quantum graphs and quantum cuntz-krieger algebrasgressman/maam2/brannan-maam2.pdfquantum...
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Quantum Graphsand
Quantum Cuntz-Krieger Algebras
Michael Brannan
Texas A&M University
The Second Mid-Atlantic Analysis MeetingOctober 18, 2020
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Today’s Goals
I To describe a generalization of graph theory within theframework of operator algebras/non-commutative geometry:Quantum Graphs.
I To explain (briefly) why one might want to study quantumgraphs.
I To introduce a new class of operator algebras that encode the“symbolic dynamics” associated to quantum graphs:Quantum Cuntz-Krieger algebras.
Joint work with:
I Kari Eifler (TAMU)
I Christian Voigt (Glasgow)
I Moritz Weber (Saarbrucken)
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Graphs
A (finite, directed) graph is a tuple G = (V,E, s, t) where
I V,E are finite sets (vertices and edges, respectively)
I s, t : E → V are functions (source and target maps).
I Thus each edge e ∈ E is thus associated to an ordered pair(s(e), t(e)) ∈ V × V . s(e) is the source of e and t(e) is thetarget of e.
I Graphs G can multiple edges: The mapE → V × V ; e 7→ (s(e), t(e)) need not be injective.
I For our purposes, we will ONLY consider graphs G with nomultiple edges.
I In this case, we simply identify E ⊆ V × V . In this case wecan consider the adjacency matrix AG ∈MV×V ({0, 1}) of G:
AG(v, w) = 1 ⇐⇒ (v, w) ∈ E.
Then G is encoded by the pair (V,AG).
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Graphs
A (finite, directed) graph is a tuple G = (V,E, s, t) where
I V,E are finite sets (vertices and edges, respectively)
I s, t : E → V are functions (source and target maps).
I Thus each edge e ∈ E is thus associated to an ordered pair(s(e), t(e)) ∈ V × V . s(e) is the source of e and t(e) is thetarget of e.
I Graphs G can multiple edges: The mapE → V × V ; e 7→ (s(e), t(e)) need not be injective.
I For our purposes, we will ONLY consider graphs G with nomultiple edges.
I In this case, we simply identify E ⊆ V × V . In this case wecan consider the adjacency matrix AG ∈MV×V ({0, 1}) of G:
AG(v, w) = 1 ⇐⇒ (v, w) ∈ E.
Then G is encoded by the pair (V,AG).
![Page 5: Quantum Graphs and Quantum Cuntz-Krieger Algebrasgressman/maam2/brannan-MAAM2.pdfQuantum Cuntz-Krieger Algebras Michael Brannan Texas A&M University The Second Mid-Atlantic Analysis](https://reader035.vdocument.in/reader035/viewer/2022081620/610c53c22df61a03533220fc/html5/thumbnails/5.jpg)
Graphs
A (finite, directed) graph is a tuple G = (V,E, s, t) where
I V,E are finite sets (vertices and edges, respectively)
I s, t : E → V are functions (source and target maps).
I Thus each edge e ∈ E is thus associated to an ordered pair(s(e), t(e)) ∈ V × V . s(e) is the source of e and t(e) is thetarget of e.
I Graphs G can multiple edges: The mapE → V × V ; e 7→ (s(e), t(e)) need not be injective.
I For our purposes, we will ONLY consider graphs G with nomultiple edges.
I In this case, we simply identify E ⊆ V × V . In this case wecan consider the adjacency matrix AG ∈MV×V ({0, 1}) of G:
AG(v, w) = 1 ⇐⇒ (v, w) ∈ E.
Then G is encoded by the pair (V,AG).
![Page 6: Quantum Graphs and Quantum Cuntz-Krieger Algebrasgressman/maam2/brannan-MAAM2.pdfQuantum Cuntz-Krieger Algebras Michael Brannan Texas A&M University The Second Mid-Atlantic Analysis](https://reader035.vdocument.in/reader035/viewer/2022081620/610c53c22df61a03533220fc/html5/thumbnails/6.jpg)
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How to “Quantize” a Graph: The NCG PerspectiveNon-commutative Geometry (NCG) is all about finding quantum(i.e., non-commutative) generalizations of familiartopological/algebraic/geometric structures by generalizing Gelfandduality:
Examples:
I LC Hausdorff spaces X ←→ abelian C∗-algebrasA = C0(X).
I Quantum Topological Spaces ←→ arbitrary C∗-algebrasA ⊂ B(H).
