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c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n
university of copenhagen
Corners and other hereditary subalgebras of graph C∗-algebras
Sara ArklintCentre for Symmetry and Deformation
Joint work with James Gabe and Efren Ruiz
Classification, Structure, Amenability and Regularity at University of Glasgow, September 1–5, 2014Slide 1/5
u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n
What is a graph C∗-algebra?DefinitionLet E = (E 0,E 1, r , s) be a (countable, directed) graph. Its graphC∗-algebra C∗(E) is the universal C∗-algebra generated by mutuallyorthogonal projections {pv | v ∈ E 0} and partial isometries {se | e ∈ E 1}satisfying the relations
1 s∗e sf = 0 if e, f ∈ E 1 and e 6= f ,2 s∗e se = pr(e) for all e ∈ E 1,3 ses∗e ≤ ps(e) for all e ∈ E 1,4 pv =
∑e∈s−1(v) ses∗e for all v ∈ E 0 with 0 < |s−1(v)| <∞.
Proposition (A-Ruiz)Let E be a graph. Then the following are equivalent:
1 E is finite graph with no sinks.2 C∗(E) is a Cuntz-Krieger algebra.3 C∗(E) is unital and
rank(K0(C∗(E))) = rank(K1(C∗(E))).
Slide 2/5 — Sara Arklint — Corners and other hereditary subalgebras of graph C∗ -algebras — C StAR, Glasgow, Sep 2014
u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n
Corners of Cuntz-Krieger algebrasTheorem (A-Ruiz)Let A be a Cuntz-Krieger algebra.
1 If p is a nonzero projection in A⊗K, then p(A⊗K)p is aCuntz-Krieger algebra.
2 If p is a nonzero projection in A, then pAp is a Cuntz-Kriegeralgebra.
CorollaryLet B be a unital C∗-algebra. If B is stably isomorphic to aCuntz-Krieger algebra, then B is a Cuntz-Krieger algebra.
Theorem (Ara-Moreno-Pardo)Let E be a row-finite graph. Any projection in C∗(E)⊗K is Murray-vonNeumann equivalent to a projection of the form∑
v∈E0
mvpv
with all but finitely many mv equal to zero.Slide 3/5 — Sara Arklint — Corners and other hereditary subalgebras of graph C∗ -algebras — C StAR, Glasgow, Sep 2014
u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n
Nonstable K -theoryTheorem (Ara-Moreno-Pardo)Let E be a row-finite graph. Any projection in C∗(E)⊗K is Murray-vonNeumann equivalent to a projection of the form∑
v∈S
mvpv
where mv ≥ 1 for all v ∈ S, and S is a finite subset of E 0.
Theorem (Hay-Loving-Montgomery-Ruiz-Todd)Let E be a graph. Any projection in C∗(E)⊗K is Murray-von Neumannequivalent to a projection of the form
∑(v,T )∈S
n(v,T )
(pv −
∑e∈T
ses∗e
)
where m(v,T ) ≥ 1 for all (v ,T ) ∈ S, and S is a finite subset of
{(v ,T ) ∈ E 0×2E1| T ⊆ s−1E (v),T is finite,T = ∅ when |s−1E (v)| <∞}.
Slide 4/5 — Sara Arklint — Corners and other hereditary subalgebras of graph C∗ -algebras — C StAR, Glasgow, Sep 2014
u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n
Hereditary subalgebras of graph C∗-algebras
Theorem (A-Gabe-Ruiz)Let E be a graph with finitely many vertices.
1 Let A be a hereditary subalgebra of C∗(E)⊗K. If A is σp-unital,then A is a graph C∗-algebra.
2 Let A be a hereditary subalgebra of C∗(E). If A is σp-unital, thenA is a graph C∗-algebra.
CorollaryLet A be a σp-unital C∗-algebra, and let E be a graph with finitely manyvertices. If A is stably isomorphic to C∗(E), then A is a graphC∗-algebra.
Slide 5/5 — Sara Arklint — Corners and other hereditary subalgebras of graph C∗ -algebras — C StAR, Glasgow, Sep 2014