Introduction to C⇤-Algebras

Thomas Sinclair

Model Theory of Operator Algebras UC Irvine

September 20, 2017

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 1 / 52 Hilbert Spaces

I H complex Hilbert space with ( , )innerproduct · · 1/2 I ⇠ := (⇠,⇠) (complete) norm k k I H⇤ continuous linear functionals : H C ! I (F. Reisz) H then ⌘ H ⇠ H((⇠)=(⇠,⌘)) 2 ⇤ 9 2 8 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 2 / 52 Hilbert Spaces

I H complex Hilbert space with ( , )innerproduct · · 1/2 I ⇠ := (⇠,⇠) (complete) norm k k I H⇤ continuous linear functionals : H C ! I (F. Reisz) H then ⌘ H ⇠ H((⇠)=(⇠,⌘)) 2 ⇤ 9 2 8 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 2 / 52 Hilbert Spaces

I H complex Hilbert space with ( , )innerproduct · · 1/2 I ⇠ := (⇠,⇠) (complete) norm k k I H⇤ continuous linear functionals : H C ! I (F. Reisz) H then ⌘ H ⇠ H((⇠)=(⇠,⌘)) 2 ⇤ 9 2 8 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 2 / 52 Hilbert Spaces

I H complex Hilbert space with ( , )innerproduct · · 1/2 I ⇠ := (⇠,⇠) (complete) norm k k I H⇤ continuous linear functionals : H C ! I (F. Reisz) H then ⌘ H ⇠ H((⇠)=(⇠,⌘)) 2 ⇤ 9 2 8 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 2 / 52 Bounded Operators

I (H, K)= T : H K : T linear, continuous B { ! } I operator norm T ⇠ T := sup k k < k k ⇠=0 ⇠ 1 6 k k I (H, K) complete B I T =sup (T ⇠,⌘) k k ⇠ = ⌘ =1 | | k k k k Proof. Wlog ⌘ T (H) 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 3 / 52 Bounded Operators

I (H, K)= T : H K : T linear, continuous B { ! } I operator norm T ⇠ T := sup k k < k k ⇠=0 ⇠ 1 6 k k I (H, K) complete B I T =sup (T ⇠,⌘) k k ⇠ = ⌘ =1 | | k k k k Proof. Wlog ⌘ T (H) 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 3 / 52 Bounded Operators

I (H, K)= T : H K : T linear, continuous B { ! } I operator norm T ⇠ T := sup k k < k k ⇠=0 ⇠ 1 6 k k I (H, K) complete B I T =sup (T ⇠,⌘) k k ⇠ = ⌘ =1 | | k k k k Proof. Wlog ⌘ T (H) 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 3 / 52 Bounded Operators

I (H, K)= T : H K : T linear, continuous B { ! } I operator norm T ⇠ T := sup k k < k k ⇠=0 ⇠ 1 6 k k I (H, K) complete B I T =sup (T ⇠,⌘) k k ⇠ = ⌘ =1 | | k k k k Proof. Wlog ⌘ T (H) 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 3 / 52 Bounded Operators

I (H):= (H, H) B B I ST S T k kk k·k k I (H) A(⇠,⌘) C : A(⇠,⌘) C ⇠ ⌘ , bilinear B ${ ! | | k k·k k } I Adjoint T (⇠,T ⌘) ⇤ $

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 4 / 52 Bounded Operators

I (H):= (H, H) B B I ST S T k kk k·k k I (H) A(⇠,⌘) C : A(⇠,⌘) C ⇠ ⌘ , bilinear B ${ ! | | k k·k k } I Adjoint T (⇠,T ⌘) ⇤ $

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 4 / 52 Bounded Operators

I (H):= (H, H) B B I ST S T k kk k·k k I (H) A(⇠,⌘) C : A(⇠,⌘) C ⇠ ⌘ , bilinear B ${ ! | | k k·k k } I Adjoint T (⇠,T ⌘) ⇤ $

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 4 / 52 Bounded Operators

I (H):= (H, H) B B I ST S T k kk k·k k I (H) A(⇠,⌘) C : A(⇠,⌘) C ⇠ ⌘ , bilinear B ${ ! | | k k·k k } I Adjoint T (⇠,T ⌘) ⇤ $

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 4 / 52 Adjoints

I (T ⇤)⇤ = T

I (T )⇤ = ¯T ⇤,(ST )⇤ = T ⇤S⇤ I T = T k ⇤k k k 2 I T T = T k ⇤ k k k Proof.

