spatial realisations of kms states on the c algebras of higher rank graphs abstract several authors have recently been studying the equilibrium or kms states on the toeplitz algebras of finite higher rank graphs for graphs of rank one that is for ordinary directed graphs there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line the algebras of higher rank graphs carry a gauge action of a higher dimensional torus and there are many potential dynamics arising from different embeddings of the real line in the torus previous results show that there is nonetheless a preferred dynamics for which the system exhibits a particularly satisfactory phase transition and that the unique kms state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph here we obtain a similar description of the kms state at the critical inverse temperature for other dynamics our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others our results are sharpest for the algebras of rank two graphs

kms states of quasi free dynamics on pimsner algebras abstract a continuous one parameter group of unitary isometries of a right hilbert c bimodule induces a quasi free dynamics on the cuntz pimsner c algebra of the bimodule and on its toeplitz extension the restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial and the corresponding kms states of the toeplitz cuntz pimsner and cuntz pimsner c algebras are characterized in terms of traces on the algebra of coefficients this generalizes and sheds light onto various earlier results about kms states of the gauge actions on cuntz algebras cuntz krieger algebras and crossed products by endomorphisms we also obtain a more general characterization in terms of kms weights for the case in which the inducing isometries are not unitary and accordingly the restriction of the quasi free dynamics to the algebra of coefficients is nontrivial

partial actions and kms states on relative graph c algebras abstract the relative graph c algebras introduced by muhly and tomforde are generalizations of both graph algebras and their toeplitz extensions for an arbitrary graph e and a subset r of the set of regular vertices of e we show that the relative graph c algebra c e r is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space given a time evolution on c e r induced by a function on the edge set we characterize the kms states and ground states using an abstract result of exel and laca guided by their work on kms states for toeplitz cuntz krieger type algebras associated to infinite matrices we obtain complete descriptions of the convex sets of kms states of finite type and of kms states of infinite type whose associated measures are supported on recurrent infinite paths this allows us to give a complete concrete description of the convex set of all kms states for a big class of graphs which includes all graphs with finitely many vertices

zappa szep products of semigroups and their c algebras abstract zappa szep products of semigroups provide a rich class of examples of semigroups that include the self similar group actions of nekrashevych we use li s construction of semigroup c algebras to associate a c algebra to zappa szep products and give an explicit presentation of the algebra we then define a quotient c algebra that generalises the cuntz pimsner algebras for self similar actions we indicate how known examples previously viewed as distinct classes fit into our unifying framework we specifically discuss the baumslag solitar groups the binary adding machine the semigroup n n and the a x b semigroup z z

equilibrium states and entropy theory for nica pimsner algebras abstract we study the equilibrium simplex of nica pimsner algebras arising from product systems of finite rank on the free abelian semigroup first we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube secondly we associate every gauge invariant part to a sub simplex of tracial states of the diagonal algebra we show how this parametrization lifts to the full equilibrium simplices of non infinite type the finite rank entails an entropy theory for identifying the two critical inverse temperatures a the least upper bound for existence of non finite type equilibrium states and b the least positive inverse temperature below which there are no equilibrium states at all we show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies modulo negative values thus phase transitions can happen only in between these two critical points and possibly at zero temperature

preferred traces on c algebras of self similar groupoids arising as fixed points abstract recent results of laca raeburn ramagge and whittaker show that any self similar action of a groupoid on a graph determines a 1 parameter family of self mappings of the trace space of the groupoid c algebra we investigate the fixed points for these self mappings under the same hypotheses that laca et al used to prove that the c algebra of the self similar action admits a unique kms state we prove that for any value of the parameter the associated self mapping admits a unique fixed point which is a universal attractor this fixed point is precisely the trace that extends to a kms state on the c algebra of the self similar action

short time fractional fourier transform and its applications the fractional fourier transform frft is a potent tool to analyze the chirp signal however it fails in locating the fractional fourier domain frfd frequency contents which is required in some applications the short time fractional fourier transform stfrft is proposed to solve this problem it displays the time and frfd frequency information jointly in the short time fractional fourier domain stfrfd two aspects of its performance are considered the 2 d resolution and the stfrfd support the time frfd bandwidth product tfbp is defined to measure the resolvable area and the stfrfd support the optimal stfrft is obtained with the criteria that maximize the 2 d resolution and minimize the stfrfd support its inverse transform properties and computational complexity are presented two applications are discussed the estimations of the time of arrival toa and pulsewidth pw of chirp signals and the stfrfd filtering simulations verify the validity of the proposed algorithms

