# Simple, Z-stable, projectionless C*-algebras

Von | September 22, 2020

Do simple, -stable, stably projectionless C*-algebras have stable rank one?

Definitions: A C*-algebra is said to be -stable if it tensorially absorbs the Jiang-Su algebra , that is, . Further, a simple C*-algebra is projectionless if it contains no nonzero projections, and it is stably projectionless if is projectionless. (In the nonsimple case, one should require that no quotient of contains a nonzero projection – see ​[1]​.) A unital C*-algebra is said to have stable rank one if the invertible elements in are dense. A nonunital C*-algebra has stable rank one if its minimal unitization does.

Background/Motivation: Rørdam showed in Theorem 6.7 in ​[2]​ that every unital, simple, finite -stable C*-algebra has stable rank one. Using this, one can show that every simple, finite, -stable C*-algebra that is not stably projectionless has stable rank one. Indeed, one may first reduce to the separable case (to show that a given element with is approximated by invertible elements, consider any separable, simple, -stable sub-C*-algebra of that is not stably projectionless and that contains .) Let be a nonzero projection. By Brown’s stabilization theorem, we have . Then the hereditary sub-C*-algebra is separable, unital, simple, finite and -stable (by Corollary 3.2 in ​[3]​) and therefore has stable rank one. Since stable rank one is invariant under stable isomorphism, it follows that has stable rank one.

Thus, a simple, -stable C*-algebra that is not stably projectionless has stable rank one if and only if it is (stably) finite. A simple, stably projectionless C*-algebra is automatically stably finite, and it is therefore natural to expect that every simple, -stable, stably projectionless C*-algebra has stable rank one.

Let be a simple, -stable, stably projectionless C*-algebra. By Corollary 3.2 in ​[1]​, almost has stable rank one, that is, every hereditary sub-C*-algebra satisfies . In particular, every element in can be approximated by invertible elements in . To show that has stable rank one, one would need to show that every element in is approximated by invertibles. In Theorem 6.13 in ​[4]​ it is shown that has stable rank at most two, that is, the tuples in such that is invertible are dense in . One can also show:

Simple, -stable, stably projectionless C*-algebras have general stable rank one.

Here, the general stable rank of a unital C*-algebra , denoted , is the least integer such that acts transitively on for all . (We use to denote the set of invertible elements in the matrix algebra . Further, denotes the set of -tuples that generate as a left ideal, that is, such that is invertible.) For a nonunital C*-algebra . In general, one has , as noted in Theorem 3.3. of the overview article ​[5]​. If is unital, then the action of , the connected component of the unit in , on has open orbits. It follows that the orbits of the action of on are also open (and hence also closed).

Let be a simple, -stable, stably projectionless C*-algebra. Let us prove that . By Theorem 6.13 in ​​[4]​, we have , and so . To verify that , we need to show that acts transitively on . So let with and . First, we find such that with and . (If , we use the identity matrix; if , we use the flip matrix; and if and we use the upper-triangular matrix with diagonal entries and upper right entry .) Given , we use that to approximate by some invertible such that . We note that is in the orbit of , that is, there exists such that . Then . Since the orbits of the action of on are closed, it follows that for some . Finally, since is finite, it follows that .

We remark that implies , but not conversely in general.

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