Showing papers in "Transactions of the American Mathematical Society in 2010"
TL;DR: In this article, the authors assign a family of rings to each graph without loops and multiple edges and classify projective modules over these rings, where g is the Kac-Moody Lie algebra associated with the graph.
Abstract: To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated with the graph.
259 citations
TL;DR: In this paper, it was shown that binormal operators, algebraic of degree two, and large classes of rank-one perturbations of normal operators are all complex symmetric.
Abstract: We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).
163 citations
TL;DR: In this article, a variant of the continuous first-order logic is proposed, which is based on the framework of open Hausdorff cats and extends Henson's logic for Banach space structures.
Abstract: We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
126 citations
TL;DR: In this article, a lower bound on the minimum number of r-cliques in graphs with n vertices and m edges is given, based on a constraint minimization of certain multilinear forms.
Abstract: Let kr (n, m) denote the minimum number of r-cliques in graphs with n vertices and m edges. We give a lower bound on kr (n, m) that approximates kr (n, m) with an error smaller than n r / n 2 − 2m � . This essentially solves a sixty year old problem. The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.
117 citations
TL;DR: In this article, a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields has been established, and the Erdos-Falconer distance conjecture does not hold in this setting.
Abstract: We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + ··· + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdos-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher-dimensional vector spaces over general finite fields.
109 citations
TL;DR: In this paper, a priori estimates and regularity results for some quasi-linear degenerate elliptic equations arising in optimal stochastic control problems are given. But they do not consider the Dirichlet problem.
Abstract: We prove a priori estimates and regularity results for some quasi-linear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Holder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.
105 citations
TL;DR: In this paper, Huang et al. studied the spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media and provided various useful estimates for spreading and generalized propagating speeds for such models.
Abstract: Spatial spread and front propagation dynamics is one of the most important dynamical issues in KPP models. Such dynamics of KPP models in time independent or periodic media has been widely studied. Recently, the author of the current paper with Huang established some theoretical foundation for the study of spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. A notion of spreading speed intervals for such models was introduced in the above-mentioned paper and was shown to be the natural extension of the classical concept of the spreading speeds for time independent or periodic KPP models and that it could be used for more general time dependent KPP models. A notion of generalized propagating speed intervals of front solutions and a notion of traveling wave solutions to time almost periodic and space periodic KPP models were also introduced, which are the generalizations of wave speeds and traveling wave solutions in time independent or periodic KPP models. The aim of the current paper is to gain some further qualitative and quantitative understanding of the spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. By applying the principal Lyapunov exponent and the principal Floquet bundle theory for time almost periodic parabolic equations, we provide various useful estimates for spreading and generalized propagating speeds for such KPP models. Under the so-called linear determinacy condition, we show that the spreading speed interval in any given direction is a singleton (called the spreading speed). Moreover, in such a case we establish a variational principle for the spreading speed and prove that there is a front solution of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction. Both the estimates and variational principle provide important and efficient tools for the spreading speeds analysis as well as the spreading speeds computation. Based on the variational principle, the influence of time and space variation of the media on the spreading speeds is also discussed in this paper. It is shown that the time and space variation cannot slow down the spatial spread and that it indeed speeds up the spatial spread except in certain degenerate cases, which provides deep insights into the understanding of the influence of the inhomogeneity of the underline media on the spatial spread in KPP models.
104 citations
TL;DR: In this paper, the authors provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space, using the special structure of the nonlinear term of the equation.
Abstract: In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.
95 citations
TL;DR: In this paper, a τ-dependent Wigner representation, Wig τ, τ E [0, 1], was introduced, which allowed us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other.
Abstract: We introduce a τ-dependent Wigner representation, Wig τ , τ E [0,1], which permits us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other. The scheme includes various types of time-frequency representations, among the others the classical Wigner and Rihaczek representations and the most common classes of pseudo-differential operators. We show further that the integral over T of Wig τ yields a new representation Q possessing features in signal analysis which considerably improve those of the Wigner representation, especially for what concerns the so-called "ghost frequencies". The relations of all these representations with respect to the generalized spectrogram and the Cohen class are then studied. Furthermore, a characterization of the L p -boundedness of both τ-pseudo-differential operators and T -Wigner representations are obtained.
94 citations
TL;DR: The double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions, and have direct application to connection probabilities for multiple-strand SLE2, SLE8, and SLE4.
Abstract: Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy’s percolation crossing probabilities, and generalize Kirchhoff’s formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials i n the matrix entries, where the coefficients have a natural algebraic interpretation and canbe computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE2, SLE8, and SLE4.