kms states on c algebras associated to a family of commuting local homeomorphisms abstract we consider a family of commuting local homeomorphisms on a compact space and build a compactly aligned product system of hilbert bimodules the nica toeplitz algebra of this system carries a gauge action of a higher dimensional torus and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus we study the kms states of these dynamics for large inverse temperatures including we describe the simplex of kms states on the nica toeplitz algebra we illustrate our main theorem by considering backward shifts on the infinite path spaces of a class of k graphs whose shift maps commute

equilibrium and ground states from cayley graphs abstract we present a general framework for the study of kms states of generalized gauge actions on the c algebra of a cayley graph which is pointed by considering the neutral element of the group as a distinguished vertex we use this framework to give a concrete description of a particular kind of kms states on the c algebras that we call abelian kms states if the group is nilpotent all kms states are abelian and our analysis gives the full picture in this case we then describe the ground states that are limits of abelian kms states when the temperature goes to zero

c algebras of toeplitz type associated with algebraic number fields we associate with the ring r of algebraic integers in a number field a c algebra mathfrak t r it is an extension of the ring c algebra mathfrak a r studied previously by the first named author in collaboration with x li in contrast to mathfrak a r it is functorial under homomorphisms of rings it can also be defined using the left regular representation of the ax b semigroup r rtimes r times on ell 2 r rtimes r times the algebra mathfrak t r carries a natural one parameter automorphism group sigma t t in mathbb r we determine its kms structure the technical difficulties that we encounter are due to the presence of the class group in the case where r is not a principal ideal domain in that case for a fixed large inverse temperature the simplex of kms states splits over the class group the partition functions are partial dedekind zeta functions we prove a result characterizing the asymptotic behavior of quotients of such partial zeta functions which we then use to show uniqueness of the beta kms state for each inverse temperature beta in 1 2

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

product system models for twisted c algebras of topological higher rank graphs abstract we use product systems of c correspondences to introduce twisted c algebras of topological higher rank graphs we define the notion of a continuous t valued 2 cocycle on a topological higher rank graph and present examples of such cocycles on large classes of topological higher rank graphs to every proper source free topological higher rank graph and continuous t valued 2 cocycle c on we associate a product system x of c 0 0 correspondences built from finite paths in we define the twisted cuntz krieger algebra c c to be the cuntz pimsner algebra o x and we define the twisted toeplitz algebra t c c to be the nica toeplitz algebra nt x we also associate to and c a product system y of c 0 correspondences built from infinite paths we prove that there is an embedding of t c c into nt y and an isomorphism between c c and o y

the c algebras of finitely aligned higher rank graphs we generalise the theory of cuntz krieger families and graph algebras to the class of finitely aligned k graphs this class contains in particular all row finite k graphs the cuntz krieger relations for non row finite k graphs look significantly different from the usual ones and this substantially complicates the analysis of the graph algebra we prove a gauge invariant uniqueness theorem and a cuntz krieger uniqueness theorem for the c algebras of finitely aligned k graphs

representations of higher rank graph c algebras associated to λ semibranching function systems abstract in this paper we discuss a method of constructing separable representations of the c algebras associated to strongly connected row finite k graphs we begin by giving an alternative characterization of the semibranching function systems introduced in an earlier paper with an eye towards constructing such representations that are faithful our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions we present a variety of new examples relying on this characterization we then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row finite source free k graph

separable representations kms states and wavelets for higher rank graphs letbe a strongly connected finite higher rank graph in this paper we construct faithful representations of c on certain separable hilbert spaces of the form l 2 x by introducing the notion of a semibranching function system a generalization of the semibranching function systems studied by marcolli and paolucci in particular we obtain a faithful representation of c on l 2 1 m where m is the perron frobenius probability measure on the infinite path space 1 recently studied by an huef laca raeburn and sims we also show how a semibranching function system gives rise to kms states for c for the higher rank graphs of robertson and steger we also obtain a faithful representation of c on l 2 x where x is a fractal subspace of 0 1 by embedding 1 into 0 1 as a fractal subset x of 0 1 in this latter case we additionally show that there exists a kms state for c whose inverse temperature is equal to the hausdorff dimension of x finally we construct a wavelet system for l 2 1 m by generalizing the work of marcolli and paolucci from graphs to higher rank graphs

on c algebras of irreversible algebraic dynamical systems abstract extending the work of cuntz and vershik we develop a general notion of independence for commuting group endomorphisms based on this concept we initiate the study of irreversible algebraic dynamical systems which can be thought of as irreversible analogues of the dynamical systems considered by schmidt to each irreversible algebraic dynamical system we associate a universal c algebra and show that it is a uct kirchberg algebra under natural assumptions moreover we discuss the structure of the core subalgebra which turns out to be closely related to generalized bunce deddens algebras in the sense of orfanos we also construct discrete product systems of hilbert bimodules for irreversible algebraic dynamical systems which allow us to view the associated c algebras as cuntz nica pimsner algebras besides we prove a decomposition theorem for semigroup crossed products of unital c algebras by semidirect products of discrete left cancellative monoids

