quantum graphs and quantum cuntz-krieger algebrasgressman/maam2/brannan-maam2.pdfquantum...

Quantum Graphsand

Quantum Cuntz-Krieger Algebras

Michael Brannan

Texas A&M University

The Second Mid-Atlantic Analysis MeetingOctober 18, 2020

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)

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).

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).

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).

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?

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?

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?

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.

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.

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.

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 .

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 .

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.

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.

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.

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.

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.

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.

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.

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, ...

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, ...

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.

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.

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

).

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

).

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

).

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!

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!

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).

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).

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).

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).

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).

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.

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).

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).

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!

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.

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.

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.

Conclusion

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

THANKS FOR LISTENING.