1,775 research outputs found
Flat dimension growth for C*-algebras
We introduce two nonnegative real-valued invariants for unital and stably
finite C*-algebras whose minimal instances coincide with the notion of
classifiability via the Elliott invariant. The first of these is defined for AH
algebras, and may be thought of as a generalisation of slow dimension growth.
The second invariant is defined for any unital and stably finite algebra, and
may be thought of as an abstract version of the first invariant. We establish
connections between both invariants and ordered K-theory, and prove that the
range of the first invariant is exhausted by simple unital AH algebras.
Consequently, the class of simple, unital, and non-Z-stable AH algebras is
uncountable.Comment: 27 pages, improvements to Propositions 2.2, 3.4, 6.2 and Theorem
3.10, Propositions 6.3 and 6.8 added, cleaned up in general, to appear in J.
Funct. Ana
Stability in the Cuntz semigroup of a commutative C*-algebra
We prove stability theorems in the Cuntz semigroup of a commutative
C*-algebra which are analogues of classical stability theorems for topological
vector bundles over compact Hausdorff spaces. Several applications to simple
unital AH algebras of slow dimension growth are then given: such algebras have
strict comparison of positive elements; their Cuntz semigroups are recovered
functorially from the Elliott invariant; the lower-semicontinuous dimension
functions are dense in the space of all dimension functions, and the latter is
a Choquet simplex.Comment: 26 pages, minor revisions, to appear in Proc. LM
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