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

fractional power spectrum in this paper by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process we consider the case associated with the random process in an explicit manner the fractional power spectral relations for the fractional fourier domain filter are derived and the expression for the fractional power spectrum in terms of the fractional correlation is obtained in addition the definitions and the properties of the fractional white noise and the chirp stationary process are presented simulation results verify the theoretical derivations and demonstrate the potential applications such as detection and parameter estimation of chirp signals fractional power spectral estimation and system identification in the fractional fourier domain

a unified framework for the fractional fourier transform the paper investigates the possibility for giving a general definition of the fractional fourier transform frt for all signal classes one dimensional 1 d and multidimensional continuous and discrete periodic and aperiodic since the definition is based on the eigenfunctions of the ordinary fourier transform ft the preliminary conditions is that the signal domain periodicity be the same as the ft domain periodicity within these classes a general frt definition is formulated and the frt properties are established in addition the multiplicity which is intrinsic in a fractional operator is clearly developed the general definition is checked in the case in which the frt is presently available and moreover to establish the frt in new classes of signals

semigroups of endomorphisms dirichlet series and phase transitions abstract we study phase transitions of c dynamical systems a in which a is the crossed product of a c algebra by a lattice semigroup of endomorphisms and is a one parameter subgroup of the dual action determined by a real valued scale on the semigroup we show that the kms equilibrium condition is equivalent to a markov type condition on the predual semigroup dynamical system and give criteria for the existence of equilibrium states and of phase transitions in terms of the semigroup action when the semigroup is an integer lattice the partition function of the real scale is a dirichlet series having an euler product expansion on their convergence region there is a phase transition whose geometry is fully determined by the ground states and is independent of the chosen scale generalizing a model recently constructed by bost and connes motivated by earlier work of julia as applications we simplify part of the proof the bost connes theorem on phase transition with spontaneous symmetry breaking and we discuss generalizations to other systems associated to subsets of primes and to number fields of class number 1

phase transition on exel crossed products associated to dilation matrices an integer matrix a md z induces a covering a of td and an endomorphism a f f a of c td for which there is a natural transfer operator l in this paper we compute the kms states on the exel crossed product c td a ln and its toeplitz extension we find that c td a ln has a unique kms state which has inverse temperature log deta its toeplitz extension on the other hand exhibits a phase transition at log deta and for larger the simplex of kms states is isomorphic to the simplex of probability measures on td

c algebras of right lcm monoids and their equilibrium states we study the internal structure of c algebras of right lcm monoids by means of isolating the core semigroup c algebra as the coecient algebra of a fock type module on which the full semigroup c algebra admits a left action if the semigroup has a generalised scale we classify the kms states for the associated time evolution on the semigroup c algebra and provide sucient conditions for uniqueness of the kms state at inverse temperature in a critical interval

amenability for fell bundles given a fell bundle b over a discrete group we construct its reduced cross sectional algebra c r b in analogy with the reduced crossed products defined for c dynamical systems when the reduced and full cross sectional algebras of b are isomorphic we say that the bundle is amenable we then formulate an approximation property which we prove to be a sufficient condition for amenability a theory of graded c algebras possessing a conditional expectation is developed with an eye on the fell bundle that one naturally associates to the grading we show for instance that all such algebras are isomorphic to c r b when the bundle is amenable we also study induced ideals in graded c algebras and obtain a generalization of results of str til and voiculescu on af algebras and of nica on quasi lattice ordered groups a brief comment is made on the relevance to our theory of a certain open problem in the theory of exact c algebras an application is given to the case of an fn grading of the cuntz krieger algebras oa recently discovered by quigg and raeburn specifically we show that the cuntz krieger bundle satisfies the approximation property and hence is amenable for all matrices a with entries in 0 1 even if a does not satisfy the well known property i studied by cuntz and krieger in their paper

spectral triples and wavelets for higher rank graphs abstract in this paper we present a new way to associate a finitely summable spectral triple to a higher rank graph via the infinite path space of moreover we prove that this spectral triple has a close connection to the wavelet decomposition of which was introduced by farsi gillaspy kang and packer in 2015 we first introduce the concept of stationary k bratteli diagrams in order to associate a family of ultrametric cantor sets and their associated pearson bellissard spectral triples to a finite strongly connected higher rank graph we then study the zeta function abscissa of convergence and dixmier trace associated to the pearson bellissard spectral triples of these cantor sets and show these spectral triples are regular in the sense of pearson and bellissard we obtain an integral formula for the dixmier trace given by integration against a measure and show that is a rescaled version of the measure m on which was introduced by an huef laca raeburn and sims finally we investigate the eigenspaces of a family of laplace beltrami operators associated to the dirichlet forms of the spectral triples we show that these eigenspaces refine the wavelet decomposition of l 2 m which was constructed by farsi et al

