67 research outputs found
Classification of homomorphisms into simple Z-stable C^*-algebras
We classify unital monomorphisms into certain simple Z-stable C^*-algebras up
to approximate unitary equivalence. The domain algebra C is allowed to be any
unital separable commutative C^*-algebra, or any unital simple separable
nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of
tracial rank zero for a UHF algebra B. The target algebra A is allowed to be
any unital simple separable Z-stable C^*-algebra such that A\otimes B has
tracial rank zero for a UHF algebra B, or any unital simple separable exact
Z-stable C^*-algebra whose projections separate traces and whose extremal
traces are finitely many
Z-actions on AH algebras and Z^2-actions on AF algebras
We consider Z-actions (single automorphisms) on a unital simple AH algebra
with real rank zero and slow dimension growth and show that the uniform
outerness implies the Rohlin property under some technical assumptions.
Moreover, two Z-actions with the Rohlin property on such a C^*-algebra are
shown to be cocycle conjugate if they are asymptotically unitarily equivalent.
We also prove that locally approximately inner and uniformly outer Z^2-actions
on a unital simple AF algebra with a unique trace have the Rohlin property and
classify them up to cocycle conjugacy employing the OrderExt group as
classification invariants.Comment: 24 page
Homology and topological full groups of etale groupoids on totally disconnected spaces
For almost finite groupoids, we study how their homology groups reflect
dynamical properties of their topological full groups. It is shown that two
clopen subsets of the unit space has the same class in H_0 if and only if there
exists an element in the topological full group which maps one to the other. It
is also shown that a natural homomorphism, called the index map, from the
topological full group to H_1 is surjective and any element of the kernel can
be written as a product of four elements of finite order. In particular, the
index map induces a homomorphism from H_1 to K_1 of the groupoid C^*-algebra.
Explicit computations of homology groups of AF groupoids and etale groupoids
arising from subshifts of finite type are also given.Comment: 34 page
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