Operator Algebras and Noncommutative Geometry

David E Evans

Cardiff University

LMS Graduate Meeting 15 November 2013

1 / 24 Overview

Gelfand Theorem

operator algebras

dimension

rotation algebra and orbifolds

Stone-von Neumann theorem

2 / 24 characters χx : A → C , f → f (x) Hom(A, C) Irred Rep (A, B(H)) Spectrum Spec(A)

recover topology of [−1, 1] from C[−1.1]

Algebra of continuous functions

A = C[−1, 1] = continuous maps f :[−1, 1] → C

||f || = sup x |f (x)|

3 / 24 Irred Rep (A, B(H)) Spectrum Spec(A)

recover topology of [−1, 1] from C[−1.1]

Algebra of continuous functions

A = C[−1, 1] = continuous maps f :[−1, 1] → C

||f || = sup x |f (x)|

characters χx : A → C , f → f (x) Hom(A, C)

3 / 24 recover topology of [−1, 1] from C[−1.1]

Algebra of continuous functions

A = C[−1, 1] = continuous maps f :[−1, 1] → C

||f || = sup x |f (x)|

characters χx : A → C , f → f (x) Hom(A, C) Irred Rep (A, B(H)) Spectrum Spec(A)

3 / 24 Algebra of continuous functions

A = C[−1, 1] = continuous maps f :[−1, 1] → C

||f || = sup x |f (x)|

characters χx : A → C , f → f (x) Hom(A, C) Irred Rep (A, B(H)) Spectrum Spec(A)

recover topology of [−1, 1] from C[−1.1]

3 / 24 2 Represent f ∈ C[−1, 1] on L (T) by multiplication operators

2 π(f ) ∈ B(L (T)) (π(f )g)(x) = f (x)g(x)

2 A = C[−1, 1] ⊂ B(L (T))

Geland-Naimark Theorem

If A is a Banach ∗-algebra where

kT ∗T k = kT k2, T ∈ A then A is a C ∗-algebra, — there exists an isometric ∗-homomorphism π : A → B(H) for a Hilbert space H.

4 / 24 Geland-Naimark Theorem

If A is a Banach ∗-algebra where

kT ∗T k = kT k2, T ∈ A then A is a C ∗-algebra, — there exists an isometric ∗-homomorphism π : A → B(H) for a Hilbert space H.

2 Represent f ∈ C[−1, 1] on L (T) by multiplication operators

2 π(f ) ∈ B(L (T)) (π(f )g)(x) = f (x)g(x)

2 A = C[−1, 1] ⊂ B(L (T))

4 / 24 H = H∗ ∈ B(H), C ∗(H, 1) ' C(σ(H)) Spec(A) = σ(H) f (H), f ∈ C(σ(H)) x → exp(itx), U(t) = exp(itH) U(t)U(s) = U(s + t)

Gelfand Theorem

If A is a commutative, unital Banach ∗-algebra with kf ∗f k = kf k2, f ∈ A then A ' C(Ω) for a compact Hausdorff Ω.

5 / 24 Gelfand Theorem

If A is a commutative, unital Banach ∗-algebra with kf ∗f k = kf k2, f ∈ A then A ' C(Ω) for a compact Hausdorff Ω.

H = H∗ ∈ B(H), C ∗(H, 1) ' C(σ(H)) Spec(A) = σ(H) f (H), f ∈ C(σ(H)) x → exp(itx), U(t) = exp(itH) U(t)U(s) = U(s + t)

5 / 24 A separable , Spec A = • =⇒ A 'K(H)

B = continuous maps from [−1, 1] → M2

Spec(B) = [−1, 1] f → f (x)

noncommutative algebra

points to matrices Mn representations are all multiples of x → x

Spectrum of Mn = singleton • Spectrum K(H) = •

6 / 24 B = continuous maps from [−1, 1] → M2

Spec(B) = [−1, 1] f → f (x)

noncommutative algebra

points to matrices Mn representations are all multiples of x → x

Spectrum of Mn = singleton • Spectrum K(H) = •

A separable , Spec A = • =⇒ A 'K(H)

6 / 24 noncommutative algebra

points to matrices Mn representations are all multiples of x → x

Spectrum of Mn = singleton • Spectrum K(H) = •

A separable , Spec A = • =⇒ A 'K(H)

