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 .) 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  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 ) 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 , 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  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 . 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 , 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.
- L. Robert, Remarks on Z-stable projectionless C*-algebras, Glasgow Math. J. (2015) 273–277. https://doi.org/10.1017/s0017089515000117.
- M. Rørdam, The stable and the real rank of -absorbing C*-algebras, Int. J. Math. (2004) 1065–1084. https://doi.org/10.1142/s0129167x04002661.
- A.S. Toms, W. Winter, Strongly self-absorbing -algebras, Trans. Amer. Math. Soc. (2007) 3999–4029. https://doi.org/10.1090/s0002-9947-07-04173-6.
- L. Robert, L. Santiago, A revised augmented Cuntz semigroup, ArXiv:1904.03690. (2019). https://arxiv.org/abs/1904.03690v1.
- B. Nica, Homotopical stable ranks for Banach algebras, Journal of Functional Analysis. (2011) 803–830. https://doi.org/10.1016/j.jfa.2011.03.001.