I Measure spaces (X,µ) ←→ abelian von Neumann algebrasM = L∞(X,µ).
I Quantum Measure spaces ←→ arbitrary von Neumannalgebras M = M ′′ ⊂ B(H).
I Quantum groups ←→ nice Hopf algebras/tensor categories.
I Quantum Riemannian manifolds ←→ Connes’ spectraltriples.
Our Problem: What is the quantum analogue of a graph?
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How to “Quantize” a Graph: The NCG PerspectiveNon-commutative Geometry (NCG) is all about finding quantum(i.e., non-commutative) generalizations of familiartopological/algebraic/geometric structures by generalizing Gelfandduality:Examples:
I LC Hausdorff spaces X ←→ abelian C∗-algebrasA = C0(X).
I Quantum Topological Spaces ←→ arbitrary C∗-algebrasA ⊂ B(H).
I Measure spaces (X,µ) ←→ abelian von Neumann algebrasM = L∞(X,µ).
I Quantum Measure spaces ←→ arbitrary von Neumannalgebras M = M ′′ ⊂ B(H).
I Quantum groups ←→ nice Hopf algebras/tensor categories.
I Quantum Riemannian manifolds ←→ Connes’ spectraltriples.
Our Problem: What is the quantum analogue of a graph?
![Page 9: Quantum Graphs and Quantum Cuntz-Krieger Algebrasgressman/maam2/brannan-MAAM2.pdfQuantum Cuntz-Krieger Algebras Michael Brannan Texas A&M University The Second Mid-Atlantic Analysis](https://reader035.vdocument.in/reader035/viewer/2022081620/610c53c22df61a03533220fc/html5/thumbnails/9.jpg)
How to “Quantize” a Graph: The NCG PerspectiveNon-commutative Geometry (NCG) is all about finding quantum(i.e., non-commutative) generalizations of familiartopological/algebraic/geometric structures by generalizing Gelfandduality:Examples:
I LC Hausdorff spaces X ←→ abelian C∗-algebrasA = C0(X).
I Quantum Topological Spaces ←→ arbitrary C∗-algebrasA ⊂ B(H).
I Measure spaces (X,µ) ←→ abelian von Neumann algebrasM = L∞(X,µ).
I Quantum Measure spaces ←→ arbitrary von Neumannalgebras M = M ′′ ⊂ B(H).
I Quantum groups ←→ nice Hopf algebras/tensor categories.
I Quantum Riemannian manifolds ←→ Connes’ spectraltriples.
Our Problem: What is the quantum analogue of a graph?
![Page 10: Quantum Graphs and Quantum Cuntz-Krieger Algebrasgressman/maam2/brannan-MAAM2.pdfQuantum Cuntz-Krieger Algebras Michael Brannan Texas A&M University The Second Mid-Atlantic Analysis](https://reader035.vdocument.in/reader035/viewer/2022081620/610c53c22df61a03533220fc/html5/thumbnails/10.jpg)
Quantizing GraphsInput: A finite directed graph G = (V,AG) with |V | = n verticesand no multiple edges.Output: A C∗-algebraic description of G that can be madenon-commutative!
I Let B = C(V ) = C∗((pv)v∈V
∣∣ p∗v = p2v = pv,
∑v pv = 1
),
the C∗-algebra of functions on V .I Equip B with the canonical trace functional ψ : B → C given
by counting measure on V . ψ is “canonical” because
ψ = TrEnd(B)
∣∣∣B
where B ↪→ End(B) = left-regular representation.
I Use (GNS) inner product 〈f |g〉 = ψ(f∗g) to turn B into aHilbert space B = L2(B) = `2(V ).
I View the adjacency matrix AG as a linear map AG ∈ End(B)in the obvious way:
AGpw =∑w∈V
AG(v, w)pv.
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Quantizing GraphsInput: A finite directed graph G = (V,AG) with |V | = n verticesand no multiple edges.Output: A C∗-algebraic description of G that can be madenon-commutative!
I Let B = C(V ) = C∗((pv)v∈V
∣∣ p∗v = p2v = pv,
∑v pv = 1
),
the C∗-algebra of functions on V .I Equip B with the canonical trace functional ψ : B → C given
by counting measure on V . ψ is “canonical” because
ψ = TrEnd(B)
∣∣∣B
where B ↪→ End(B) = left-regular representation.