B = continuous maps from [−1, 1] → M2

Spec(B) = [−1, 1] f → f (x)

6 / 24 Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

7 / 24 Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

7 / 24 Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

7 / 24 Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

7 / 24  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

Z2 action on B = C([−1, 1], M2)

7 / 24 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U

7 / 24 Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2)

orbifold

[−1, 1] under Z2 action x → −x

Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C

7 / 24 orbifold

[−1, 1] under Z2 action x → −x

Z2 action on B = C([−1, 1], M2)  0 1  by x → −x and U()U where U = 1 0 f → g(x) = Uf (−x)U 2 BZ2 ⊂ B : f (x) ∈ M2, x ∈ (0, 1] and f (0) = Uf (0)U i.e. f (0) ∈ C Spec(BZ2 ) =

∗ BZ2 = C ([−1, 1]/Z2) 7 / 24 Iterating

C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

Nn Nn 2 M2 ' M2n ' End( C )

Nn M2 ' M2n

the Fermion algebra

M2n → M2n+2n = M2n+1 ' M2n ⊗ M2  a 0  a → ' a ⊗ 1 0 a

8 / 24 the Fermion algebra

M2n → M2n+2n = M2n+1 ' M2n ⊗ M2  a 0  a → ' a ⊗ 1 0 a Iterating

C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

Nn Nn 2 M2 ' M2n ' End( C )

Nn M2 ' M2n

8 / 24 C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

X n Trace(a) = aii /2

 a 0  a → (1) 0 a

e = e∗ = e2 : 1 2 3 dim(e) = trace(e) ∈ {0, 2n , 2n , 2n ..., 1} → N[1/2]

K0(⊗NM2) = Z[1/2]

Dimensions of Operator Algebras

Projections e = e2 = e∗ in A ⊂ B(H) correspond to decompositions H = eH ⊕ eH⊥

9 / 24 e = e∗ = e2 : 1 2 3 dim(e) = trace(e) ∈ {0, 2n , 2n , 2n ..., 1} → N[1/2]

K0(⊗NM2) = Z[1/2]

Dimensions of Operator Algebras

Projections e = e2 = e∗ in A ⊂ B(H) correspond to decompositions H = eH ⊕ eH⊥

C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

X n Trace(a) = aii /2

 a 0  a → (1) 0 a

9 / 24 K0(⊗NM2) = Z[1/2]

Dimensions of Operator Algebras

Projections e = e2 = e∗ in A ⊂ B(H) correspond to decompositions H = eH ⊕ eH⊥

C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

X n Trace(a) = aii /2

 a 0  a → (1) 0 a

e = e∗ = e2 : 1 2 3 dim(e) = trace(e) ∈ {0, 2n , 2n , 2n ..., 1} → N[1/2]

9 / 24 Dimensions of Operator Algebras

Projections e = e2 = e∗ in A ⊂ B(H) correspond to decompositions H = eH ⊕ eH⊥

C → M2 → M4 → M8 → · · · → M2n → M2n+1 → · · ·

X n Trace(a) = aii /2

 a 0  a → (1) 0 a

e = e∗ = e2 : 1 2 3 dim(e) = trace(e) ∈ {0, 2n , 2n , 2n ..., 1} → N[1/2]

K0(⊗NM2) = Z[1/2]

9 / 24 discrete rotation matrices

 0 1 0   0 0    V =  ··     · 1  1 0 0  1 0   0 ω    U =  ··     ·  0 0 ωn−1 ωn = 1

VUV ∗ = ωU

r Ver = er−1 Uer = ω er 10 / 24 −1 ∗ ∼ Ad V = V ( · )V leaves C (U) = C(T) invariant and acts by rotation through θ on the circle

if θ is irrational, Aθ is simple, unique 2 cf. C(T ) when θ = 0.

rotation algebra

2 multiplication and translation operators on L (T),

Uh(z) = zh(z)

Vh(z) = h(exp(2πiθ)z)

VU = exp(2πiθ)UV

∗ Aθ = C (U, V ) VUV −1 = exp(2πiθ)U;

11 / 24 if θ is irrational, Aθ is simple, unique 2 cf. C(T ) when θ = 0.

rotation algebra

2 multiplication and translation operators on L (T),

Uh(z) = zh(z)

