118 research outputs found
Just-infinite C*-algebras and their invariants
Just-infinite C*-algebras, i.e., infinite dimensional C*-algebras, whose
proper quotients are finite dimensional, were investigated in
[Grigorchuk-Musat-Rordam, 2016]. One particular example of a just-infinite
residually finite dimensional AF-algebras was constructed in that article. In
this paper we extend that construction by showing that each infinite
dimensional metrizable Choquet simplex is affinely homeomorphic to the trace
simplex of a just-infinite residually finite dimensional C*-algebras. The trace
simplex of any unital residually finite dimensional C*-algebra is hence
realized by a just-infinite one. We determine the trace simplex of the
particular residually finite dimensional AF-algebras constructed in the above
mentioned article, and we show that it has precisely one extremal trace of type
II_1.
We give a complete description of the Bratteli diagrams corresponding to
residually finite dimensional AF-algebras. We show that a modification of any
such Bratteli diagram, similar to the modification that makes an arbitrary
Bratteli diagram simple, will yield a just-infinite residually finite
dimensional AF-algebra.Comment: 22 pages. A more detailed proof of Proposition 2.2 is included in
this version, and a missing condition in Proposition 2.2 (and Corollary 2.3)
is added. To appear in Int. Math. Res. Not. IMR
A purely infinite AH-algebra and an application to AF-embeddability
We show that there exists a purely infinite AH-algebra. The AH-algebra arises
as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs
the Cuntz algebra O_\infty tensorially. Thus one can reach an
O_\infty-absorbing C*-algebra as an inductive limit of the finite and
elementary C*-algebras C_0([0,1),M_k).
As an application we give a new proof of a recent theorem of Ozawa that the
cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a
concrete AF-algebra into which this class of C*-algebras can be embedded.Comment: 20 pages, revised January 2004, to appear in Israel J. Mat
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 same Elliott invariant as the complex
numbers. We show that:
(i) The Cuntz semigroup W(A) of equivalence classes of positive elements in
matrix algebras over A is weakly unperforated.
(ii) If A is exact, then A is purely infinite if and only if A is traceless.
(iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty
if and only if A is traceless.
(iv) If A is simple and unital, then the stable rank of A is one if and only
if A is finite.
We also characterise when A is of real rank zero.Comment: 24 pages. Minor revisions August 2004. To appear in International J.
Mat
AF-embeddings into C*-algebras of real rank zero
It is proved that every separable -algebra of real rank zero contains an
AF-sub--algebra such that the inclusion mapping induces an isomorphism of
the ideal lattices of the two -algebras and such that every projection in
a matrix algebra over the large -algebra is equivalent to a projection in
a matrix algebra over the AF-sub--algebra. This result is proved at the
level of monoids, using that the monoid of Murray-von Neumann equivalence
classes of projections in a -algebra of real rank zero has the refinement
property. As an application of our result, we show that given a unital
-algebra of real rank zero and a natural number , then there is a
unital -homomorphism for some
natural numbers with for all if and only if
has no representation of dimension less than .Comment: 28 page
Strongly Self-Absorbing C*-algebras which contain a nontrivial projection
It is shown that a strongly self-absorbing C*-algebra is of real rank zero
and absorbs the Jiang-Su algebra if it contains a nontrivial projection. We
also consider cases where the UCT is automatic for strongly self-absorbing
C*-algebras, and K-theoretical ways of characterizing when Kirchberg algebras
are strongly self-absorbing.Comment: 10 page
Universal properties of group actions on locally compact spaces
We study universal properties of locally compact G-spaces for countable
infinite groups G. In particular we consider open invariant subsets of the
\beta-compactification of G (which is a G-space in a natural way), and their
minimal closed invariant subspaces. These are locally compact free G-spaces,
and the latter are also minimal. We examine the properies of these G-spaces
with emphasis on their universal properties.
As an example of our resuts, we use combinatorial methods to show that each
countable infinite group admits a free minimal action on the locally compact
non-compact Cantor set.Comment: 42 page
- …