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Quantum Graphs and Quantum Cuntz-Krieger Algebras Michael Brannan Texas A&M University The Second Mid-Atlantic Analysis Meeting October 18, 2020

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Page 1: 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

Quantum Graphsand

Quantum Cuntz-Krieger Algebras

Michael Brannan

Texas A&M University

The Second Mid-Atlantic Analysis MeetingOctober 18, 2020

Michael
Pencil
Page 2: 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

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)

Page 3: 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

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 4: 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

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

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
Michael
Pencil
Page 7: 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

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 8: 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

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

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

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.

Page 11: 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

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.

Page 12: 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

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.

Page 13: 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

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 .

Page 14: 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

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 .

Michael
Pencil
Page 15: 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

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.

Page 16: 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

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.

Page 17: 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

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.

Page 18: 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

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.

Page 19: 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

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.

Page 20: 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

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.

Page 21: 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

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.

Page 22: 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

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, ...

Page 23: 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

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, ...

Page 24: 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

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.

Michael
Pencil
Page 25: 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

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.

Page 26: 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

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

).

Page 27: 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

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

).

Page 28: 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

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

).

Page 29: 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

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!

Page 30: 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

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!

Page 31: 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

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).

Page 32: 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

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).

Page 33: 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

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).

Page 34: 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

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).

Page 35: 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

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).

Page 36: 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

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.

Page 37: 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

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).

Page 38: 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

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).

Page 39: 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

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!

Page 40: 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

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.

Page 41: 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

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.

Page 42: 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

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.

Page 43: 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

Conclusion

Quantum graphs and Quantum Cuntz-Krieger Algebras seem tohave some new and interesting properties! Lots of work to bedone!

THANKS FOR LISTENING.