Vh(z) = h(exp(2πiθ)z)

VU = exp(2πiθ)UV

∗ Aθ = C (U, V ) VUV −1 = exp(2πiθ)U; −1 ∗ ∼ Ad V = V ( · )V leaves C (U) = C(T) invariant and acts by rotation through θ on the circle

11 / 24 rotation algebra

2 multiplication and translation operators on L (T),

Uh(z) = zh(z)

Vh(z) = h(exp(2πiθ)z)

VU = exp(2πiθ)UV

∗ Aθ = C (U, V ) VUV −1 = exp(2πiθ)U; −1 ∗ ∼ Ad V = V ( · )V leaves C (U) = C(T) invariant and acts by rotation through θ on the circle

if θ is irrational, Aθ is simple, unique 2 cf. C(T ) when θ = 0.

11 / 24 trace on rotation algebra

2 T action α on Aθ by

2 (t1, t2) ∈ T

U → t1U, V → t2V

RR unique trace τ = αs,t ds dt, Z n τ(f (U)V ) = f (t) dt δn,0

K-group of dimensions generated by trace of projections

12 / 24 ∼ K0(Aθ) = Z + θZ (m, n) n + θm > 0 ∼ 2 2 K1(Aθ) = Z , If θ = 0, K1(C(T )) is generated by the coordinate unitary maps (z1, z2) → zi , i = 1, 2, i.e. U and V .

13 / 24  F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

14 / 24  F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

14 / 24  F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

14 / 24  F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

14 / 24 −1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

 F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

14 / 24 −1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

p g continuous function = F (1 − F )χ[a,b] Z τ(eθ) = f = θ

projections in rotation algebras

 F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

 F  f = χ + [b,c] 1 − F

14 / 24 projections in rotation algebras

 F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

 F  f = χ + [b,c] 1 − F p g continuous function = F (1 − F )χ[a,b] 14 / 24 Z τ(eθ) = f = θ projections in rotation algebras

 F pF (1 − F )  e = χ + [b,c] pF (1 − F ) 1 − F

−1 ∗ = V g + f + gV ∈ C (U, V ) ≡ Aθ ,

Z τ(eθ) = f = θ

15 / 24 toroidal orbifold Z/2 action on T2

2 The flip (z1, z2) → (z1, z2) on T

 180◦

2 The orbifold T /(Z/2) yields a sphere.

16 / 24 2 Toroidal orbifold, T /(Z/2)

2 C(T ) o (Z/2) = M2 valued functions on the sphere, restricted to 2 C at 4 points

toroidal orbifold

17 / 24 2 Toroidal orbifold, T /(Z/2)

2 C(T ) o (Z/2) = M2 valued functions on the sphere, restricted to 2 C at 4 points

toroidal orbifold

17 / 24 2 C(T ) o (Z/2) = M2 valued functions on the sphere, restricted to 2 C at 4 points

toroidal orbifold

2 Toroidal orbifold, T /(Z/2)

17 / 24 2 C(T ) o (Z/2) = M2 valued functions on the sphere, restricted to 2 C at 4 points

toroidal orbifold

2 Toroidal orbifold, T /(Z/2)

17 / 24 toroidal orbifold

2 Toroidal orbifold, T /(Z/2)

2 C(T ) o (Z/2) = M2 valued functions on the sphere, restricted to 2 C at 4 points 17 / 24 non-commutative toroidal orbifold

symmetry U, V → U−1, V −1 preserves relation VU = e2πiθUV .

Z/2 Aθ ⊂ Aθ ⊂ Aθ o (Z/2)

2 C(T/Z) = C = C(T /R)

2 C(T) o Z ∼Morita C(T ) o R

18 / 24 Aθ = lim→Mr (C(T)) ⊕ Ms (C(T))

Z2 Aθ = lim→Mr1 (C) ⊕ Mr2 (C) ⊕ · · ·

non-commutative toroidal orbifold

19 / 24 Z2 Aθ = lim→Mr1 (C) ⊕ Mr2 (C) ⊕ · · ·

non-commutative toroidal orbifold

Aθ = lim→Mr (C(T)) ⊕ Ms (C(T))

19 / 24 non-commutative toroidal orbifold

Aθ = lim→Mr (C(T)) ⊕ Ms (C(T))

Z2 Aθ = lim→Mr1 (C) ⊕ Mr2 (C) ⊕ · · ·

19 / 24 almost Mathieu operator

Z/2 ∗ −1 −1 Aθ = C (U + U , V + V ) is AF Hamiltonians

H = U + U−1 + λ(V + V −1)

Uh(z) = zh(z), Vh(z) = h(exp(2πiθ)z). almost Mathieu operator:

f → f (n + 1) + f (n − 1) + 2λ cos(2πnθ)f (n)

has Cantor spectrum for θ irrational, λ 6= 0.