I Use (GNS) inner product 〈f |g〉 = ψ(f∗g) to turn B into aHilbert space B = L2(B) = `2(V ).
I View the adjacency matrix AG as a linear map AG ∈ End(B)in the obvious way:
AGpw =∑w∈V
AG(v, w)pv.
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Quantizing GraphsInput: A finite directed graph G = (V,AG) with |V | = n verticesand no multiple edges.Output: A C∗-algebraic description of G that can be madenon-commutative!
I Let B = C(V ) = C∗((pv)v∈V
∣∣ p∗v = p2v = pv,
∑v pv = 1
),
the C∗-algebra of functions on V .I Equip B with the canonical trace functional ψ : B → C given
by counting measure on V . ψ is “canonical” because
ψ = TrEnd(B)
∣∣∣B
where B ↪→ End(B) = left-regular representation.
I Use (GNS) inner product 〈f |g〉 = ψ(f∗g) to turn B into aHilbert space B = L2(B) = `2(V ).
I View the adjacency matrix AG as a linear map AG ∈ End(B)in the obvious way:
AGpw =∑w∈V
AG(v, w)pv.
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Quantizing Adjacency MatricesSo far:
G (B = C(V ), ψ = TrEnd(B)
∣∣∣B, AG ∈ End(B)
).
Problem: How to intrinsically capture the fact that AG is anadjacency matrix at the level of the function algebra B?
Answer: AG ∈Mn({0, 1}) ⇐⇒ AG is idempotent with respectto Schur multiplication in End(B)!
I Let m : B ⊗B → B be the algebra multiplication,m∗ : B → B ⊗B its Hilbertian adjoint.
m(pv ⊗ pw) = δv,wpv, m∗(pv) = (pv ⊗ pv).
I Simple calculation: Given X,Y ∈ End(B),
Schur Multiplication : X ? Y = m(X ⊗ Y )m∗
Conclusion: AG ∈Mn({0, 1}) ⇐⇒ m(AG ⊗AG)m∗ = AG .
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Quantizing Adjacency MatricesSo far:
G (B = C(V ), ψ = TrEnd(B)
∣∣∣B, AG ∈ End(B)
).
Problem: How to intrinsically capture the fact that AG is anadjacency matrix at the level of the function algebra B?Answer: AG ∈Mn({0, 1}) ⇐⇒ AG is idempotent with respectto Schur multiplication in End(B)!
I Let m : B ⊗B → B be the algebra multiplication,m∗ : B → B ⊗B its Hilbertian adjoint.
m(pv ⊗ pw) = δv,wpv, m∗(pv) = (pv ⊗ pv).
I Simple calculation: Given X,Y ∈ End(B),
Schur Multiplication : X ? Y = m(X ⊗ Y )m∗
Conclusion: AG ∈Mn({0, 1}) ⇐⇒ m(AG ⊗AG)m∗ = AG .
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Quantum Graphs
We now go non-commutative! Let B be a finite-dimensionalC∗-algebra equipped with its canonical trace functional
ψ : B → C; ψ = TrEnd(B)
∣∣∣B
(If B =⊕
kMn(k)(C)), ψ =∑
k n(k)Trn(k)(·) = Plancharel trace).
DefinitionA linear map A ∈ End(B) is called a quantum adjacency matrix if
m(A⊗A)m∗ = A (A is a “quantum Schur idempotent”)
We call the triple G = (B,ψ,A) a quantum graph.
I Classical graphs ⇐⇒ quantum graphs with abelian B.
I For certain applications, we can consider quantum graphsG = (B,ψ,A) with non-tracial states ψ...but not today.
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Quantum Graphs
We now go non-commutative! Let B be a finite-dimensionalC∗-algebra equipped with its canonical trace functional
ψ : B → C; ψ = TrEnd(B)
∣∣∣B
(If B =⊕
kMn(k)(C)), ψ =∑
k n(k)Trn(k)(·) = Plancharel trace).
DefinitionA linear map A ∈ End(B) is called a quantum adjacency matrix if
m(A⊗A)m∗ = A (A is a “quantum Schur idempotent”)
We call the triple G = (B,ψ,A) a quantum graph.
I Classical graphs ⇐⇒ quantum graphs with abelian B.
I For certain applications, we can consider quantum graphsG = (B,ψ,A) with non-tracial states ψ...but not today.
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Some Basic ExamplesFix any pair (B,ψ).
