arXiv subjects

Ken Goodearl

Publications and source records attributed to Ken Goodearl.

6 recordsLinked to original sources

arXiv2026

Positive definite matrices and involutions: the manners of their infinite cousins

Matrices and involutions in/on the algebras $B(\mathbb{R})$ and $B(\mathbb{C})$ of real (resp., complex) row- and column-finite $\omega\times\omega$ matrices are studied. It is proved that any positive definite $\mathbb{R}$-algebra involution on $B(\mathbb{R})$ (resp., any positive definite conjugate-linear involution on $B(\mathbb{C})$) is given by conjugating the transpose involution (resp., the conjugate-transpose involution) with a positive definite matrix from the algebra in question. All positive definite matrices in these algebras have Cholesky factorizations within the algebra. Examples are constructed to show that such positive definite matrices need not have any eigenvalues for their canonical action on column-finite column vectors, and they need not have any square roots in $B(\mathbb{C})$.

math.RA↗

arXiv2026

Poisson fields of two variables

We study invariants and structures of Poisson fields of rational functions in two variables. For four particular families, we classify the members, establish criteria for isomorphisms and, with the exception of the Weyl Poisson field, describe the automorphism groups. Embeddings are also investigated, along with an analog of the Dixmier Conjecture: For which Poisson fields is every Poisson endomorphism an automorphism? The answer is negative for the first family, but positive answers are obtained for several subclasses of the other families. Finally, we exhibit a Poisson field which is not isomorphic to any Poisson field $\Bbbk(x,y)$ for which $\{x,y\}$ is a polynomial in $\Bbbk[x,y]$.

math.RA↗

arXiv2024

Levels of cancellation for monoids and modules

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

math.GR↗

arXiv2024

Regular Ring Properties Degraded Through Inverse Limits

We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to answer the long standing Separativity Problem (in the negative).

math.RA↗

arXiv2024

The separativity problem in terms of varieties and diagonal reduction

We provide two new formulations of the separativity problem. First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By refining this information, we characterize separative regular rings in terms of a special type of inner inverse operation, which is defined via a single identity. This shows that the separative regular rings form a subvariety of the regular rings. The separativity problem reduces to the question of whether every element of the form $(1-aa')bac(1-a'a)$ in a regular ring is unit-regular, where $a'$ is an inner inverse for $a$. Second, it is known that separativity in exchange rings is equivalent to regular matrices over corner rings being reducible to diagonal matrices via elementary row and column operations. We show that for $2\times 2$ invertible matrices, four elementary operations are sufficient, and in general also necessary. Dropping the invertibility hypothesis, but specializing to separative regular rings, we show that three elementary row operations together with three elementary column operations are sufficient, and again in general also necessary. The separativity problem can subsequently be reframed in terms of the explicit number of operations needed to diagonally reduce.

math.RA↗

arXiv2008

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 $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many nonsimple Leavitt path algebras is purely infinite.

math.RA↗