Non-simple purely infinite rings

Abstract

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 non-simple Leavitt path algebras is purely infinite.

A purely infinite Cuntz-like Banach ⁎-algebra with no purely infinite ultrapowers

“…Furthermore, given a lack of nice "permanence" properties for purely infinite Banach algebras, it seems that knowing O p 2 is purely infinite is no direct help in showing that A/J is purely infinite, or vice versa. We remark that similar questions around "permanence properties" are raised at the end of [3]. To represent A d /J d on ℓ p , we simply replace "2n" by "dn", for example, defining The main aim of this section is to prove that there exist non-zero bounded traces on A/J (Theorem 4.3), while there are no non-zero bounded traces on O p 2 (Theorem 4.10).…”

“…In a more general direction, the notion of a simple ring being purely infinite was studied in [2], where it is taken as definition that a simple ring R is purely infinite if every right (or equivalently, left) ideal of R contains an infinite idempotent. Consideration of what it means for a non-simple ring to be purely infinite is given in [3].…”

Amenability of coarse spaces and $$\mathbb {K}$$ K -algebras

“…The following lemma was shown in the row-finite case in [ 13 , Lemma 7.3]. We include the identical proof for completeness.…”

The type semigroup, comparison, and almost finiteness for ample groupoids

“…We will use some results from [6, 7, 30]. Recall that an element x of a monoid M is weakly divisible if it can be written as for .…”

“…An element x of a conical monoid M is said to be irreducible if it is non-zero and given any decomposition in M , we have that either a or b are zero. If follows easily from [7, Theorem 6.7] that an order-unit u of a conical refinement monoid M is weakly divisible if and only if is not irreducible in any simple quotient of M .…”