2 sup (T ⇤T ⇠,⌘) =sup (T ⇠,T ⌘) =sup T ⇠ ⇠ = ⌘ =1 | | ⇠ = ⌘ =1 | | ⇠ =1 k k k k k k k k k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 5 / 52 Adjoints

I (T ⇤)⇤ = T

I (T )⇤ = ¯T ⇤,(ST )⇤ = T ⇤S⇤ I T = T k ⇤k k k 2 I T T = T k ⇤ k k k Proof.

2 sup (T ⇤T ⇠,⌘) =sup (T ⇠,T ⌘) =sup T ⇠ ⇠ = ⌘ =1 | | ⇠ = ⌘ =1 | | ⇠ =1 k k k k k k k k k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 5 / 52 Adjoints

I (T ⇤)⇤ = T

I (T )⇤ = ¯T ⇤,(ST )⇤ = T ⇤S⇤ I T = T k ⇤k k k 2 I T T = T k ⇤ k k k Proof.

2 sup (T ⇤T ⇠,⌘) =sup (T ⇠,T ⌘) =sup T ⇠ ⇠ = ⌘ =1 | | ⇠ = ⌘ =1 | | ⇠ =1 k k k k k k k k k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 5 / 52 Adjoints

I (T ⇤)⇤ = T

I (T )⇤ = ¯T ⇤,(ST )⇤ = T ⇤S⇤ I T = T k ⇤k k k 2 I T T = T k ⇤ k k k Proof.

2 sup (T ⇤T ⇠,⌘) =sup (T ⇠,T ⌘) =sup T ⇠ ⇠ = ⌘ =1 | | ⇠ = ⌘ =1 | | ⇠ =1 k k k k k k k k k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 5 / 52 C⇤-Algebras

Definition A concrete C -algebra is a norm closed, adjoint closed, subalgebra of (H) ⇤ B for some Hilbert space H.

Definition

AC⇤-algebra is a: I complex Banach algebra - ab a b k kk k·k k I with involution - (a)⇤ = ¯a⇤,(a⇤)⇤ = a

I which satisfies the C⇤-identity

2 a⇤a = a . k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 6 / 52 C⇤-Algebras

Definition A concrete C -algebra is a norm closed, adjoint closed, subalgebra of (H) ⇤ B for some Hilbert space H.

Definition

AC⇤-algebra is a: I complex Banach algebra - ab a b k kk k·k k I with involution - (a)⇤ = ¯a⇤,(a⇤)⇤ = a

I which satisfies the C⇤-identity

2 a⇤a = a . k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 6 / 52 C⇤-Algebras

Definition A concrete C -algebra is a norm closed, adjoint closed, subalgebra of (H) ⇤ B for some Hilbert space H.

Definition

AC⇤-algebra is a: I complex Banach algebra - ab a b k kk k·k k I with involution - (a)⇤ = ¯a⇤,(a⇤)⇤ = a

I which satisfies the C⇤-identity

2 a⇤a = a . k k k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 6 / 52 C⇤-Algebras

Theorem (Big Theorem) Every C -algebra is isometrically -isomorphic to a concrete C -algebra. ⇤ ⇤ ⇤ Done by Gelfand and Naimark in 1943 with extra axioms. It took almost two decades and the work of many others (Segal, Kaplansky, etc.) to reach this form.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 7 / 52 C⇤-Algebras

Theorem (Big Theorem) Every C -algebra is isometrically -isomorphic to a concrete C -algebra. ⇤ ⇤ ⇤ Done by Gelfand and Naimark in 1943 with extra axioms. It took almost two decades and the work of many others (Segal, Kaplansky, etc.) to reach this form.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 7 / 52 Theorem

Every finite dimensional C⇤-algebra is of this form.

Examples - Finite Dimension

Here are some examples to finite-dimensional C⇤-algebras: I C 2 I Mn(C)= (` ) B n I Mn1 (C) Mnk (C) ···

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 8 / 52 Theorem

Every finite dimensional C⇤-algebra is of this form.

Examples - Finite Dimension

Here are some examples to finite-dimensional C⇤-algebras: I C 2 I Mn(C)= (` ) B n I Mn1 (C) Mnk (C) ···

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 8 / 52 Theorem

Every finite dimensional C⇤-algebra is of this form.

Examples - Finite Dimension

Here are some examples to finite-dimensional C⇤-algebras: I C 2 I Mn(C)= (` ) B n I Mn1 (C) Mnk (C) ···

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 8 / 52 Examples - Finite Dimension

Here are some examples to finite-dimensional C⇤-algebras: I C 2 I Mn(C)= (` ) B n I Mn1 (C) Mnk (C) ··· Theorem

Every finite dimensional C⇤-algebra is of this form.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 8 / 52 Examples - Abelian

X is a set. For f ` (X ), M (`2(X )) by 2 1 f 2B M ⇠(x):=f (x)⇠(x), x X f 8 2 Theorem f M gives an isometric -embedding ` (X ) , (`2(X )) 7! f ⇤ 1 !B Proof.