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

c envelopes of semicrossed products by lattice ordered abelian semigroups abstract a semicrossed product is a non selfadjoint operator algebra encoding the action of a semigroup on an operator or c algebra we prove that when the positive cone of a discrete lattice ordered abelian group acts on a c algebra the c envelope of the associated semicrossed product is a full corner of a crossed product by the whole group by constructing a c cover that itself is a full corner of a crossed product and computing the shilov ideal we obtain an explicit description of the c envelope this generalizes a result of davidson fuller and kakariadis from z n to the class of all discrete lattice ordered abelian groups

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

partial dynamical systems and the kms condition given a countably infinite 0 1 matrix a without identically zero rows let a be the cuntz krieger algebra recently introduced by the authors and a be the toeplitz extension of a once the latter is seen as a cuntz pimsner algebra as recently shown by szymanski we study the kms equilibrium states of c dynamical systems based on a and a with dynamics satisfying for the canonical generating partial isometries s x and arbitrary real numbers n x 1 the kms states on both a and a are completely characterized for certain values of the inverse temperature according to the position of relative to three critical values defined to be the abscissa of convergence of certain dirichlet series associated to a and the n x our results for a are derived from those for a by virtue of the former being a covariant quotient of the latter when the matrix a is finite these results give theorems of olesen and pedersen for n and of enomoto fujii and watatani for a as particular cases

optimal filtering in fractional fourier domains for time invariant degradation models and stationary signals and noise the classical fourier domain wiener filter which can be implemented in o n log n time gives the minimum mean square error estimate of the original undistorted signal for time varying degradations and nonstationary processes however the optimal linear estimate requires o n sup 2 time for implementation we consider filtering in fractional fourier domains which enables significant reduction of the error compared with ordinary fourier domain filtering for certain types of degradation and noise especially of chirped nature while requiring only o n log n implementation time thus improved performance is achieved at no additional cost expressions for the optimal filter functions in fractional domains are derived and several illustrative examples are given in which significant reduction of the error by a factor of 50 is obtained

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

kms states on the toeplitz algebras of higher rank graphs abstract the toeplitz algebra t c for a finite k graph is equipped with a continuous one parameter group r for each r r k obtained by composing the map r t e i t r 1 e i t r k t k with the gauge action on t c in this paper we give a complete description of the kms states for the c dynamical system t c r for all finite k graphs and all values of r and r r k

kms states on nica toeplitz hbox c c algebras given a quasi lattice ordered group g p and a compactly aligned product system x of essential hbox c correspondences over the monoid p we show that there is a bijection between the gauge invariant hbox kms beta states on the nica toeplitz algebra mathcal nt x of x with respect to a gauge type dynamics on one side and the tracial states on the coefficient algebra a satisfying a system in general infinite of inequalities on the other this strengthens and generalizes a number of results in the literature in several directions we do not make any extra assumptions on p and x and our result can in principle be used to study kms states at any finite inverse temperature beta under fairly general additional assumptions we show that there is a critical inverse temperature beta c such that for beta beta c all hbox kms beta states are of gibbs type hence gauge invariant in which case we have a complete classification of hbox kms beta states in terms of tracial states on a while at beta beta c we have a phase transition manifesting itself in the appearance of hbox kms beta states that are not of gibbs type in the case of right angled artin monoids we show also that our system of inequalities for traces on a can be reduced to a much smaller system a finite one when the monoid is finitely generated most of our results generalize to arbitrary quasi free dynamics on mathcal nt x

kms states entropy and the variational principle in full c dynamical systems to any periodic and full c dynamical system an invertible operator s acting on the banach space of trace functionals of the fixed point algebra is canonically associated kms states correspond to positive eigenvectors of s a perron frobenius type theorem asserts the existence of kms states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s extremal kms states examples arising from subshifts in symbolic dynamics self similar sets in fractal geometry and noncommutative metric spaces are discussed

semigroup c algebras of ax b semigroups we study semigroup c algebras of ax b semigroups over integral domains the goal is to generalize several results about c algebras of ax b semigroups over rings of algebraic integers we prove results concerning k theory and structural properties like the ideal structure or pure infiniteness our methods allow us to treat ax b semigroups over a large class of integral domains containing all noetherian integrally closed domains and coordinate rings of affine varieties over infinite fields

