maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

controlling chaos and deterministic directed transport in inertia ratchets using backstepping control we address the problem of controlling chaotic motion and deterministic directed transport in inertia ratchets we employ a recursive backstepping nonlinear control technique to control intermittent chaos and then track a desired trajectory by means of the same technique for the parameter regime where two non identical attractors coexist in phase space we propose a new backstepping control scheme that is capable of controlling the directed transport exhibited by these attractors numerical simulations show that the controllers are singularity free and the closed loop systems are globally stable

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

stability of group relations under small hilbert schmidt perturbations abstract if matrices almost satisfying a group relation are close to matrices exactly satisfying the relation then we say that a group is matricially stable here almost and close are in terms of the hilbert schmidt norm using tracial 2 norm on i i 1 factors we similarly define i i 1 factor stability for groups our main result is that all 1 relator groups with non trivial center are i i 1 factor stable many of them are also matricially stable and rfd for amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters each character must be a pointwise limit of traces of finite dimensional representations this allows us to prove matricial stability for the discrete heisenberg group h 3 and for all virtually abelian groups for non amenable groups the same approximation property is a necessary condition for being matricially stable we study this approximation property and show that rf groups with character rigidity have it

graph c algebras with real rank zero given a row finite directed graph e a universal c algebra c e generated by a family of partial isometries and projections subject to the relations determined by e is associated to the graph e the cuntz krieger algebras are those graph c algebras associated to some finite graphs we prove that a graph c algebra c e has real rank zero in the sense that the set of invertible self adjoint elements is dense in the set of all self adjoint elements in c e or in the unitization of c e if c e is nonunital if and only if e satisfies a loop condition k that is analogous to the condition for a finite 0 1 matrix a under which cuntz analyzed the ideal structure of the cuntz krieger algebra oa

purely infinite partial crossed products let a g be a partial dynamical system we show that there is a bijective correspondence between g invariant ideals of a and ideals in the partial crossed product a rg provided the action is exact and residually topologically free assuming in addition a technical condition automatic when a is abelian we show that a rg is purely infinite if and only if the positive nonzero elements in a are properly infinite in a rg as an application we verify pure infiniteness of various partial crossed products including realisations of the cuntz algebras on oa on and oz as partial crossed products

the nuclear dimension of graph c algebras abstract consider a graph c algebra c e with a purely infinite ideal i possibly all of c e such that i has only finitely many ideals and c e i is approximately finite dimensional we prove that the nuclear dimension of c e is 1 if i has infinitely many ideals then the nuclear dimension of c e is either 1 or 2

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

general stability analysis of synchronized dynamics in coupled systems we consider the stability of synchronized states including equilibrium point periodic orbit or chaotic attractor in arbitrarily coupled dynamical systems maps or ordinary differential equations we develop a general approach based on the master stability function and gershgorin disk theory to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics systems with specific coupling schemes are used as examples to illustrate our general method

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

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

synchronization multistability and basin crisis in coupled pendula abstract the synchronization dynamics of two linearly coupled pendula is studied in this paper based on the lyapunov stability theory and linear matrix inequality lmi some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k th for the on set of full synchronization is obtained the numerical value of k th determined from the average energies of the systems is in good agreement with theoretical analysis prior to the on set of synchronization the boundary crisis of the chaotic attractor is identified in the bistable states where two asymmetric periodic attractors co exist it is shown that the coupled pendula can attain multistable states via a new dynamical transition the basin crisis that occur prior to the on set of stable synchronization the essential feature of basin crisis is that the two co existing attractors are destroyed while new three or more co existing attractors of the same or different periodicity are created in addition the linear perturbation technique and the routh hurwitz criteria are employed to investigate the stability of steady states and clearly identify the different types of bifurcations likely to be encountered finally two parameter phase plots show various regions of chaos hyperchaos and periodicity

weakly clean rings abstract we study the class of weakly clean rings which were introduced in 15 it is known that weakly clean rings are a subclass of exchange rings and that they contain clean rings as a proper subclass in this paper we prove that weakly clean rings also contain some other important examples of exchange rings such as regular rings and c algebras of real rank zero further we prove that many classes of weakly clean rings can be viewed as corners of clean rings this for example implies that every regular ring and every c algebra of real rank zero is a corner of a clean ring lastly we study the question when the ideal extension of weakly clean rings is weakly clean and we give an example of a non weakly clean exchange ring answering the question in 15

af embeddings into c algebras of real rank zero abstract it is proved that every separable c algebra of real rank zero contains an af sub c algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two c algebras and such that every projection in a matrix algebra over the large c algebra is equivalent to a projection in a matrix algebra over the af sub c algebra this result is proved at the level of monoids using that the monoid of murray von neumann equivalence classes of projections in a c algebra of real rank zero has the refinement property as an application of our result we show that given a unital c algebra a of real rank zero and a natural number n then there is a unital homomorphism mn1 mnr a for some natural numbers r n1 nr with nj n for all j if and only if a has no representation of dimension less than n

