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

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

purely infinite simple c algebras that are principal groupoid c algebras abstract from a suitable groupoid g we show how to construct an amenable principal groupoid whose c algebra is a kirchberg algebra which is kk equivalent to c g using this construction we show by example that many uct kirchberg algebras can be realised as the c algebras of amenable principal groupoids

kumjian pask algebras of locally convex higher rank graphs the kumjian pask algebra of a higher rank graph generalises the leavitt path algebra of a directed graph we extend the definition of kumjian pask algebra to row finite higher rank graphs with sources which satisfy a local convexity condition after proving versions of the graded uniqueness theorem and the cuntz krieger uniqueness theorem we study the kumjian pask algebra of rank 2 bratteli diagrams by studying certain finite subgraphs which are locally convex we show that the desourcification procedure of farthing and webster yields a row finite higher rank graph without sources such that the kumjian pask algebras of and are morita equivalent we then use the morita equivalence to study the ideal structure of the kumjian pask algebra of by pulling the appropriate results across the equivalence

kumjian pask algebras of finitely aligned higher rank graphs abstract we extend the definition of kumjian pask algebras to include algebras associated to finitely aligned higher rank graphs we show that these kumjian pask algebras are universally defined and have a graded uniqueness theorem we also prove the cuntz krieger uniqueness theorem to do this we use a groupoid approach as a consequence of the graded uniqueness theorem we show that every kumjian pask algebra is isomorphic to the steinberg algebra associated to its boundary path groupoid we then use steinberg algebra results to prove the cuntz krieger uniqueness theorem and also to characterize simplicity and basic simplicity

approximating commuting operators the problem of approximating m tuples of commuting n n complex matrices by commuting m tuples of generic matrices is studied we narrow the gap for commuting triples by showing that they can be perturbed if n 29

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

category equivalences involving graded modules over path algebras of quivers abstract let q be a finite quiver with vertex set i and arrow set q 1 k a field and k q its path algebra with its standard grading this paper proves some category equivalences involving the quotient category qgr k q gr k q fdim k q of graded k q modules modulo those that are the sum of their finite dimensional submodules namely qgr k q mod s q gr l q mod l q 0 qgr k q n here s q lim end k i k q 1 n is a direct limit of finite dimensional semisimple algebras q is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from q l q is the leavitt path algebra of q l q 0 is its degree zero component and q n is the quiver whose incidence matrix is the n th power of that for q it is also shown that all short exact sequences in qgr k q the full subcategory of finitely presented objects in qgr k q split consequently qgr k q can be given the structure of a triangulated category with suspension functor the serre degree twist 1 this triangulated category is equivalent to the singularity category d b d perf where is the radical square zero algebra k q k q 2 and d b is the bounded derived category of finite dimensional left modules

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

polynomials in a matrix and its commutant abstract given any norm u on the space of linear transformations matrices over a finite dimensional complex vector space and given commuting transformations t s and any e 0 there is a transformation t e such that u t e t p e such that p e t e s this result has implications for the theory of the numerical radius and other operator radii its proof involves the following fact of interest in itself if the jordan canonical form of s has k blocks then the transformations with k distinct eigenvalues are dense in the commutant of s

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

functions of sheaves and drinfeld modules in this paper we apply dwork s p adic methods to study the meromorphic continuation and rationality of various l functions arising from 7r adic galois representations drinfeld modules and p sheaves as a consequence we prove some conjectures of goss about the rationality of the local l function and the meromorphic continuation of the global l function attached to a drinfeld module let fq be a finite field of q elements with characteristic p let 7r be a prime of the polynomial ring a fq t let a be the completion of the ring a at 7r this is an analogue of the classical ring 7p of p adic integers let x be an irreducible algebraic variety defined over fq and let 7r x be the arithmetic fundamental group of x ifq with respect to some base point the group 7r x may be regarded as the galois groiip of a separable closure of the function field of x fq modulo the inertia groups at the closed points of x lq suppose now that we are given a continuous 7r adic representation