von neumann algebras of strongly connected higher rank graphs we investigate the factor types of the extremal kms states for the preferred dynamics on the toeplitz algebra and the cuntz krieger algebra of a strongly connected finite k graph for inverse temperatures above 1 all of the extremal kms states are of type mathrm i infty at inverse temperature 1 there is a dichotomy if the k graph is a simple k dimensional cycle we obtain a finite type mathrm i factor otherwise we obtain a type iii factor whose connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph

type iii1 equilibrium states of the toeplitz algebra of the affine semigroup over the natural numbers abstract we complete the analysis of kms states of the toeplitz algebra t n n of the affine semigroup over the natural numbers recently studied by raeburn and the first author by showing that for every inverse temperature in the critical interval 1 2 the unique kms state is of type iii1 we prove this by reducing the type classification from t n n to that of the symmetric part of the bost connes system with a shift in inverse temperature to carry out this reduction we first obtain a parametrization of the nica spectrum of n n in terms of an adelic space combining a characterization of traces on crossed products due to the second author with an analysis of the action of n n on the nica spectrum we can also recover all the kms states of t n n originally computed by raeburn and the first author our computation sheds light on why there is a free transitive circle action on the extremal kms states for 2 that does not ostensibly come from an action of t on the c algebra

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

kms states on c algebras associated to higher rank graphs consider a higher rank graph of rank k both the cuntz krieger algebra and the toeplitz cuntz krieger algebra of the graph carry natural gauge actions of the torus tk and restricting these gauge actions to one parameter subgroups of tk gives dynamical systems involving actions of the real line we study the kms states of these dynamical systems we find that for large inverse temperatures the simplex of kms states on the toeplitz cuntz krieger algebra has dimension d one less than the number of vertices in the graph we also show that there is a preferred dynamics for which there is a critical inverse temperature c for larger than c there is a d dimensional simplex of kms states when c and the one parameter subgroup is dense there is a unique kms state and this state factors through the cuntz krieger algebra as in previous studies for k 1 our main tool is the perron frobenius theory for irreducible non negative matrices though here we need a version of the theory for commuting families of matrices

structure of free semigroupoid algebras abstract a free semigroupoid algebra is the wot closure of the algebra generated by a tck family of a graph we obtain a structure theory for these algebras analogous to that of free semigroup algebra we clarify the role of absolute continuity and wandering vectors these results are applied to obtain a lebesgue von neumann wold decomposition of tck families along with reflexivity a kaplansky density theorem and classification for free semigroupoid algebras several classes of examples are discussed and developed including self adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence

uct kirchberg algebras have nuclear dimension one we prove that every kirchberg algebra in the uct class has nuclear dimension 1 we first show that kirchberg 2 graph algebras with trivial k0 and finite k1 have nuclear dimension 1 by adapting a technique developed by winter and zacharias for cuntz algebras we then prove that every kirchberg algebra in the uct class is a direct limit of 2 graph algebras to obtain our main theorem

tensor algebras of product systems and their c envelopes abstract let g p be an abelian lattice ordered group and let x be a compactly aligned product system over p with coefficients in a we show that the c envelope of the nica tensor algebra n t x coincides with both sehnem s covariance algebra a x p and the co universal c algebra n o x r for injective gauge compatible nica covariant representations of carlsen larsen sims and vittadello we give several applications of this result on both the selfadjoint and non selfadjoint operator algebra theory first we guarantee the existence of n o x r thus settling a problem of carlsen larsen sims and vittadello which was open even for abelian lattice ordered groups as a second application we resolve a problem posed by skalski and zacharias on dilating isometric representations of product systems to unitary representations as a third application we characterize the c envelope of the tensor algebra of a finitely aligned higher rank graph which also holds for topological higher rank graphs as a final application we prove reduced hao ng isomorphisms for generalized gauge actions of discrete groups on c algebras of product systems this generalizes recent results that were obtained by various authors in the case where g p z n

fractional convolution and correlation via operator methods and an application to detection of linear fm signals using operator theory methods together with our previously introduced unitary fractional operator we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner via operator manipulations we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation through simulation examples we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation it is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function af we also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear fm signals

kms states of self similar k graph c algebras abstract let g be a countable discrete amenable group and be a strongly connected finite k graph if g is a pseudo free and locally faithful self similar action which satisfies the finite state condition then the structure of the kms simplex of the c algebra o g associated to g is described it is either empty or affinely isomorphic to the tracial state space of the c algebra of the periodicity group per g of g depending on whether the perron frobenius eigenvector of is g equivariant as applications of our main results we also exhibit several classes of important examples

method for defining a class of fractional operations the fractional fourier transform frt permits a variety of associated fractional operations this correspondence proposes a systematic method based on the structure of the frt which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection the approach also exposes the possible paths for implementing such operations