1. The complete quantum graph over B is K(B) = (B,ψ,A),
A = ψ(·)1B Classical case: A =
1 · · · 1... 1
...1 · · · 1
2. The trivial quantum graph over B is T (B) = (B,ψ,A) where
A = idB→B Classical case: A =
1 · · · 0... 1
...0 · · · 1
3. If B =
⊕kMn(k)(C) and d = (d(k))k ∈
⊕kDn(k) ⊂ B is
diagonal with Trn(k)(d(k)) = 1 for all k, then
Ad(x) = dx is a quantum adjacency matrix.
We call D(B) = (B,ψ,Ad) a diagonal quantum graph over B.
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More Examples: Quantum Edge Space PictureWriting down interesting examples of quantum adjacency matricesmight at first seem not so easy.
Alternate Approach [Nik Weaver]: Recall that a graph G iscompletely described by its edge relation E ⊂ V × V .I Consider the linear subspace
SG = span{ex,y : (x, y) ∈ E}} ⊂ B(`2(V )).
I Then SG is a bimodule over C(V ) = C(V )′ ⊂ B(`2(V )).I (Weaver) Every C(V )′ − C(V )′-bimodule S ⊂ B(`2(V )) is of
the form SG for some unique G = (V,E).
Definition (Weaver 2010)
Let B ⊆ B(H) be a finite dimensional C∗-algebra. A quantumgraph on B is B′-B′-bimodule
B′SB′ ⊆ B(H).
Think of B′SB′ as the collection of all “edge operators” that“connect quantum vertices” in the graph.
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More Examples: Quantum Edge Space PictureWriting down interesting examples of quantum adjacency matricesmight at first seem not so easy.Alternate Approach [Nik Weaver]: Recall that a graph G iscompletely described by its edge relation E ⊂ V × V .I Consider the linear subspace
SG = span{ex,y : (x, y) ∈ E}} ⊂ B(`2(V )).
I Then SG is a bimodule over C(V ) = C(V )′ ⊂ B(`2(V )).I (Weaver) Every C(V )′ − C(V )′-bimodule S ⊂ B(`2(V )) is of
the form SG for some unique G = (V,E).
Definition (Weaver 2010)
Let B ⊆ B(H) be a finite dimensional C∗-algebra. A quantumgraph on B is B′-B′-bimodule
B′SB′ ⊆ B(H).
Think of B′SB′ as the collection of all “edge operators” that“connect quantum vertices” in the graph.
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More Examples: Quantum Edge Space PictureWriting down interesting examples of quantum adjacency matricesmight at first seem not so easy.Alternate Approach [Nik Weaver]: Recall that a graph G iscompletely described by its edge relation E ⊂ V × V .I Consider the linear subspace
SG = span{ex,y : (x, y) ∈ E}} ⊂ B(`2(V )).
I Then SG is a bimodule over C(V ) = C(V )′ ⊂ B(`2(V )).I (Weaver) Every C(V )′ − C(V )′-bimodule S ⊂ B(`2(V )) is of
the form SG for some unique G = (V,E).
Definition (Weaver 2010)
Let B ⊆ B(H) be a finite dimensional C∗-algebra. A quantumgraph on B is B′-B′-bimodule
B′SB′ ⊆ B(H).
Think of B′SB′ as the collection of all “edge operators” that“connect quantum vertices” in the graph.
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Quantum Edge Space vs Quantum Adjacency Matrices
Both approaches to quantum graphs are essentially the same:
I Fact 1: Given B ⊆ B(H), a bimodule B′SB′ ⊆ B(H) isuniquely determined by a self-adjoint projection
p =∑i
ai⊗bopi ∈ B⊗Bop;S = p·B(H) =
{∑i
aiXbi : X ∈ B(H)}.
I Fact 2: Any such p = p∗ = p2 ∈ B ⊗Bop arises as the“Choi-Jamio lkowski” matrix of a completely positivequantum adjacency matrix A ∈ End(B) given by
p = (A⊗ 1)m∗(1B).
Conclusion: Quantum graphs G = (B,ψ,A) (with completelypositive A) ←→ operator spaces S ⊆ B(H) that areB′-B′-bimodules.
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Quantum Graphs in “Nature”Quantum graphs arise naturally in problems in quantuminformation theory and representation theory.I Zero-error channel capacity: Given finite sets V,W and a
noisy channel with transition probabilities (p(w|v))v∈V,w∈W ,can form the confusability graph GΦ = (V,E) where
(v1, v2) ∈ E ⇐⇒ ∃w ∈W s.t. p(w|v1)p(w|v2) 6= 0.