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

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

the interplay between steinberg algebras and skew rings abstract we study the interplay between steinberg algebras and skew rings for a partial action of a group in a hausdorff locally compact totally disconnected topological space we realize the associated partial skew group ring as a steinberg algebra over the transformation groupoid attached to the partial action we then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings over commutative algebras in terms of continuous orbit equivalence of the associated partial actions finally we show that any steinberg algebra associated to a hausdorff ample groupoid can be seen as a partial skew inverse semigroup ring

a variety of commuting triples abstract the problem of approximating triples of commuting n n matrices by generic matrices is equivalent to the problem of whether the variety c 3 n of commuting triples is irreducible the answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well all the proofs are given over an arbitrary algebraically closed field of characteristic zero

semigroupoid c algebras abstract a semigroupoid is a set equipped with a partially defined associative operation given a semigroupoid we construct a c algebra o from it we then present two main examples of semigroupoids namely the markov semigroupoid associated to an infinite 0 1 matrix and the semigroupoid associated to a row finite higher rank graph without sources in both cases the semigroupoid c algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object namely the cuntz krieger algebras and the higher rank graph c algebras respectively in the case of a higher rank graph d it follows that the dimension function d is superfluous for defining the corresponding c algebra

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

jordan forms for mutually annihilating nilpotent pairs abstract in this paper we completely characterize all possible pairs of jordan canonical forms for mutually annihilating nilpotent pairs i e pairs a b of nilpotent matrices such that ab ba 0

on varieties of commuting triples abstract the problem of approximating triples of commuting n n matrices by triples of generic matrices is equivalent to that whether the variety c 3 n of triples of commuting matrices is irreducible it is known that the variety is irreducible for n 7 and reducible for n 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that c 3 8 is irreducible

primitive ideal space of higher rank graph c algebras and decomposability abstract in this paper we describe primitive ideal space of the c algebra c associated to any locally convex row finite k graph to do this we will apply the farthing s desourcifying method on a recent result of carlsen kang shotwell and sims we also characterize certain maximal ideals of c furthermore we study the decomposability of c we apply the description of primitive ideals to show that if i is a direct summand of c then it is gauge invariant and isomorphic to a certain k graph c algebra so we may characterize decomposable higher rank c algebras by giving necessary and sufficient conditions for the underlying k graphs moreover we determine all such c algebras which can be decomposed into a direct sum of finitely many indecomposable c algebras

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

commuting pairs in the centralizers of 2 regular matrices inmn k kan algebraically closed field we call a matrixl regular if each eigenspace is at mostl dimensional we prove that the variety of commuting pairs in the centralizer of a 2 regular matrix is the direct product of various affine spaces and various determinantal varieties zl mobtained from matrices over truncated polynomial rings we prove that these varieties zl mare irreducible and apply this to the case of thek algebra generated by three commuting matrices we show that if one of the three matrices is 2 regular then the algebra has dimension at mostn we also show that such an algebra is always contained in a commutative subalgebra ofmn k of dimension exactlyn

finite simple labeled graph c algebras of cantor minimal subshifts abstract it is well known that a simple graph c algebra is either af or purely infinite in this paper we address the question of whether this is the case for labeled graph c algebras which include all graph c algebras and matsumoto algebras of subshifts there have been various c algebra constructions associated with subshifts and some of them are known to have the crossed products c x t z of cantor minimal subshifts x t as their quotient algebras we show that such a simple crossed product c x t z can be realized as a labeled graph c algebra since this c algebra is known to be an a t algebra and has z as its k 1 group our result provides a family of simple finite non af unital labeled graph c algebras

uniformizable families of t motives abelian t modules and the dual notion of t motives were introduced by anderson as a generalization of drinfeld modules for such anderson defined and studied the important concept of uniformizability it is an interesting question and the main objective of the present article to see how uniformizability behaves in families since uniformizability is an analytic notion we have to work with families over a rigid analytic base we provide many basic results and in fact a large part of this article concentrates on laying foundations for studying the above question building on these we obtain a generalization of a uniformizability criterion of anderson and among other things we establish that the locus of uniformizability is berkovich open

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

global l functions over function fields let c be a smooth projective curve over a finite field k with a marked point and let a be the ring of regular functions on c r furthermore let c be the completion of an algebraic closure of the function field k of c at a place above in this situation goss attaches a global l function to any family of drinfeld a modules via an infinite euler product on the domain s c zp cf 11 as in the classical situation it converges on some half plane of s a similar procedure yields for any closed point v of c and any family m of a motives a global lfunction l v m s which converges on a half plane of a suitably defined domain sv 12 8 the theme of the current article is to derive some consequences for such global l functions from the theory of crystals over function fields introduced by r pink and the current author in 2 this theory encompasses a motives as defined by anderson 1 indeed any family of a motives on a scheme x is represented by an a crystal on x furthermore for any compactifiable morphism f y x there is a functor direct image with compact support from a crystals on y to a crystals on x at least in the derived context for such f a trace formula is given in 2 for l functions of crystals it is this trace formula and the relations between goss global l functions and l functions of a crystals which we will mainly exploit slightly generalizing goss definition we obtain for any a scheme x of finite type over k any a crystal f on x and any closed point v of c a vadic l function l v f s as an euler product which converges on some half plane of sv cf definition 2 8 our principal goal is to prove that any such function has a meromorphic essentially algebraic continuation to all of sv in the sense of 12 8 along the way we will prove some interesting

