Lia Vas

Canonical Traces and Directly Finite Leavitt Path Algebras

Motivated by the study of traces on graph C∗-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph C∗-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as canonical since they are uniquely determined by their values on the vertices. A desirable property of a ℂ

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

\mathbb {C}

TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

-valued trace on a C∗-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra LK(E) with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, K-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, ℂ

Extending Ring Derivations to Right and Symmetric Rings and Modules of Quotients

\mathbb {C}

Baer and Baer *-ring characterizations of Leavitt path algebras

-valued (algebra) traces on Lℂ(E)

A Note on (α, β)-higher Derivations and their Extensions to Modules of Quotients

L_{\mathbb {C}}(E)

-Clean Rings; Some Clean and Almost Clean Baer -rings and von Neumann Algebras

of a countable graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph C∗-algebra C∗(E). With the direct finite condition (i.e xy=1 implies yx=1) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of “local” Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.

Dimension and torsion theories for a class of Baer *-rings

ABSTRACT The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜. Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K 0(𝒜) → K 0 (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ℬ of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ℬ-module.

Differentiability of torsion theories

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Class of Baer ∗-rings defined by a relaxed set of axioms

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.