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We show that a directed graph
$E$
is a finite graph with no sinks if and only if, for each commutative unital ring
$R$
, the Leavitt path algebra
$L_{R}(E)$
is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the
$C^{\ast }$
-algebra
$C^{\ast }(E)$
is unital and
$\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$
. Let
$k$
be a field and
$k^{\times }$
be the group of units of
$k$
. When
$\text{rank}(k^{\times })<\infty$
, we show that the Leavitt path algebra
$L_{k}(E)$
is isomorphic to an algebraic Cuntz–Krieger algebra if and only if
$L_{k}(E)$
is unital and
$\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$
. We also show that any unital
$k$
-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.
Let
$A=\bigoplus _{i\in \mathbb{Z}}A_{i}$
be a finite-dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree
$d$
. We prove that if
$d\neq 0$
then
$A_{-d}$
contains the Higman ideal
$H(A)$
and
$\dim H(A)\leq \dim A_{0}$
, and provide a semisimplicity criterion for
$A$
in terms of the centralizer of
$A_{0}$
.
The purpose of this note is to prove the following. Suppose
$\mathfrak{R}$
is a semiprime unity ring having an idempotent element
$e$
(
$e\neq 0,~e\neq 1$
) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on
$\mathfrak{R}$
is a generalized derivation.
For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.
Nakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.
Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology
$\mathbb{G}$
of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at
$\mathbb{G}$
fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology
$\mathbb{G}$
, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all
$\mathbb{G}$
-separated
$\mathbb{G}$
-complete left R-modules.
In this article, we consider a twisted partial action
$\unicode[STIX]{x1D6FC}$
of a group
$G$
on an associative ring
$R$
and its associated partial crossed product
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
. We provide necessary and sufficient conditions for the commutativity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
when the twisted partial action
$\unicode[STIX]{x1D6FC}$
is unital. Moreover, we study necessary and sufficient conditions for the simplicity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
in the following cases: (i)
$G$
is abelian; (ii)
$R$
is maximal commutative in
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
; (iii)
$C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$
is simple; (iv)
$G$
is hypercentral. When
$R=C_{0}(X)$
is the algebra of continuous functions defined on a locally compact and Hausdorff space
$X$
, with complex values that vanish at infinity, and
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$
is the associated partial skew group ring of a partial action
$\unicode[STIX]{x1D6FC}$
of a topological group
$G$
on
$C_{0}(X)$
, we study the simplicity of
$C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$
by using topological properties of
$X$
and the results about the simplicity of
$R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$
.
The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra
$A_{1}$
(over a field of characteristic zero) is an automorphism, i.e., if
$PQ-QP=1$
for some
$P,Q\in A_{1}$
, then
$A_{1}=K\langle P,Q\rangle$
. The Weyl algebra
$A_{1}$
is a
$\mathbb{Z}$
-graded algebra. We prove that the Dixmier Conjecture holds if the elements
$P$
and
$Q$
are sums of no more than two homogeneous elements of
$A_{1}$
(there is no restriction on the total degrees of
$P$
and
$Q$
).
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.
We determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.
We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.
We use concepts of continuous higher randomness, developed in Bienvenu et al. [‘Continuous higher randomness’, J. Math. Log.17(1) (2017).], to investigate
$\unicode[STIX]{x1D6F1}_{1}^{1}$
-randomness. We discuss lowness for
$\unicode[STIX]{x1D6F1}_{1}^{1}$
-randomness, cupping with
$\unicode[STIX]{x1D6F1}_{1}^{1}$
-random sequences, and an analogue of the Hirschfeldt–Miller characterization of weak 2-randomness. We also consider analogous questions for Cohen forcing, concentrating on the class of
$\unicode[STIX]{x1D6F4}_{1}^{1}$
-generic reals.
Let R be a graded ring. We introduce the concepts of Ding gr-injective and Ding gr-projective R-modules, which are the graded analogues of Ding injective and Ding projective modules. Several characterizations and properties of Ding gr-injective and Ding gr-projective modules are obtained. In addition, we investigate the relationships among Gorenstein gr-flat, Ding gr-injective and Ding gr-projective modules.
Let
$\Bbbk$
be a field of characteristic zero. For any positive integer
$n$
and any scalar
$a\in \Bbbk$
, we construct a family of Artin–Schelter regular algebras
$R(n,a)$
, which are quantizations of Poisson structures on
$\Bbbk [x_{0},\ldots ,x_{n}]$
. This generalizes an example given by Pym when
$n=3$
. For a particular choice of the parameter
$a$
we obtain new examples of Calabi–Yau algebras when
$n\geqslant 4$
. We also study the ring theoretic properties of the algebras
$R(n,a)$
. We show that the point modules of
$R(n,a)$
are parameterized by a bouquet of rational normal curves in
$\mathbb{P}^{n}$
, and that the prime spectrum of
$R(n,a)$
is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe
$\operatorname{Spec}R(n,a)$
as a union of commutative strata.
In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.
Given a partial action
$\unicode[STIX]{x1D703}$
of a group on a set with an algebraic structure, we construct a reflector of
$\unicode[STIX]{x1D703}$
in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if
$\unicode[STIX]{x1D703}$
is a partial action on an algebra from a variety
$\mathsf{V}$
, then we show that the problem reduces to the embeddability of a certain generalized amalgam of
$\mathsf{V}$
-algebras associated with
$\unicode[STIX]{x1D703}$
. As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.
Let
${\mathcal{A}}$
be a unital torsion-free algebra over a unital commutative ring
${\mathcal{R}}$
. To characterise Lie
$n$
-higher derivations on
${\mathcal{A}}$
, we give an identity which enables us to transfer problems related to Lie
$n$
-higher derivations into the same problems concerning Lie
$n$
-derivations. We prove that: (1) if every Lie
$n$
-derivation on
${\mathcal{A}}$
is standard, then so is every Lie
$n$
-higher derivation on
${\mathcal{A}}$
; (2) if every linear mapping Lie
$n$
-derivable at several points is a Lie
$n$
-derivation, then so is every sequence
$\{d_{m}\}$
of linear mappings Lie
$n$
-higher derivable at these points; (3) if every linear mapping Lie
$n$
-derivable at several points is a sum of a derivation and a linear mapping vanishing on all
$(n-1)$
th commutators of these points, then every sequence
$\{d_{m}\}$
of linear mappings Lie
$n$
-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all
$(n-1)$
th commutators of these points. We also give several applications of these results.
Let
$A_{2}$
be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of
$A_{2}$
and an algorithm to determine whether a given noninjective endomorphism of
$A_{2}$
has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.
We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane
provided
$r>2$
. More exactly, we show that the above operator maps
$L^{p}\times L^{q}\rightarrow L^{s}$
whenever
$p,q,s^{\prime }$
are in the ‘local
$L^{r^{\prime }}$
’ range, that is,