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.
- [1]L. Robert, Remarks on Z-stable projectionless C*-algebras, Glasgow Math. J. (2015) 273–277. https://doi.org/10.1017/s0017089515000117.
- [2]M. Rørdam, The stable and the real rank of
-absorbing C*-algebras, Int. J. Math. (2004) 1065–1084. https://doi.org/10.1142/s0129167x04002661.
- [3]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.
- [4]L. Robert, L. Santiago, A revised augmented Cuntz semigroup, ArXiv:1904.03690. (2019). https://arxiv.org/abs/1904.03690v1.
- [5]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.