additive maps preserving jordan zero products on nest algebras abstract let alg n and alg m be nest algebras associated with the nests n and m on banach spaces assume that n n and m m are complemented whenever n n and m m let alg n alg m be a unital additive surjection it is shown that preserves jordan zero products in both directions that is a b b a 0 ab ba 0 if and only if is either a ring isomorphism or a ring anti isomorphism particularly all unital additive surjective maps between hilbert space nest algebras which preserves jordan zero products are characterized completely

the hasse principle for systems of diagonal cubic forms we establish the hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r times r submatrix thereby achieving the theoretical limit of the circle method for such systems

diagonality inc algebras soit a une c algebre et a sa les elements hermitiens soit i un ideal ferme separable dans une c algebre a et on suppose que les elements de a sa de spectres finis sont denses dans a sa si a a sa et e 0 alors il existe b a sa diagonal relativement a i et il existe x i sa tel que x e et a b x

on infinite simple c algebras abstract we show that simple c algebras with an infinite element contain nontrivial projections we also prove that various apparently different hypotheses purely infinite on simple c algebras are all equivalent

classifying c algebras with both finite and infinite subquotients abstract we give a classification result for a certain class of c algebras a over a finite topological space x in which there exists an open set u of x such that u separates the finite and infinite subquotients of a we apply our results to c algebras arising from graphs

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

bilinear forms on matrix algebras vanishing on zero products of xy and yx abstract let d be a division algebra finite dimensional over its center c m m d the m m matrix algebra over d and v be a vector space over c we characterize all n linear forms on in terms of reduced traces and elementary operators for m 1 it is proved that a bilinear form b v vanishes on zero products of xy and yx if and only if there exist linear maps g h v such that b x y g x y h y x for all x y as an application a bilinear form b is completely characterized if b x y 0 whenever x y satisfy x y y x 0 where is a fixed nonzero element in c

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

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

controlling current reversals in synchronized underdamped ratchets a pair of underdamped ratchets coupled via a perturbed asymmetric potential is shown to make a transition to a fully synchronized state wherein stable controlled transport is achieved when the coupling strength exceeds a threshold at which the collective dynamics is attained this transition to collective transport is connected to a chaos periodic quasiperiodic bifurcation in which current reversal is completely eliminated based on the lyapunov stability theory and linear matrix inequalities we give some necessary and sufficient criteria for stable controlled transport and obtain an exact analytic estimate of the threshold kth for the onset of stable controlled current

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

k theory of certain purely infinite crossed products it was shown by rordam and the second named author that a countable group g admits an action on a compact space such that the crossed product is a kirchberg algebra if and only if g is exact and non amenable this construction allows a certain amount of choice we show that different choices can lead to different algebras at least with the free group

subconvexity for additive equations pairs of undenary cubic forms we investigate pairs of diagonal cubic equations with integral coefficients for a class of such diophantine systems with 11 or more variables we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude

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

on waring s problem for cubes and smooth weyl sums non trivial estimates for fractional moments of smooth cubic weyl sums are developed complemented by bounds for such sums of use on both the major and minor arcs in a hardy littlewood dissection these estimates are applied to derive an upper bound for the s th moment of the smooth cubic weyl sum of the expected order of magnitude as soon as s ge 7 691 related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers all of whose prime divisors are at most exp c log n log log n 1 2 for a suitable positive number c this conclusion improves a previous result of g harcos in which nine cubes are required 1991 mathematics subject classification 11p05 11l15 11p55

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

jordan zero product preserving additive maps on operator algebras let a b be an additive surjective map between some operator algebras such that ab ba 0 implies a b b a 0 we show that under some mild conditions is a jordan homomorphism multiplied by a central element such operator algebras include von neumann algebras c algebras and standard operator algebras etc particularly if h and k are infinite dimensional real or complex hilbert spaces and a b h and b b k then there exists a nonzero scalar c and an invertible linear or conjugate linear operator u h k such that either a cuau 1 for all a b h or a cua u 1 for all a b h

on the ideal of compact operators on a hilbert c module abstract this note studies certain properties of the ideal k e consisting of compact operators on e where e is a hilbert c module over a c algebra a it is proved that this ideal has p unit simpleness separability real rank zero and stable rank one under certain conditions an counterexample is given to illustrate that two hilbert c modules may be not unitarily equivalent when their ideals of compact operators are isomorphic hereditary c subalgebras of such an ideal are also characterized

the hasse principle for pairs of diagonal cubic forms by means of the hardy littlewood method we apply a new mean value theorem for exponential sums to confirm the truth over the rational numbers of the hasse principle for pairs of diagonal cubic forms in thirteen or more variables

bifurcation and chaos in coupled ratchets exhibiting synchronized dynamics the bifurcation and chaotic behavior of unidirectionally coupled deterministic ratchets is studied as a function of the driving force amplitude a and frequency a classification of the various types of bifurcations likely to be encountered in this system was done by examining the stability of the steady state in linear response as well as constructing a two parameter phase diagram in the a plane numerical explorations revealed varieties of bifurcation sequences including quasiperiodic route to chaos besides the familiar perioddoubling and crises route to chaos exhibited by the one dimensional ratchet were also found in addition the coupled ratchets display symmetry breaking saddle nodes and bubbles of bifurcations chaotic behavior is characterized by using the lyapunov exponent spectrum while a perusal of the phase space projected in the poincare cross section confirms some of the striking features

