The generator problem for Z-stable C*-algebras

@article{Thiel2012TheGP,
  title={The generator problem for Z-stable C*-algebras},
  author={Hannes Thiel and Wilhelm Winter},
  journal={arXiv: Operator Algebras},
  year={2012}
}
The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebras A is singly generated, which means that there exists an element x in A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a… 
The generator rank of C⁎-algebras
FRA¨´ E LIMITS OF C*-ALGEBRAS
We realize the Jiang-Su algebra, all UHF algebras, and the hy- perfinite II1 factor as Fra¨osse limits of suitable classes of structures. Moreover by means of Fra¨osse theory we provide new examples
The generator rank of subhomogeneous C*-algebras
We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed
Generators in $\mathcal{Z}$-stable C*-algebras of real rank zero.
We show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. It follows that in every classifiable, simple, nuclear
FRAÏSSÉ LIMITS OF C*-ALGEBRAS
TLDR
The Jiang-Su algebra, all UHF algebras, and the hyperfinite II1 factor are realized as Fraïssé limits of suitable classes of structures and Ramsey-theoretic results about the class of full-matrix alge bras are deduced.
C∗-algebras, Abstract, and Concrete
  • I. Farah
  • Mathematics
    Springer Monographs in Mathematics
  • 2019
In this chapter we introduce the abstract C∗-algebras and work towards the Gelfand–Naimark–Segal Theorem (Theorem 1.10.1). Along the way we discuss abelian C∗-algebras and Gelfand–Naimark and Stone
CONTINUOUS LOGIC AND BOREL EQUIVALENCE RELATIONS
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the
Stably projectionless Fra\"iss\'e limits
. We realise the algebra W , the algebra Z 0 and the algebras Z 0 ⊗ A , where A is a unital separable UHF algebra as Fra¨ıss´e limits of suitable classes of structures. In doing so, we show that such
...
...

References

SHOWING 1-10 OF 48 REFERENCES
Some C ∗ -Algebras with a Single Generator
This paper grew out of the following question: If X is a compact subset of Cn, is C(X) ? Mn (the C*-algebra of n x n matrices with entries from C(X)) singly generated? It is shown that the answer is
The stable and the real rank of Z-absorbing C*-algebras
Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the
Nuclear dimension and -stability of pure C ∗ -algebras
In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially)
GENERATORS AND DIRECT INTEGRAL DECOMPOSITIONS OF {W^ * }-ALGEBRAS
Let A be a W*-algebra on separable Hubert space H. A is singly generated if there is an operator TeA such that A is the smallest W*algebra containing T. It has long been conjectured that every A is
Single generation and rank of C*-algebras
We mainly treat a separable C*-algebra A in this article. Let S be a subset of Asa. We call S a generator of A when any C*-subalgebra B of A containing S is equal to A, and we denote A = C∗(S). If S
THE STABLE AND THE REAL RANK OF Z-ABSORBING C -ALGEBRAS
Suppose that A is a C -algebra for which A = A Z, where Z is the Jiang{Su algebra: a unital, simple, stably nite, separable, nuclear, innite-dimensional C -algebra with the same Elliott invariant as
The Jiang–Su algebra revisited
Abstract We give a number of new characterizations of the Jiang–Su algebra 𝒵, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the way
On generators for von Neumann algebras
1. I t has been conjectured that every von Neumann algebra on a separable Hubert space has a single generator. The conjecture is true for type I algebras [3] and for hyperfinite algebras [7, Theorem
On simplicity of reduced C*-algebras of groups
A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of C*-simple groups which
...
...