The zero-error capacity of Φ: Is the Shannon capacity of GΦ.I In Quantum Information Theory, one considers quantum
channelsΦ : L1(B,ψ)→Mk(C).
The Zero-error capacity of Φ: study Shannon capacity ofthe non-commutative confusability graph SΦ of Φ:SΦ ⊂ B(L2(B)) is a B′-B′-bimodule.
I Qgraphs G = (B,ψ,A) appear in the theory of non-localgames (graph isomorphism games), quantum teleportationand superdense coding schemes in QIT, representation theoryof quantum symmetry groups of graphs, ...
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Quantum Graphs in “Nature”Quantum graphs arise naturally in problems in quantuminformation theory and representation theory.I Zero-error channel capacity: Given finite sets V,W and a
noisy channel with transition probabilities (p(w|v))v∈V,w∈W ,can form the confusability graph GΦ = (V,E) where
(v1, v2) ∈ E ⇐⇒ ∃w ∈W s.t. p(w|v1)p(w|v2) 6= 0.
The zero-error capacity of Φ: Is the Shannon capacity of GΦ.I In Quantum Information Theory, one considers quantum
channelsΦ : L1(B,ψ)→Mk(C).
The Zero-error capacity of Φ: study Shannon capacity ofthe non-commutative confusability graph SΦ of Φ:SΦ ⊂ B(L2(B)) is a B′-B′-bimodule.
I Qgraphs G = (B,ψ,A) appear in the theory of non-localgames (graph isomorphism games), quantum teleportationand superdense coding schemes in QIT, representation theoryof quantum symmetry groups of graphs, ...
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Cuntz-Krieger Algebras of graphsLet G = (V,A) be a finite graph without multiple edges.
The Cuntz-Krieger algebra O(G) = OA is the universalC∗-algebra generated by partial isometries (sv)v∈V satisfying therelations
(CK1)∑
v∈V svs∗v = 1.
(CK2) s∗vsv =∑
w∈V A(v, w)sws∗w.
Why study these operator algebras?I These algebras turn out to encode the symbolic dynamics
associated to a graph G: They provide new conjugacyinvariants for the topological Markov chain (XA, σA)associated to G.
XA = {(vi)i∈N : A(vi, vi+1) = 1}} space of infinite paths,
σA : XA → XA; σA(v1, v2, . . .) = (v2, v3, . . .).
I They provide a rich class of C∗-algebras that are amenable toclassification.
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Cuntz-Krieger Algebras of graphsLet G = (V,A) be a finite graph without multiple edges.
The Cuntz-Krieger algebra O(G) = OA is the universalC∗-algebra generated by partial isometries (sv)v∈V satisfying therelations
(CK1)∑
v∈V svs∗v = 1.
(CK2) s∗vsv =∑
w∈V A(v, w)sws∗w.
Why study these operator algebras?I These algebras turn out to encode the symbolic dynamics
associated to a graph G: They provide new conjugacyinvariants for the topological Markov chain (XA, σA)associated to G.
XA = {(vi)i∈N : A(vi, vi+1) = 1}} space of infinite paths,
σA : XA → XA; σA(v1, v2, . . .) = (v2, v3, . . .).
I They provide a rich class of C∗-algebras that are amenable toclassification.
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Examples1. G = Kn the complete graph on n vertices. Then O(Kn) has
generators (si)ni=1 satisfying
(CK1)∑i
sis∗i = 1 & (CK2) s∗i si
(=∑j
sjs∗j
)= 1.
This is the famous Cuntz algebra On!
2. G = Tn (trivial graph on n vertices, with self-loops). ThenO(Tn) has generators (si)
ni=1 satisfying∑
i
sis∗i = 1, s∗i si = sis
∗i =⇒ O(Tn) = C(T)⊕n.
3. Let G ↔ A =
(1 11 0
). Then
O(G) = C∗(s1, s2 | s1s∗1+s2s∗2=1
s∗1s1=1, s∗2s2=s1s∗1
).
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Examples1. G = Kn the complete graph on n vertices. Then O(Kn) has
generators (si)ni=1 satisfying
(CK1)∑i
sis∗i = 1 & (CK2) s∗i si
(=∑j
sjs∗j
)= 1.
This is the famous Cuntz algebra On!