20 / 24 PQ − QP = −i

irreducible Schr¨odingerrepresentation is

2 Q = x , P = −id/dx on L (R)

Stone-von Neumann Theorem

U(m) = Um, V (n) = V n U(m)V (n) = exp(2πimnθ)V (n)U(m) , m, n ∈ Z Continuous version – CCR Canonical Commutation Relations U(t)V (s) = exp(2πistθ)V (s)U(t) , s, t ∈ R U(t) = exp(itP), V (s) = exp(isQ)

21 / 24 Stone-von Neumann Theorem

U(m) = Um, V (n) = V n U(m)V (n) = exp(2πimnθ)V (n)U(m) , m, n ∈ Z Continuous version – CCR Canonical Commutation Relations U(t)V (s) = exp(2πistθ)V (s)U(t) , s, t ∈ R U(t) = exp(itP), V (s) = exp(isQ)

PQ − QP = −i

irreducible Schr¨odingerrepresentation is

2 Q = x , P = −id/dx on L (R)

21 / 24 Mackey-Rieffel picture:

Reps of CCR ∼ Reps of C0(R) o R ∼Morita C

∗ C0(G/H) o G ∼Morita C (H)

Stone-von Neumann theorem

There is an unique irreducible representation of the Canonical Commutation Relations.

22 / 24 ∗ C0(G/H) o G ∼Morita C (H)

Stone-von Neumann theorem

There is an unique irreducible representation of the Canonical Commutation Relations.

Mackey-Rieffel picture:

Reps of CCR ∼ Reps of C0(R) o R ∼Morita C

22 / 24 Stone-von Neumann theorem

There is an unique irreducible representation of the Canonical Commutation Relations.

Mackey-Rieffel picture:

Reps of CCR ∼ Reps of C0(R) o R ∼Morita C

∗ C0(G/H) o G ∼Morita C (H)

22 / 24 References

Akeman, C. and Weaver, N. (2004) Consistency of a counter example to Naimark’s problem. Proc. Nat. Acad. Sci. USA. 101, 7522-7525 Bratteli, O. and Kishimoto, A. (1992). Non-commutative spheres, III, Irrational rotations. Commun, in Mathematical Physics, 147, 605–624. Connes, A. (1994). Noncommutative geometry. Academic Press, New York. Elliott, G. A. and Evans, D. E. (1993). The structure of the irrational rotation C ∗-algebra. Annals of Mathematics, 138, 477–501. Evans, D.E. (1992). Some non-commutative orbifolds. Ideas and methods in mathematical analysis, stochastics, and applications. eds S.Albeverio, J.E. Fenstad, H. Holden, T. Lindstrom. Cambridge University Press. 344–364. Evans, D. E. and Kawahigashi, Y. (1998). Quantum Symmetries on Operator Algebras. Oxford University Press. Mackey, G.W. (1949). A theorem of Stone and von Neumann. Duke Math. J. 16, 313-326. Rieffel, M.A. (1972). On the uniqueness of the Heisenberg commutation relations. Duke Math. J. 39 745-752. Rieffel, M. A. (1981). C ∗-algebras associated with irrational rotations. Pacific Journal of Mathematics, 93, 415–429. Rosenberg, J. A. (2004). Selective History of the Stone-von Neumann Theorem. Operator algebras, quantization, and noncommutative geometry,Contemp. Math., 365, Amer. Math. Soc., Providence, RI, 123-158. Stone M. H. (1930). Linear transformations in Hilbert space, III: operational methods and group theory, Proc. Nat. Acad. Sci. U.S.A. 16 172-175. von Neumann, J. (1931). Die Eindeutigkeit der Schr¨odingerischenOperatoren. Math. Ann. 104 570-578. 23 / 24