The graded Grothendieck group and the classification of Leavitt path algebras

This paper is an attempt to show that, parallel to Elliott's classification of AF $C^*$-algebras by means of $K$-theory, the graded $K_0$-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and comet and polycephaly graphs (graphs which each head is connected to a cycle or a collection of loops).Comment: 30 pages more polishe

SK1 of Azumaya algebras over hensel pairs

Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).Comment: 7 page

Comparability in the graph monoid

Let $\Gamma$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $\Gamma$-action instead. Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^\Gamma$ with a $\Gamma$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^\Gamma$ is comparable, periodic, graphs such that every nonzero element of $M_E^\Gamma$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^\Gamma$ is periodic, and graphs such that no element of $M_E^\Gamma$ is aperiodic. The Graded Classification Conjecture can be formulated to state that $M_E^\Gamma$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^\Gamma.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.Comment: This version contains some modifications based on the input of a referee for the New York Journal of Mathematic

Baer and Baer *-ring characterizations of Leavitt path algebras

We characterize Leavitt path algebras which are Rickart, Baer, and Baer $*$-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer $*$-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well. Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer $*$-ring, a Rickart $*$-ring which is not Baer, or a Baer and not a Rickart $*$-ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their $C^*$-algebra counterparts. For example, while a graph $C^*$-algebra is Baer (and a Baer $*$-ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer $*$-ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops.Comment: Some typos present in the first version are now correcte

K-theory Classification of Graded Ultramatricial Algebras with Involution

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $\Gamma$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial $*$-algebras over a $\Gamma$-graded $*$-field $A$ such that (1) each nontrivial graded component of $A$ has a unitary element in which case we say that $A$ has enough unitaries, and (2) the zero-component $A_0$ is 2-proper (for any $a,b\in A_0,$ $aa^*+bb^*=0$ implies $a=b=0$) and $*$-pythagorean (for any $a,b\in A_0,$ $aa^*+bb^*=cc^*$ for some $c\in A_0$). If the involutive structure is not considered, our result implies that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial algebras over any graded field $A.$ If the grading is trivial and the involutive structure is not considered, we obtain some well known results as corollaries. If $R$ and $S$ are graded matricial $*$-algebras over a $\Gamma$-graded $*$-field $A$ with enough unitaries and $f: K_0^{\operatorname{gr}}(R)\to K_0^{\operatorname{gr}}(S)$ is a contractive $\mathbb Z[\Gamma]$-module homomorphism, we present a specific formula for a graded $*$-homomorphism $\phi: R\to S$ with $K_0^{\operatorname{gr}}(\phi) = f.$ If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if $E$ and $F$ are countable, row-finite, no-exit graphs in which every path ends in a sink or a cycle and $K$ is a 2-proper and $*$-pythagorean field, then the Leavitt path algebras $L_K(E)$ and $L_K(F)$ are isomorphic as graded rings if any only if they are isomorphic as graded $*$-algebras.Comment: Some typos present in the second version are now correcte