2. G = Tn (trivial graph on n vertices, with self-loops). ThenO(Tn) has generators (si)
ni=1 satisfying∑
i
sis∗i = 1, s∗i si = sis
∗i =⇒ O(Tn) = C(T)⊕n.
3. Let G ↔ A =
(1 11 0
). Then
O(G) = C∗(s1, s2 | s1s∗1+s2s∗2=1
s∗1s1=1, s∗2s2=s1s∗1
).
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Examples1. G = Kn the complete graph on n vertices. Then O(Kn) has
generators (si)ni=1 satisfying
(CK1)∑i
sis∗i = 1 & (CK2) s∗i si
(=∑j
sjs∗j
)= 1.
This is the famous Cuntz algebra On!
2. G = Tn (trivial graph on n vertices, with self-loops). ThenO(Tn) has generators (si)
ni=1 satisfying∑
i
sis∗i = 1, s∗i si = sis
∗i =⇒ O(Tn) = C(T)⊕n.
3. Let G ↔ A =
(1 11 0
). Then
O(G) = C∗(s1, s2 | s1s∗1+s2s∗2=1
s∗1s1=1, s∗2s2=s1s∗1
).
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General properties of Cuntz-Krieger algebras
I O(G) 6= 0 as long as AG 6= 0.
I O(G) is always nuclear.
I O(G) is always unital.
I O(G) are classified, up to isomorphism.
I Include many naturally occuring examples of C∗-algebras.
I Can be studied using graph theoretical/symbolic dynamicaltools.
Goal: Generalize Cuntz-Krieger Algebras to Quantum Graphs!
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General properties of Cuntz-Krieger algebras
I O(G) 6= 0 as long as AG 6= 0.
I O(G) is always nuclear.
I O(G) is always unital.
I O(G) are classified, up to isomorphism.
I Include many naturally occuring examples of C∗-algebras.
I Can be studied using graph theoretical/symbolic dynamicaltools.
Goal: Generalize Cuntz-Krieger Algebras to Quantum Graphs!
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Quantum Cuntz-Krieger Algebras
Now let G = (B,ψ,A) be a quantum graph. We want to build anoperator algebra FO(G) out of G as above. Here’s our attempt.
Definition (B-Eifler-Voigt-Weber)
The quantum Cuntz-Krieger Algebra associated to G is theuniversal C∗-algebra FO(G) generated by the range of a linear mapS : B → FO(G) satisfying the relations
1. µ(1⊗ µ)(S ⊗ S∗ ⊗ S)(1⊗m∗)m∗ = S
2. µ(S∗ ⊗ S)m∗ = µ(S ⊗ S∗)m∗Awhere µ : FO(G)⊗ FO(G)→ FO(G) is the multiplication map andS∗(b) = S(b∗)∗.
Why is this a reasonable definition?
TheoremIf G is classical (i.e., B = C(V ) is abelian), then FO(G) is theusual Cuntz-Krieger algebra O(G) associated to G (up to aKK-equivalence).
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Quantum Cuntz-Krieger Algebras
Now let G = (B,ψ,A) be a quantum graph. We want to build anoperator algebra FO(G) out of G as above. Here’s our attempt.
Definition (B-Eifler-Voigt-Weber)
The quantum Cuntz-Krieger Algebra associated to G is theuniversal C∗-algebra FO(G) generated by the range of a linear mapS : B → FO(G) satisfying the relations
1. µ(1⊗ µ)(S ⊗ S∗ ⊗ S)(1⊗m∗)m∗ = S
2. µ(S∗ ⊗ S)m∗ = µ(S ⊗ S∗)m∗Awhere µ : FO(G)⊗ FO(G)→ FO(G) is the multiplication map andS∗(b) = S(b∗)∗.
Why is this a reasonable definition?
TheoremIf G is classical (i.e., B = C(V ) is abelian), then FO(G) is theusual Cuntz-Krieger algebra O(G) associated to G (up to aKK-equivalence).
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Quantum Cuntz-Krieger Algebras
Now let G = (B,ψ,A) be a quantum graph. We want to build anoperator algebra FO(G) out of G as above. Here’s our attempt.
Definition (B-Eifler-Voigt-Weber)
The quantum Cuntz-Krieger Algebra associated to G is theuniversal C∗-algebra FO(G) generated by the range of a linear mapS : B → FO(G) satisfying the relations
1. µ(1⊗ µ)(S ⊗ S∗ ⊗ S)(1⊗m∗)m∗ = S
2. µ(S∗ ⊗ S)m∗ = µ(S ⊗ S∗)m∗Awhere µ : FO(G)⊗ FO(G)→ FO(G) is the multiplication map andS∗(b) = S(b∗)∗.
