Faktor fisik perairan dan antropogenik pada Terumbu Karang Kima di Perairan Bunati, Kabupaten Tanah Bumbu, Kalimantan Selatan (original) (raw)
The Leavitt path algebras of arbitrary graphs
2016
If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L K (E). We show that the involution on L K (E) is proper if the involution on K is positive definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L K (E) to be * -regular (i.e. regular with proper involution). This characterization of * -regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph-theoretic properties of E alone. As a corollary, we show that Handelman's conjecture (stating that every * -regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.
Purely infinite simple Leavitt path algebras
Journal of Pure and Applied Algebra, 2006
We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L(E) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz-Krieger C *-algebra C * (E) given in [7].
The Leavitt path algebra of a graph
Leavitt, 2024
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.
*-Regular Leavitt path algebras of arbitrary graphs
2013
If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L_K(E). We show that the involution on L_K(E) is proper if the involution on K is positive definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L_K(E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L_K(E) to be ^∗-regular (i.e. regular with proper involution). This characterization of ^∗-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph-theoretic properties of E alone. As a corollary, we show that Handelman's conjecture (stating that every ^∗-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its ge...
Journal of Algebra, 2021
We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra LK (E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C * -algebra of a countable graph is isomorphic to the C * -algebra of an ample groupoid.
The classification question for Leavitt path algebras
Journal of Algebra, 2008
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Zgraded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the K 0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K 0 , classify the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.
Primitive Ideal Space of Ultragraph C * -algebras
In this paper, we describe the primitive ideal space of the C *-algebra C * (G) associated to the ultragraph G. We investigate the structure of the closed ideals of the quotient ultragraph C *-algebra C * (G/(H, S)) which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the Hong and Szyma´nskiSzyma´nski's description of the primitive ideal space of a graph C *-algebra by a simpler method.
Noetherian Leavitt Path Algebras and Their Regular Algebras
Mediterranean Journal of Mathematics, 2013
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 C (E) and L C (F) are noetherian Leavitt path algebras, then L C (E) ∼ = L C (F) as rings implies C * (E) ∼ = C * (F) as *-algebras.
Socle theory for Leavitt path algebras of arbitrary graphs
Revista Matemática Iberoamericana, 2000
The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra, extending so the results for row-finite graphs in the previous paper (but with different methods). A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field.