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 jjf jj = sup x jf (x)j 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 jjf jj = sup x jf (x)j 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 jjf jj = sup x jf (x)j 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 jjf jj = sup x jf (x)j 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 2 C[−1; 1] on L (T) by multiplication operators 2 π(f ) 2 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 2 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 2 A then A is a C ∗-algebra, | there exists an isometric ∗-homomorphism π : A ! B(H) for a Hilbert space H. 2 Represent f 2 C[−1; 1] on L (T) by multiplication operators 2 π(f ) 2 B(L (T)) (π(f )g)(x) = f (x)g(x) 2 A = C[−1; 1] ⊂ B(L (T)) 4 / 24 H = H∗ 2 B(H), C ∗(H; 1) ' C(σ(H)) Spec(A) = σ(H) f (H), f 2 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 2 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 2 A then A ' C(Ω) for a compact Hausdorff Ω. H = H∗ 2 B(H), C ∗(H; 1) ' C(σ(H)) Spec(A) = σ(H) f (H), f 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 M2; x 2 (0; 1] and f (0) = Uf (0)U i.e. f (0) 2 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) 2 f0; 2n ; 2n ; 2n :::; 1g ! 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) 2 f0; 2n ; 2n ; 2n :::; 1g ! 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) 2 f0; 2n ; 2n ; 2n :::; 1g ! 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) 2 f0; 2n ; 2n ; 2n :::; 1g ! N[1=2] K0(⊗NM2) = Z[1=2] 9 / 24 discrete rotation matrices 0 0 1 0 1 B 0 0 C B C V = B ·· C B C @ · 1 A 1 0 0 0 1 0 1 B 0 ! C B C U = B ·· C B C @ · A 0 0 !n−1 !n = 1 VUV ∗ = !U r Ver = er−1 Uer = ! 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.