 0 of regular rings which the author gave in an earlier paper can be extended to a wider class of rings which includes c algebras of real rank zero among others also the framework of pre ordered groups developed here allows other potential applications

metrically complete regular rings this paper is concerned with the structure of those von neumann regular rings r which are complete with respect to the weakest metric derived from the pseudo rank functions on r known as the n metric it is proved that this class of regular rings includes all regular rings with bounded index of nilpotence and all 80 continuous regular rings the major tool of the investigation is the partially ordered grothendieck group ko r which is proved to be an archimedean normcomplete interpolation group such a group has a precise representation as affine continuous functions on a choquet simplex from earlier work of the author and d e handelman and additional aspects of its structure are derived here these results are then translated into ring theoretic results about the structure of r for instance it is proved that the simple homomorphic images of r are right and left self injective rings and r is a subdirect product of these simple self injective rings also the isomorphism classes of the finitely generated projective r modules are determined by the isomorphism classes modulo the maximal two sided ideals of r as another example of the results derived it is proved that if all simple artinian homomorphic images of r are n x n matrix rings for some fixed positive integer n then r is an n x n matrix ring all rings in this paper are associative with 1 and all modules are unital right modules for the overall theory of regular rings we refer the reader to 2 for the general development of ko of regular rings as partially ordered abelian groups and the theory of partially ordered abelian groups via their state spaces we refer the reader to 2 4 in particular these references should be consulted for more detail on definitions and concepts which are just sketched here i n completeness completeness of a regular ring with respect to a rank function or with respect to a family of pseudo rank functions implies that the ring is right and left self injective 2 theorems 19 7 and 20 8 hence a considerable amount of structure theory is available for such rings 2 chapters 9 12 the purpose of this paper is to investigate a much broader class of regular rings namely those which are complete with respect to the pseudo metric obtained from the supremum n of all pseudo rank functions on the ring in particular all regular rings complete with respect to a family of pseudo rank functions are n complete but also as we prove later in this section all regular rings with bounded index of nilpotence and all 8 0 continuous regular rings are n complete received by the editors january 13 1981 and in revised form april 3 1981 1980 mathematics subject classification primary 16a30 16a54 06f20

simple regular rings satisfying weak comparability abstract open problem 3 in goodearl s book on von neumann regular rings asks if a directly finite simple regular ring must be unit regular this paper gives a positive answer in the case where the ring satisfies a very weak comparability condition no directly finite simple regular rings are known to fail this comparability condition

recasting the elliott conjecture let a be a simple unital finite and exact c algebra which absorbs the jiang su algebra mathcal z tensorially we prove that the cuntz semigroup of a admits a complete order embedding into an ordered semigroup which is obtained from the elliott invariant in a functorial manner we conjecture that this embedding is an isomorphism and prove the conjecture in several cases in these same cases mathcal z stable algebras all we prove that the elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker outside the class of mathcal z stable c algebras this weaker conjecture has no known counterexamples and it is plausible that none exist thus we reconcile the still intact principle of elliott s classification conjecture that mathrm k theoretic invariants will classify separable and nuclear c algebras with the recent appearance of counterexamples to its strongest concrete form

the cancellation theorem for projective modules over irrational rotation c algebras on montre que deux modules projectifs sur a qui sont isomorphes de facon stable sont en fait isomorphes

an infinite family of non isomorphic c algebras with identical k theory we exhibit a countably infinite family of simple separable nuclear and mutually non isomorphic c algebras which agree on k theory and traces the algebras do not absorb the jiang su algebra z tensorially answering a question of n c phillips they are also pairwise shape and morita equivalent confirming a conjecture from our earlier work the distinguishing invariant is the radius of comparison a non stable invariant of the cuntz semigroup

adaptive output feedback stabilization of linear 2 2 hyperbolic systems using anti collocated sensing and control abstract we construct a control law that manages to adaptively stabilize a class of linear 2 2 hyperbolic systems of partial differential equations pdes from a single boundary sensing anti collocated with the boundary where actuation takes place we do this by introducing a series of invertible transformations that bring the system into an observer canonical form for which adaptive control design becomes feasible we establish pointwise boundedness of all signals in the closed loop system and pointwise convergence of the system states to zero the theory is demonstrated in a simulation

products of idempotent linear transformations in 1966 j m howie characterised the transformations of an arbitrary set that can be written as a product under composition of idempotent transformations of the same set in 1967 j a erdos considered the analogous problem for linear transformations of a finite dimensional vector space and in 1983 r j dawlings investigated the corresponding idea for bounded operators on a separable hilbert space in this paper we study the case of arbitrary vector spaces