87 citations
TL;DR: In this article, the authors introduce the notion of admissible functions p and then develop a theory of localized Hardy spaces associated with p, which includes several maximal function characterizations of H 1 ρ (X), the relations between the Hardy space and the classical Hardy space H 1 (X) via constructing a kernel function related to ρ, and the boundedness of certain localized singular integrals.
Abstract: Let X be an RD-space, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions p and then develop a theory of localized Hardy spaces H 1 ρ (X) associated with p, which includes several maximal function characterizations of H 1 ρ (X), the relations between H 1 ρ (X) and the classical Hardy space H 1 (X) via constructing a kernel function related to ρ, the atomic decomposition characterization of H 1 ρ (X) , and the boundedness of certain localized singular integrals on H 1 ρ (X) via a finite atomic decomposition characterization of some dense subspace of H 1 ρ (X). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrodinger operator or the degenerate Schrodinger operator on ℝ n , or to the sub-Laplace Schrodinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrodinger operators considered here are associated with nonnegative potentials satisfying the reverse Holder inequality.
TL;DR: In this article, a family of power ideals arising naturally from a hyperplane arrangement A were investigated and their Hilbert series is determined by the combinatorics of A and can be computed from its Tutte polynomial.
Abstract: We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope.
TL;DR: In this article, a Gaussian dual Brunn-Minkowski inequality for Gauss measure is proved, together with precise equality conditions, and shown to be best possible from several points of view.
Abstract: A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coecients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.
TL;DR: For d > 2, this paper showed that block gluing shifts factor onto all full shifts of strictly smaller entropy, i.e., no non-trivial full shift is a factor.
Abstract: For d > 2 we exhibit mixing ℤ d shifts of finite type and sofic shifts with large entropy but poorly separated subsystems (in the sofic examples, the only minimal subsystem is a single point). These examples consequently have very constrained factors; in particular, no non-trivial full shift is a factor. We also provide examples to distinguish certain mixing conditions and develop the natural class of "block gluing" shifts. In particular, we show that block gluing shifts factor onto all full shifts of strictly smaller entropy.
TL;DR: In this paper, Morse Index Theorems for elliptic boundary value problems in multi-dimensions are proved for star-shaped domains and are based on the idea of measuring the "oscillation" of the trace of the set of solutions on a shrinking boundary.
Abstract: Morse Index Theorems for elliptic boundary value problems in multi-dimensions are proved under various boundary conditions. The theorems work for star-shaped domains and are based on a new idea of measuring the "oscillation" of the trace of the set of solutions on a shrinking boundary. The oscillation is measured by formulating a Maslov index in an appropriate Sobolev space of functions on this boundary. A fundamental difference between the cases of Dirichlet and Neumann boundary conditions is exposed through a monotonicity that holds only in the former case.
TL;DR: In this article, the authors define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M, �) as the categorical groups whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopies between based loops.
Abstract: We define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M,�) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E ! G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M,�) ! C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.
TL;DR: In this paper, the authors consider the Petrie conjecture in the case of a circle acting in a Hamiltonian fashion on a compact symplectic manifold (M, ω) which satisfies H 2 i (M ; ℝ) = H 2i (ℂℙ n,ℝ), for all i.
Abstract: Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold (M, ω) which satisfies H 2i (M ; ℝ) = H 2i (ℂℙ n ,ℝ) for all i. Is H j (M ; ℤ) = H j (ℂℙ n ;ℤ) for all j? Is the total Chern class of M determined by the cohomology ring H * (M; Z)? We answer these questions in the six-dimensional case by showing that H j (M ; ℤ) is equal to H j (ℂℙ 3 ; ℤ) for all j, by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if H*(M; ℤ) is isomorphic to H*(ℂℙ 3 ; ℤ) or H*(G 2 (ℝ 5 ); ℤ), then the representations at the fixed components are compatible with one of the standard actions; in the remaining two cases, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: Do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.
TL;DR: In this paper, the Hermitian weighted composition operators on H 2 and their spectral measures are derived and used to find the polar decomposition, absolute value, and the Aluthge transform of some composition operators.
Abstract: Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, we identify the Hermitian weighted composition operators on H 2 and compute their spectral measures. Some relevant semigroups are studied. The resulting ideas can be used to find the polar decomposition, the absolute value, and the Aluthge transform of some composition operators on H 2 .
TL;DR: In this article, the authors study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of a reductive linear algebraic group over an algebraically closed field of characteristic p > 0.
Abstract: Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.
TL;DR: In this paper, the authors study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. They show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently if A is of tame (domestic or tubular) representation type.
Abstract: We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.
TL;DR: In this article, the authors introduce the notion of cohomological support varieties for the finite dimensional supermodules for a classical Lie superalgebra g = g 0 ⊕ g 1 which are completely reducible over g 0.
