On prime nonprimitive von Neumann regular algebras

Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.

In this paper, we present results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.

In this work, we realize Leavitt path algebras as partial skew groupoid rings. This yields a free path groupoid grading on Leavitt path algebras. Using this grading, we characterize free path groupoid graded isomorphisms of Leavitt path algebras that preserves generators.

Extensions of simple modules over Leavitt path algebras

In the past, it has been shown that the Leavitt path algebra L(E) = L K (E) of a graph E over a field K is left and right noetherian if and only if the graph E is finite and no cycle of E has an exit. If Q(E) = Q K (E) denotes the regular algebra over L(E), we prove that these two conditions are further equivalent with any of the following: L(E) contains no infinite set of orthogonal idempotents, L(E) has finite uniform dimension, L(E) is directly finite, Q(E) is directly finite, Q(E) is unit-regular, Q(E) is left (right) self-injective and a few more equivalences. In addition, if the involution on the field K is positive definite, these conditions are equivalent with the following: the involution * extends from L(E) to Q(E), Q(E) is *-regular, Q(E) is finite, Q(E) is the maximal (total or classical) symmetric ring of quotients of L(E), the maximal right ring of quotients of L(E) is the same as the total (or classical) left ring of quotients of L(E), every finitely generated nonsingular L(E)-module is projective, and the matrix ring M n (L(E)) is strongly Baer for every n. It may not be surprising that a noetherian Leavitt path algebra has these properties, but a more interesting fact is that these properties hold only if a Leavitt path algebra is noetherian (i.e. E is a finite no-exit graph). Using some of these equivalences, we give a specific description of the inverse of the isomorphism V (L(E)) → V (Q(E)) of monoids of equivalence classes of finitely generated projective modules of L(E) and Q(E) for noetherian Leavitt path algebras. We also prove that two noetherian Leavitt path algebras are isomorphic as rings if and only if they are isomorphic as *-algebras. This answers in affirmative the Isomorphism Conjecture for the class of noetherian Leavitt path algebras: if $${L_{\mathbb{C}}(E)}$$ and $${L_{\mathbb{C}}(F)}$$ are noetherian Leavitt path algebras, then $${L_{\mathbb{C}}(E) \cong L_{\mathbb{C}}(F)}$$ as rings implies $${C^{*} (E) \cong C^{*} (F)}$$ as *-algebras.

The multiplicative ideal theory of Leavitt path algebras

Let E E be an arbitrary graph and K K be any field. It is shown that the Leavitt path algebra L K ( E ) L_{K}(E) is a Zorn ring if and only if the graph E E satisfies Condition (L). As a consequence, every homomorphic image of L K ( E ) L_{K}(E) is a Zorn ring if and only if L K ( E ) L_{K}(E) is a weakly regular ring. The corresponding statement for the graph C*-algebra C ∗ ( E ) C^{\ast }(E) is also investigated.

The graded structure of algebraic Cuntz-Pimsner rings

In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a K -vector space is found for Z(L(E)) when L(E) enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra L(E) is central (i.e. the center reduces to the base field K ) when L(E) is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.

For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\mathbb {C}}(E)$ and the graph $C^*$-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring $*$-homomorphisms, algebra homomorphisms, and algebra $*$-homomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$. We prove that in certain situations isomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$ yield $*$-isomorphisms between the corresponding $C^*$-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that $*$-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\mathbb {C}}(E)$ and $L_{\mathbb {C}}(F)$ in specific cases. The relationship between Leavitt path algebras and graph $C^*$-algebras is also explored in the context of Morita equivalence.

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.

We introduce a notion of ideal-related K-theory for rings, and use it to prove that if two complex Leavitt path algebras are Morita equivalent (respectively, isomorphic), then the ideal-related K-theories (respectively, the unital ideal-related K-theories) of the corresponding graph C*-algebras are isomorphic. This has consequences for the "Morita equivalence conjecture" and "isomorphism conjecture" for graph algebras, and allows us to prove that when E and F belong to specific collections of graphs whose C*-algebras are classified by ideal-related K-theory, Morita equivalence (respectively, isomorphism) of the Leavitt path algebras implies strong Morita equivalence (respectively, isomorphism) of the graph C*-algebras. We state a number of corollaries that describe various classes of graphs where these implications hold. In addition, we conclude with a classification of Leavitt path algebras of amplified graphs similar to the existing classification for graph C*-algebras of amplified graphs.

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.

In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra L is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras L in which every ideal admits such factorizations and also in which every ideal is one of these special type of ideals.