well posedness of nonlinear diffusion equations with nonlinear conservative noise we prove the pathwise well posedness of stochastic porous media and fast diffusion equations driven by nonlinear conservative noise as a consequence the generation of a random dynamical system is obtained this extends results of the second author and souganidis who considered analogous spatially homogeneous and first order equations and earlier works of lions perthame and souganidis

the boundary quotient for algebraic dynamical systems abstract we introduce the notion of accurate foundation sets and the accurate refinement property for right lcm semigroups for right lcm semigroups with this property we derive a more explicit presentation of the boundary quotient in the context of algebraic dynamical systems we also analyse finiteness properties of foundation sets which lead us to a very concrete presentation based on starling s recent work we provide sharp conditions on certain algebraic dynamical systems for pure infiniteness and simplicity of their boundary quotient

kms states on the c algebra of a higher rank graph and periodicity in the path space we study the kms states of the c algebra of a strongly connected finite k graph we find that there is only one 1 parameter subgroup of the gauge action that can admit a kms state the extreme kms states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite path space of the graph we deduce that there is a unique kms state if and only if the k graph c algebra is simple giving a complete answer to a question of yang when the k graph c algebra is not simple our results reveal a phase change of an unexpected nature in its toeplitz extension

co universal algebras associated to product systems and gauge invariant uniqueness theorems let x be a product system over a quasi lattice ordered group under mild hypotheses we associate to x a c algebra which is co universal for injective nica covariant toeplitz representations of x which preserve the gauge coaction under appropriate amenability criteria this co universalc algebra coincides with the cuntz nica pimsner algebra introduced by sims and yeend we prove two key uniqueness theorems and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co universal algebras in each case it is an easy corollary that the cuntz nica pimsner algebra is isomorphic to the corresponding full crossed product

a class of fractional integral transforms a generalization of the fractional fourier transform the paper presents a systematic and unified approach to fractional integral transforms we introduce a new class of fractional integral transforms that includes the fractional fourier and hankel transforms and the fractional integration and differentiation operators as special cases these fractional transforms may also be viewed as angular transforms indexed by an angular parameter spl alpha since their kernels are obtained by taking the limits of analytic functions in the unit disc along a radius making an angle spl alpha with the x axis

speed of propagation for hamilton jacobi equations with multiplicative rough time dependence and convex hamiltonians we show that the initial value problem for hamilton jacobi equations with multiplicative rough time dependence typically stochastic and convex hamiltonians satisfies finite speed of propagation we prove that in general the range of dependence is bounded by a multiple of the length of the skeleton of the path that is a piecewise linear path obtained by connecting the successive extrema of the original one when the driving path is a brownian motion we prove that its skeleton has almost surely finite length we also discuss the optimality of the estimate

a class of c algebras generalizing both graph algebras and homeomorphism c algebras i fundamental results we introduce a new class of c algebras which is a generalization of both graph algebras and homeomorphism c algebras this class is very large and also very tractable we prove the so called gauge invariant unique ness theorem and the cuntz krieger uniqueness theorem and compute the k groups of our algebras

applications of entropy of product systems higher rank graphs abstract we consider c algebras of finite higher rank graphs along with their rotational action we show how the entropy theory of product systems with finite frames applies to identify the phase transitions of the dynamics we compute the positive inverse temperatures where symmetry breaks and in particular we identify the subharmonic parts of the gauge invariant equilibrium states our analysis applies to positively weighted rotational actions through a recalibration of the entropies

extrapolation of bandlimited signals in linear canonical transform domain the linear canonical transform lct has been shown to be a powerful analyzing tool in signal processing many results of this transform are already known including bandlimited extrapolation the existing algorithm for solving the problem of lct bandlimited extrapolation is based on signal expansion into a series of generalized prolate spheroidal wave functions gpswfs however the requirement to compute and store the gpswfs and the errors due to the series truncation render this algorithm ill suitable for a practical implementation in this correspondence we first propose a new formulation of the gerchberg papoulis gp algorithm for lct bandlimited extrapolation then we present a fast convergence algorithm for the new formulation the classical gp algorithm related to fourier bandlimited signals is noted as a special case moreover the comparison between the proposed algorithm and the one based on gpswfs expansion is provided and the validity of the theoretical derivations is demonstrated via simulations several potential applications of the achieved theory are also presented

unified fractional fourier transform and sampling theorem the fractional fourier transform frt is an extension of the ordinary fourier transform ft applying the language of the unified ft we develop frt expressions for discrete and continuous signals introducing a particular form of periodicity chirp periodicity the frt sampling theorem is derived as an extension of its ordinary counterpart