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

zero product preserving additive maps on symmetric operator spaces and self adjoint operator spaces in this note we characterize the additive maps on the space of symmetric operators and the space of self adjoint operators which preserve zero products in both directions and the additive maps on the space of self adjoint operators which preserve jordan zero products in both directions we also give a complete classification of additive maps on the von neumann algebra of all bounded linear operators acting on a hilbert space which preserve square zero in both directions or preserve operators annihilated by a polynomial in both directions

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

the asymptotic formulae in waring s problem for cubes write rs n for the number of representations of a natural number as the sum of s positive integral cubes we investigate the higher moments of rs n establishing asymptotic formulae for n n rs n h when s 5 and h 3 when s 6 and h 28 5 and when s 7 and h 12 hitherto such an asymptotic formula was available only when s 4 and h 2

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

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

bifurcation analysis of two coupled periodically driven duffing oscillators bifurcation diagrams and phase diagrams of two coupled periodically driven identical duffing oscillators are presented it is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single periodically driven strictly dissipative oscillators the subpattern itself however is different from that of a single duffing oscillator due in particular to hopf bifurcations that are newly added to the bifurcation scenario

groups quasidiagonality and interpolation by multiplier projections we relate the following conditions on a a unital c algebra a with the fs property a k1 a 0 b every projection in m a a lifts c the general weyl von neumann theorem holds in m a any selfadjoint element h in m a can be written as h a lpi a for some selfadjoint element a in a some bounded real sequence ai and some mutually orthogonal projections pi in a with ez1 pi 1 d m a has fs and e interpolation by multiplier projections holds for any closed projections p and q in a with pq o there is a projection r in m a such that p b 4 a and that a x b if in addition a is stable combining the above results we obtain counterexamples to the conjecture of g k pedersen a has fs m a has fs for example the stabilized buncededdens algebras hence the generalized weyl von neumann theorem does not generally hold in l ha for a unital c algebras with fs

rational solutions of pairs of diagonal equations one cubic and one quadratic we obtain an essentially optimal estimate for the moment of order 32 3 of the exponential sum having argumentx 3 x 2 subject to modest local solubility hypotheses we thereby establish that pairs of diagonal diophantine equations one cubic and one quadratic possess non trivial integral solutions whenever the number of variables exceeds 10

when almost multiplicative morphisms are close to homomorphisms absrract it is shown that approximately multiplicative contractive positive morphisms from c x with dim x 2 into a simple c algebra a of real rank zero and of stable rank one are close to homomorphisms provided that certain k theoretical obstacles vanish as a corollary we show that a homomorphism h c x a is approximated by homomorphisms with finite dimensional range if h gives no k theoretical obstacle

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

on maps preserving zeros of lie polynomials of small degrees we describe maps preserving zeros of multilinear lie polynomials of degrees 3 and 4 on prime algebras and matrices over unital algebras in particular our theorems generalize several results related to commutativity preserving maps

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

exponential rank and exponential length of operators on hilbert c modules in this article we consider unitary operators on the hilbert c module ha associated with a large class of c algebras let l x denote the algebra consisting of all bounded linear operators on a separable hilbert space 7 and let k denote the algebra of all compact operators on r halmos and kakutani proved that every unitary operator on r is a product of four symmetries cf ha 143 haka for the case when x is a compact hausdorff space ringrose r recently considered the c algebra c x l q consisting of all norm continuous maps from x to l x and proved that every unitary element of c x l x can be written as a product of at most three exponentials in other words a norm continuous map u from x to the unitary group of l x forms a product of three norm continuous maps written as exp ihi exp ih2 exp ih3 where hl h2 and h3 are maps from x to the space of self adjoint operators in l x phillips and ringrose phr conjectured that every unitary of c x l x can even be approximated in the norm by products of two exponentials using techniques in the theory of c algebras but by a different approach we generalized ringrose s result to hilbert c modules associated with all a unital in particular all unital c algebras cf z8 we also proved in z8 similar results for the c algebra consisting of norm continuous maps from x to a purely infinite simple c algebra the purpose of this article is to settle the phillips ringrose conjecture in the affirmative and to prove a generalized version of the halmos kakutani theorem in haka we consider the group of unitary operators on hilbert c modules associated with the following two specific classes which together comprise the majority of interesting c algebras