Why is this a reasonable definition?
TheoremIf G is classical (i.e., B = C(V ) is abelian), then FO(G) is theusual Cuntz-Krieger algebra O(G) associated to G (up to aKK-equivalence).
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Quantum CK-Algebras: Unpacking the Definition
G = (B,ψ,A) FO(G) = C∗(
(S(b))b∈B
∣∣∣ S:B→OG linearµ(1⊗µ)(S⊗S∗⊗S)(1⊗m∗)m∗=Sµ(S∗⊗S)m∗=µ(S⊗S∗)m∗A
)Write B =
⊕mk=1Mn(k)(C) with standard matrix unit basis {e(k)
ij }.For 1 ≤ k ≤ m, put f
(k)ij = n(k)−1e
(k)ij and
S(k) = [S(f(k)ij )] ∈Mn(k)(FO(G)).
Then µ(1⊗ µ)(S ⊗ S∗ ⊗ S)(1⊗m∗)m∗ = S
⇐⇒ S(k)S(k)∗S(k) = S(k), k = 1, . . . ,m
⇐⇒ each S(k)is a partial isometry in Mn(k)(FO(G)).
µ(S∗ ⊗ S)m∗ = µ(S ⊗ S∗)m∗A
⇐⇒ S(k)∗S(k) = (S(1)S(1)∗, . . . , S(m)S(m)∗)A
where Axyzijk = n(z)n(k)A
xyzijk (reweighted adjacency matrix).
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Quantum CK-Algebras: Unpacking the Definition
G = (B,ψ,A) FO(G) = C∗(
(S(b))b∈B
∣∣∣ S:B→OG linearµ(1⊗µ)(S⊗S∗⊗S)(1⊗m∗)m∗=Sµ(S∗⊗S)m∗=µ(S⊗S∗)m∗A
)Write B =
⊕mk=1Mn(k)(C) with standard matrix unit basis {e(k)
ij }.For 1 ≤ k ≤ m, put f
(k)ij = n(k)−1e
(k)ij and
S(k) = [S(f(k)ij )] ∈Mn(k)(FO(G)).
Then µ(1⊗ µ)(S ⊗ S∗ ⊗ S)(1⊗m∗)m∗ = S
⇐⇒ S(k)S(k)∗S(k) = S(k), k = 1, . . . ,m
⇐⇒ each S(k)is a partial isometry in Mn(k)(FO(G)).
µ(S∗ ⊗ S)m∗ = µ(S ⊗ S∗)m∗A
⇐⇒ S(k)∗S(k) = (S(1)S(1)∗, . . . , S(m)S(m)∗)A
where Axyzijk = n(z)n(k)A
xyzijk (reweighted adjacency matrix).
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Quantum CK-Algebras: Unpacking the Definition
Summary: Given G = (B,ψ,A), B =⊕m
k=1Mn(k), the quantumCuntz-Krieger Algebra OG is:
1. Generated by the coefficients of m matrix partialisometries S(k) ∈Mn(k)(FO(G)),
2. Subject to the quantum Cuntz-Krieger relations
S(k)∗S(k) = (S(1)S(1)∗, . . . , S(m)S(m)∗)A.
3. BUT: There seems to be no natural analogue of the relation
“∑k
S(k)(S(k))∗ = 1′′ (incompatible matrix sizes)
=⇒ O(G) might not be unital in general.
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Basic Examples, Basic PropertiesExample 1: Trivial Quantum Graph G = T (Mn).
FO(T (Mn)) = C∗(Sij , 1 ≤ i, j ≤ n | S=[Sij ] is a partial isometry∑
l S∗liSlj=
∑l SilS
∗jl ⇐⇒ S∗S=SS∗
)
I FO(T (Mn)) is non-unital, non-nuclear.
I We have quotient maps
OT (Mn) → C(T)?n (non-unital free product)
Sij 7→ δijzi.
OT (Mn) → Unc(n)
Sij 7→ uij .
where U (nc)(n) = C∗(uij , 1 ≤ i, j ≤ n
∣∣∣ U = [uij ] unitary)
is Brown’s Universal noncommutative unitary algebra.