transform order division multiplexing multiplexing in the fractional fourier domain frfd only transmits the signals in a single direction in the time frequency plane and the transform order of the fractional fourier transform frft is not used sufficiently in this paper we apply the perfect reconstruction transmultiplexer in the frfd to develop the transform order division multiplexing todm which transmits the original information in the multidirections in the time frequency plane simultaneously the proposed todm systems can be divided into two types one needs the multitransmission of the original information while the other needs the modifications of the original information by the interpolation and the inverse discrete frft the performance analysis of the todm systems is also presented including the system capacity the reconstruction accuracy the security and the limitation compared with the multiplexing in the frfd the todm is more flexible and the security of the system is highly strengthened finally simulation results confirm the obtained analytical results

boundary quotients and ideals of toeplitz c algebras of artin groups abstract we study the quotients of the toeplitz c algebra of a quasi lattice ordered group g p which we view as crossed products by a partial actions of g on closed invariant subsets of a totally disconnected compact hausdorff space the nica spectrum of g p our original motivation and our main examples are drawn from right angled artin groups but many of our results are valid for more general quasi lattice ordered groups we show that the nica spectrum has a unique minimal closed invariant subset which we call the boundary spectrum and we define the boundary quotient to be the crossed product of the corresponding restricted partial action the main technical tools used are the results of exel laca and quigg on simplicity and ideal structure of partial crossed products which depend on amenability and topological freeness of the partial action and its restriction to closed invariant subsets when there exists a generalised length function or controlled map defined on g and taking values in an amenable group we prove that the partial action is amenable on arbitrary closed invariant subsets the topological freeness of the boundary action depends on topological freeness of the restriction to a certain lattice subgroup of g the core of g p which often turns out to be trivial our main results are obtained for right angled artin groups with trivial centre that is those with no cyclic direct factor they include a presentation of the boundary quotient in terms of generators and relations that generalises cuntz s presentation of o n a proof that the boundary quotient is purely infinite and simple and a parametrisation of the ideals of the toeplitz c algebra in terms of subsets of the standard generators of the artin group

the temperature state space of a c dynamical system ii on etudie l espace des etats kms aux temperatures inverses finies d un systeme dynamique a ou a est une c algebre simple initiale et une representation d un groupe de lie comme automorphismes de a on etudie le cas ou l algebre de lie de est r n

the noncommutative choquet boundary let s be an operator system a self adjoint linear sub space of a unital c algebra a such that 1 s and a c s is generated by s a boundary representation for s is an irreducible rep resentationof c s on a hilbert space with the property that s has a unique completely positive extension to c s the set s of all unitary equivalence classes of boundary representations is the non commutative counterpart of the choquet boundary of a function system s c x that separates points of x it is known that the closure of the choquet boundary of a function system s is the y boundary of x relative to s the corresponding noncommutative problem of whether every operator system has suf ficiently many boundary representations was formulated in 1969 but has remained unsolved despite progress on related issues in particu lar it was unknown if s 6 for generic s in this paper we show that every separable operator system has sufficiently many boundary representations our methods use separability in an essential way

path dependent convex conservation laws abstract for convex scalar conservation laws in 1 d driven by a rough path z t in the sense of lions perthame and souganidis in 32 we show that it is possible to replace z t by a piecewise linear path and still obtain the same solution at a given time this result is connected to the spatial regularity of solutions we show that solutions are spatially lipschitz continuous for an a priori set of times depending on the path and the initial data fine properties of the map z u for a prescribed time are studied we provide a detailed description of the properties of the rough path z t that influence the solution this description is extracted by a factorization of the solution operator at time in a companion paper 24 we make use of the observations herein to construct computationally efficient numerical methods

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

regularization by noise for stochastic hamilton jacobi equations we study regularizing effects of nonlinear stochastic perturbations for fully nonlinear pde more precisely path by path l infty bounds for the second derivative of solutions to such pde are shown these bounds are expressed as solutions to reflected sde and are shown to be optimal

topological aperiodicity for product systems over semigroups of ore type we prove a version of uniqueness theorem for cuntz pimsner algebras of discrete product systems over semigroups of ore type to this end we introduce doplicher roberts picture of cuntz pimsner algebras and the semigroup dual to a product system of regular c correspondences under a certain aperiodicity condition on the latter we obtain the uniqueness theorem and a simplicity criterion for the algebras in question these results generalize the corresponding ones for crossed products by discrete groups due to archbold and spielberg and for exel s crossed products due to exel and vershik they also give interesting conditions for topological higher rank graphs and p graphs and apply to the new cuntz c algebra qn arising from the ax b semigroup over n

