Quasitraces on exact C*-algebras are traces

By Uffe Haagerup. 2014

136 citations recorded by the source corpus.

It is shown that all 2-quasitraces on a unital exact C-algebra are traces. As consequences one gets: (1) Every stably finite exact unital Calgebra has a tracial state, and (2) if an AW -factor of type II1 is generated (as an AW -algebra) by an exact C-subalgebra, then it is a von Neumann II1-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that RR(A) = 0 for every simple non-commutative torus of any dimension.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fcaa9a43a3849f30f7789b3532d91cc834497f26

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Citations in the corpus

Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “Further exactness is only used in Theorems D and E in order to access Haagerup’s result that quasitraces are traces 4In fact each fibre is the hyperfinite II1 factor. for exact algebras ([40]).”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, result): “Of course, by now two beautiful alternative proofs without using automorphisms have been obtained by Haagerup and by Popa, but for our purposes it will be more suitable to focus on the original strategy involving approximately inner flips.”
Quasidiagonality of nuclear C*-algebras (2015, background): “It is a well-known consequence of [4] and [33] that if A is stably finite then it has a trace;(28) the converse is trivial (and no use of the UCT is required for either direction).”
A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, background): “Since A is stably finite and assumed to be amenable, therefore exact, any positive state of K0(A) is the restriction of a tracial state of A ([5] and [42]).”
Nuclear dimension of simple stably projectionless C∗-algebras (2019, background): “29(i)] for how to deduce the general case from [32].”
Sequentially split ∗-homomorphisms between C*-algebras (2015, background): “This implies that T (A) = QT (A) by Haagerup’s theorem [26].”
Subalgebras of simple AF-algebras (2018, background): “12 in [36], and then τ |A is an invariant trace on A.”
STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, unknown): “DonHadwin (1987).”
On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “Furthermore, by [13], A ⊗ Q is stably finite and nuclear and so by [1] and [11], T(A) 6= Ø.”
Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, background): “Since A is stably finite and assumed to be nuclear, therefore exact, any positive state of K0(A) is the restriction of a tracial state of A ([5] and [38]).”