Abstract: Unlike Lie algebras, the finite dimensional complex representations of a simple Lie superalgebra are usually not semisimple. As a consequence, despite over thirty years of study, these remain mysterious objects. In this paper we introduce a new tool: the notion of cohomological support varieties for the finite dimensional supermodules for a classical Lie superalgebra g = g 0 ⊕ g 1 which are completely reducible over g 0 . They allow us to provide a new, functorial description of the previously combinatorial notions of defect and atypicality. We also introduce the detecting subalgebra of g. Its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic. Using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra.
TL;DR: In this article, the standard modules of a complex reflection group were studied using a commutative subalgebra t of H discovered by Dunkl and Opdam, and it was shown that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ.
Abstract: The rational Cherednik algebra ℍ is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for ℍ. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on ℂ n with values in the representation S λ . For G(1,1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of ℍ so that the rational Cherednik algebra for G(r,p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.
TL;DR: In this paper, it was shown that for any fixed ε > 0, there are infinitely many positive solutions whose energy can be made arbitrarily large, whereas for ε small, all solutions are constants.
Abstract: We consider the following nonlinear Neumann problem: {―Au + μu = u N+2 N―2 , u > 0 in Ω, ∂u ∂n = 0 on ∂Ω, where Ω C ℝ N is a smooth and bounded domain, μ > 0 and n denotes the outward unit normal vector of ∂Ω. Lin and Ni (1986) conjectured that for μ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains Ω. Furthermore, we prove that for any fixed μ, there are infinitely many positive solutions, whose energy can be made arbitrarily large. This seems to be a new phenomenon for elliptic problems in bounded domains.
TL;DR: In this article, the authors studied the computational complexity of the word problem in free solvable groups S r,d, where r > 2 is the rank and d ≥ 2 the solvability class of the group.
Abstract: We study the computational complexity of the Word Problem (WP) in free solvable groups S r,d , where r > 2 is the rank and d ≥ 2 is the solvability class of the group. It is known that the Magnus embedding of S r,d into matrices provides a polynomial time decision algorithm for WP in a fixed group S r,d . Unfortunately, the degree of the polynomial grows together with d, so the uniform algorithm is not polynomial in d. In this paper we show that WP has time complexity O(rn log 2 n) in S r,2 , and O(n 3 rd) in S r,d for d ≥ 3. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S r,2 is NP-complete. We prove also that one can compute Fox derivatives of elements from S r,d in time O(n 3 rd); in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.
TL;DR: Real Paley-Wiener theory for Fourier transform on ℝ d for Schwartz functions, L p -functions and distributions, in an elementary treatment based on the inversion theorem is studied in this paper.
Abstract: We systematically develop real Paley-Wiener theory for the Fourier transform on ℝ d for Schwartz functions, L p -functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley—Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley― Wiener theory is included.
TL;DR: In this article, the Lyapunov exponents of a hyperbolic ergodic measure are approximated by the LyAPunov exponent of hyperbola atomic measures on periodic orbits.
Abstract: Lyapunov exponents of a hyperbolic ergodic measure are approximated by Lyapunov exponents of hyperbolic atomic measures on periodic orbits.
TL;DR: In this paper, the second author was supported in part by NSF grants DMS-0404729 and NMS-0706728, and the first author was also supported by the NSF.
Abstract: The research of the second author was supported in part by NSF grants DMS-0404729 and DMS-0706728.
TL;DR: In this paper, the best approximation in L1(ℝ) by entire functions of exponential type, for a class of even functions that includes e−λ|x|, where λ>0, log |x| and |x |α, where −1<α<1.
Abstract: We obtain the best approximation in L1(ℝ), by entire functions of exponential type, for a class of even functions that includes e−λ|x|, where λ>0, log |x| and |x|α, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.
TL;DR: In this article, the authors give an algebraic description of normal automorphisms of relatively hyperbolic groups and prove that Out(G) is residually finite for every finite group G with more than one end, if G is non-elementary and has no non-trivial finite normal subgroups.
Abstract: An automorphism of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group G, Inn(G) has finite index in the subgroup Aut_n(G) of normal automorphisms. If, in addition, G is non-elementary and has no non-trivial finite normal subgroups, then Aut_n(G)=Inn(G). As an application, we show that Out(G) is residually finite for every finitely generated residually finite group G with more than one end.
TL;DR: For arbitrary unital rings, the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action as discussed by the authors, and it is shown that if the globalization exists, it is unique up to a certain equivalence relation and moreover, the crossed product corresponding to the twisted action is Morita equivalent to that corresponding to its globalization.
Abstract: Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish a criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.