on c algebras associated with c correspondences abstract we study c algebras arising from c correspondences which were introduced by the author we prove the gauge invariant uniqueness theorem and obtain conditions for our c algebras to be nuclear exact or satisfy the universal coefficient theorem we also obtain a 6 term exact sequence of k groups involving the k groups of our c algebras

kms states on pimsner algebras associated with c dynamical systems abstract we examine the theory of the kms states on pimsner algebras arising from multivariable unital c dynamical systems as an application we show that pimsner algebras of piecewise conjugate classical systems attain the same kms states even though it is still an open problem whether or not they are isomorphic

phase transition on the toeplitz algebra of the affine semigroup over the natural numbers abstract we show that the group q q of orientation preserving affine transformations of the rational numbers is quasi lattice ordered by its subsemigroup n n the associated toeplitz c algebra t n n is universal for isometric representations which are covariant in the sense of nica we give a presentation of t n n in terms of generators and relations and use this to show that the c algebra q n recently introduced by cuntz is the boundary quotient of q q n n in the sense of crisp and laca the toeplitz algebra t n n carries a natural dynamics which induces the one considered by cuntz on the quotient q n and our main result is the computation of the kms equilibrium states of the dynamical system t n n r for all values of the inverse temperature for 1 2 there is a unique kms state and the kms 1 state factors through the quotient map onto q n giving the unique kms state discovered by cuntz at 2 there is a phase transition and for 2 the kms states are indexed by probability measures on the circle there is a further phase transition at where the kms states are indexed by the probability measures on the circle but the ground states are indexed by the states on the classical toeplitz algebra t n

equilibrium states on the cuntz pimsner algebras of self similar actions we consider a family of cuntz pimsner algebras associated to self similar group actions and their toeplitz analogues both families carry natural dynamics implemented by automorphic actions of the real line and we investigate the equilibrium states the kms states for these dynamical systems we find that for all inverse temperatures above a critical value the kms states on the toeplitz algebra are given in a very concrete way by traces on the full group algebra of the group at the critical inverse temperature the kms states factor through states of the cuntz pimsner algebra if the self similar group is contracting then the cuntz pimsner algebra has only one kms state we apply these results to a number of examples including the self similar group actions associated to integer dilation matrices and the canonical self similar actions of the basilica group and the grigorchuk group

equilibrium states on operator algebras associated to self similar actions of groupoids on graphs abstract we consider self similar actions of groupoids on the path spaces of finite directed graphs and construct examples of such self similar actions using a suitable notion of graph automaton self similar groupoid actions have a cuntz pimsner algebra and a toeplitz algebra both of which carry natural dynamics lifted from the gauge actions we study the equilibrium states the kms states on the resulting dynamical systems above a critical inverse temperature the kms states on the toeplitz algebra are parametrised by the traces on the full c algebra of the groupoid and we describe a program for finding such traces the critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph and at the critical temperature the kms states on the toeplitz algebra factor through states of the cuntz pimsner algebra under a verifiable hypothesis on the self similar action there is a unique kms state on the cuntz pimsner algebra we discuss an explicit method of computing the values of this kms state and illustrate with examples

kms states on the c algebras of finite graphs abstract we consider a finite directed graph e and the gauge action on its toeplitz cuntz krieger algebra viewed as an action of r for inverse temperatures larger than a critical value c we give an explicit construction of all the kms states if the graph is strongly connected then there is a unique kms c state and this state factors through the quotient map onto c e our approach is direct and relatively elementary

atomic representations of rank 2 graph algebras abstract we provide a detailed analysis of atomic representations of rank 2 graphs on a single vertex they are completely classified up to unitary equivalence and decomposed into a direct sum or direct integral of irreducible atomic representations the building blocks are described as the minimal dilations of defect free representations modelled on finite groups of rank 2

on k theoretic invariants of semigroup c algebras attached to number fields abstract we show that semigroup c algebras attached to a x b semigroups over rings of integers determine number fields up to arithmetic equivalence under the assumption that the number fields have the same number of roots of unity for finite galois extensions this means that the semigroup c algebras are isomorphic if and only if the number fields are isomorphic

discrete product systems and their c algebras continuous product systems or product systems over r were recently introduced by arveson to develope an index theory for continuous semigroups of endomorphisms of b h in this paper we continue his work by studying discrete product systems or product systems over g where g is a countable dense subgroup of r we give examples of discrete product systems and compute their automorphism groups to each discrete product system e over g we associate a separable c algebra oe g whose non degenerate representations correspond bijectively to representations of e this family of c algebras may be regarded as analogues of cuntz algebras or car algebras in some degenerate cases these c algebras are the generalized toeplitz algebras we show that oe g is simple and study its gauge automorphisms quasi free automorphisms and fock state