generalised morphisms of k graphs k morphs in a number of recent papers k l graphs have been constructed from k graphs by inserting new edges in the last l dimensions these constructions have been motivated by c algebraic considerations so they have not been treated systematically at the level of higher rank graphs themselves here we introduce k morphs which provide a systematic unifying framework for these various constructions we think of k morphs as the analogue at the level of k graphs of c correspondences between c algebras to make this analogy explicit we introduce a category whose objects are k graphs and whose morphisms are isomorphism classes of k morphs we show how to extend the assignment c to a functor from this category to the category whose objects are c algebras and whose morphisms are isomorphism classes of c correspondences

on the representations of leavitt path algebras abstract given a graph e we define e algebraic branching systems show their existence and how they induce representations of the associated leavitt path algebra we also give sufficient conditions to guarantee faithfulness of the representations associated to e algebraic branching systems and to guarantee equivalence of a given representation or a restriction of it to a representation arising from an e algebraic branching system

non simple purely infinite rings in this paper we introduce the concept of purely infinite rings which in the simple case agrees with the already existing notion of pure infiniteness we establish various permanence proper ties of this notion with respect to passage to matrix rings corners and behaviour under extensions so being purely infinite is preserved under morita equivalence we show that a wealth of examples falls into this class including important analogues of constructions commonly found in operator al gebras in particular for any s unital k algebra having enough nonzero idempotents for example for a von neumann regular algebra its tensor product over k with many non simple leavitt path algebras is purely infinite introduction the notion of pure infiniteness has proved key in the the ory of operator algebras since its conception in the early eighties by j cuntz see 27 this was done for simple algebras and provided a huge list of exam ples one of the milestones of the theory became the classification of separable nuclear unital purely infinite simple algebras by means of k theoretic invariants due to kirchberg 41 and phillips 58

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

on twisted higher rank graph c algebras we define the categorical cohomology of a k graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper this leads to an alternative characterisation of the twisted k graph c algebras introduced there we prove a gauge invariant uniqueness theorem and use it to show that every twisted k graph c algebra is isomorphic to a twisted groupoid c algebra we deduce criteria for simplicity prove a cuntz krieger uniqueness theorem and establish that all twisted k graph c algebras are nuclear and belong to the bootstrap class

a generalized cuntz krieger uniqueness theorem for higher rank graphs we present a uniqueness theorem for k graph c algebras that requires neither an aperiodicity nor a gauge invariance assumption specifically we prove that for the injectivity of a representation of a k graph c algebra it is sufficient that the representation be injective on a distinguished abelian c subalgebra a crucial part of the proof is the application of an abstract uniqueness theorem which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient c algebra each of which is the unique extension of a state on the distinguished abelian c subalgebra

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

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

af embeddability of 2 graph algebras and quasidiagonality of k graph algebras abstract we characterise quasidiagonality of the c algebra of a cofinal k graph in terms of an algebraic condition involving the coordinate matrices of the graph this result covers all simple k graph c algebras in the special case of cofinal 2 graphs we further prove that af embeddability quasidiagonality and stable finiteness of the 2 graph algebra are all equivalent

period spaces for hodge structures in equal characteristic we develop the analog in equal positive characteristic of fontaine s theory for crystalline galois representations of a p adic eld in particular we describe the analog of fontaine s functor which assigns to a crystalline galois representation an isocrystal with a hodge ltration in equal characteristic the role of isocrystals and hodge ltrations is played by z isocrystals and hodge pink structures the latter were invented by pink our rst main result in this article is the analog of the colmez fontaine theorem that weakly admissible implies admissible next we construct period spaces for hodge pink structures on a xed z isocrystal these period spaces are analogs of the rapoport zink period spaces for fontaine s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces for our period spaces we prove the analog of a conjecture of rapoport and zink stating the existence of a universal local system on a berkovich open subspace of the period space as a consequence of weakly admissible implies admissible this berkovich open subspace contains every classical rigid analytic point of the period space as the principal tool to demonstrate these results we use the analog of kedlaya s slope filtration theorem which we also formulate and prove here

some thoughts on gerstenhaber s theorem abstract we weigh the current evidence for and against an extension of gerstenhaber s 1961 theorem to three commuting matrices over a field