M. Siles Molina

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a∊J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a∊P(a)2J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a∊J with generalized inverse b the subtriple generated by {a, b} is strongly associative

Goldie theorems for associative pairs

We develop a Goldie theory for associative pairs and characterize associative pairs which are orders in semiprime associative pairs coinciding with their socle, and those which are orders in semiprime artinian associative pairs

NON-SIMPLE PURELY INFINITE RINGS

In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-) unital

Morita Invariance and Maximal Left Quotient Rings

K

Left Quotient Associative Pairs and Morita Invariant Properties

-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over

Generalized Regularity Conditions for Leavitt Path Algebras over Arbitrary Graphs

K

Local Rings of Rings of Quotients

with many non-simple Leavitt path algebras is purely infinite.

Jordan canonical form for finite rank elements in Jordan algebras

Abstract We prove that under conditions of regularity the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. We show that if R and S are two non-unital Morita equivalent rings then their maximal left quotient rings are not necessarily Morita equivalent. This situation contrasts with the unital case. However we prove that the ideals generated by two Morita equivalent idempotent rings inside their own maximal left quotient rings are Morita equivalent.

Martindale Algebras of Quotients of Graded Algebras

In this paper, we prove that left nonsingularity and left nonsingularity plus finite left local Goldie dimension are two Morita invariant properties for idempotent rings without total left or right zero divisors. Moreover, two Morita equivalent idempotent rings, semiprime and left local Goldie, have Fountain–Gould left quotient rings that are Morita equivalent too. These results can be obtained from others concerning associative pairs. We introduce the notion of (general) left quotient pair of an associative pair and show the existence of a maximal left quotient pair for every semiprime or left nonsingular associative pair. Moreover, we characterize those associative pairs for which their maximal left quotient pair is von Neumann regular and give a Gabriel-like characterization of associative pairs whose maximal left quotient pair is semiprime and artinian.

Compact graph C ∗ -algebras

Let E be an arbitrary graph, and let K be any field. We show that many generalized regularity conditions for the Leavitt path algebra L K (E) are equivalent and that this happens exactly when the graph E satisfies Condition (K).