I In fact Mn(FO(T (Mn))+) ∼= Mn ? (C(S1)⊕ C).
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Basic Examples, Basic PropertiesExample 1: Trivial Quantum Graph G = T (Mn).
FO(T (Mn)) = C∗(Sij , 1 ≤ i, j ≤ n | S=[Sij ] is a partial isometry∑
l S∗liSlj=
∑l SilS
∗jl ⇐⇒ S∗S=SS∗
)
I FO(T (Mn)) is non-unital, non-nuclear.
I We have quotient maps
OT (Mn) → C(T)?n (non-unital free product)
Sij 7→ δijzi.
OT (Mn) → Unc(n)
Sij 7→ uij .
where U (nc)(n) = C∗(uij , 1 ≤ i, j ≤ n
∣∣∣ U = [uij ] unitary)
is Brown’s Universal noncommutative unitary algebra.
I In fact Mn(FO(T (Mn))+) ∼= Mn ? (C(S1)⊕ C).
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Basic Examples, Basic Properties
Example 2: Diagonal Quantum Graph G = D(M2)Ad(eij) = δi,12eij .
FO(D(M2)) = C∗(Sij , 1 ≤ i, j ≤ 2 | S=[Sij ] is a partial isometry∑
l S∗liSlj=δ{i=j=1}
∑l 2S1lS
∗1l
)
I Get S∗l2Sl2 = 0 =⇒ Sl2 = 0.
I Thus, the canonical map S : B → OG need not be injective!
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Basic Examples, Basic PropertiesExample 3: Complete Quantum Graph G = K(Mn).
FO(K(Mn)) = C∗(Sij , 1 ≤ i, j ≤ n |
S=[Sij ] is a partial isometry∑l S∗liSlj=δi,j
(n∑
x,l SxlS∗xl
))
I What is this algebra? Is it unital? Nuclear? ...I We have a quotient map
FO(K(Mn))→ On2 = O(Kn2) (the Cuntz algebra)
Sij 7→ n−1/2Sij (Sij = standard Cuntz isometry).
I Very surprising/mysterious theorem: The abovehomomorphism is actually an isomorphismFO(K(Mn)) ∼= On2 .
I So FO(K(Mn)) is indeed unital, nuclear, etc.I The proof: Uses the notion of a quantum isomorphism
between quantum graphs (aka superdense coding in QIT):K(Mn) and Kn2 turn out to be quantum isomorphic.
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Basic Examples, Basic PropertiesExample 3: Complete Quantum Graph G = K(Mn).
FO(K(Mn)) = C∗(Sij , 1 ≤ i, j ≤ n |
S=[Sij ] is a partial isometry∑l S∗liSlj=δi,j
(n∑
x,l SxlS∗xl
))I What is this algebra? Is it unital? Nuclear? ...I We have a quotient map
FO(K(Mn))→ On2 = O(Kn2) (the Cuntz algebra)
Sij 7→ n−1/2Sij (Sij = standard Cuntz isometry).
I Very surprising/mysterious theorem: The abovehomomorphism is actually an isomorphismFO(K(Mn)) ∼= On2 .
I So FO(K(Mn)) is indeed unital, nuclear, etc.I The proof: Uses the notion of a quantum isomorphism
between quantum graphs (aka superdense coding in QIT):K(Mn) and Kn2 turn out to be quantum isomorphic.
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Basic Examples, Basic PropertiesExample 3: Complete Quantum Graph G = K(Mn).
FO(K(Mn)) = C∗(Sij , 1 ≤ i, j ≤ n |
S=[Sij ] is a partial isometry∑l S∗liSlj=δi,j
(n∑
x,l SxlS∗xl
))I What is this algebra? Is it unital? Nuclear? ...I We have a quotient map
FO(K(Mn))→ On2 = O(Kn2) (the Cuntz algebra)
Sij 7→ n−1/2Sij (Sij = standard Cuntz isometry).
I Very surprising/mysterious theorem: The abovehomomorphism is actually an isomorphismFO(K(Mn)) ∼= On2 .
I So FO(K(Mn)) is indeed unital, nuclear, etc.I The proof: Uses the notion of a quantum isomorphism
between quantum graphs (aka superdense coding in QIT):K(Mn) and Kn2 turn out to be quantum isomorphic.
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Conclusion
Quantum graphs and Quantum Cuntz-Krieger Algebras seem tohave some new and interesting properties! Lots of work to bedone!
THANKS FOR LISTENING.