I Mf¯ = Mf⇤ I Mfg = Mf Mg = Mg Mf I Mf =sup (Mf ⇠,⌘) =sup (, f ) =sup f (x) k k ⇠ = ⌘ =1 | | =1 | | x | | k k2 k k2 k k1

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 9 / 52 Examples - Abelian

X is a set. For f ` (X ), M (`2(X )) by 2 1 f 2B M ⇠(x):=f (x)⇠(x), x X f 8 2 Theorem f M gives an isometric -embedding ` (X ) , (`2(X )) 7! f ⇤ 1 !B Proof.

I Mf¯ = Mf⇤ I Mfg = Mf Mg = Mg Mf I Mf =sup (Mf ⇠,⌘) =sup (, f ) =sup f (x) k k ⇠ = ⌘ =1 | | =1 | | x | | k k2 k k2 k k1

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 9 / 52 Examples - Abelian

X is a set. For f ` (X ), M (`2(X )) by 2 1 f 2B M ⇠(x):=f (x)⇠(x), x X f 8 2 Theorem f M gives an isometric -embedding ` (X ) , (`2(X )) 7! f ⇤ 1 !B Proof.

I Mf¯ = Mf⇤ I Mfg = Mf Mg = Mg Mf I Mf =sup (Mf ⇠,⌘) =sup (, f ) =sup f (x) k k ⇠ = ⌘ =1 | | =1 | | x | | k k2 k k2 k k1

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 9 / 52 Fact C(X ) is a closed -subalgebra of ` (Xˇ) (`2(Xˇ)) ⇤ 1 ⇢B

I “Xˇ”isX considered as a (discrete) set.

Examples - Abelian

X compact, Hausdor↵.

C(X ):= f : X C : f is continuous { ! }

Fact

C(X ) is an (abstract) C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 10 / 52 I “Xˇ”isX considered as a (discrete) set.

Examples - Abelian

X compact, Hausdor↵.

C(X ):= f : X C : f is continuous { ! }

Fact

C(X ) is an (abstract) C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

Fact C(X ) is a closed -subalgebra of ` (Xˇ) (`2(Xˇ)) ⇤ 1 ⇢B

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 10 / 52 Examples - Abelian

X compact, Hausdor↵.

C(X ):= f : X C : f is continuous { ! }

Fact

C(X ) is an (abstract) C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

Fact C(X ) is a closed -subalgebra of ` (Xˇ) (`2(Xˇ)) ⇤ 1 ⇢B

I “Xˇ”isX considered as a (discrete) set.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 10 / 52 Examples - Abelian

X locally compact, Hausdor↵. C0(X )isallf : X C continuous s.t. ! ✏>0 K X compact( f (x) ✏ x K) 8 9 ⇢ | | ) 2

I C0(X )isan(abstract)C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

I C (X ) , C(X ), X = Stone–Cechˇ compactification. 0 ! I C0(X )isunitali↵X is compact.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 11 / 52 Examples - Abelian

X locally compact, Hausdor↵. C0(X )isallf : X C continuous s.t. ! ✏>0 K X compact( f (x) ✏ x K) 8 9 ⇢ | | ) 2

I C0(X )isan(abstract)C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

I C (X ) , C(X ), X = Stone–Cechˇ compactification. 0 ! I C0(X )isunitali↵X is compact.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 11 / 52 Examples - Abelian

X locally compact, Hausdor↵. C0(X )isallf : X C continuous s.t. ! ✏>0 K X compact( f (x) ✏ x K) 8 9 ⇢ | | ) 2

I C0(X )isan(abstract)C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

I C (X ) , C(X ), X = Stone–Cechˇ compactification. 0 ! I C0(X )isunitali↵X is compact.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 11 / 52 Examples - Abelian

X locally compact, Hausdor↵. C0(X )isallf : X C continuous s.t. ! ✏>0 K X compact( f (x) ✏ x K) 8 9 ⇢ | | ) 2

I C0(X )isan(abstract)C⇤-algebra under pointwise multiplication, pointwise conjugation, and “sup-norm”.

I C (X ) , C(X ), X = Stone–Cechˇ compactification. 0 ! I C0(X )isunitali↵X is compact.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 11 / 52 Abelian C⇤-Algebras

Theorem (Gelfand–Naimark, 1943) An abelian C -algebra is isometrically -isomorphic to C (X ) for some X ⇤ ⇤ 0 locally compact, Hausdor↵.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 12 / 52 More Examples

Definition An operator T (H)isfiniterankif 2B n T = ( ,⌘ )⇠ · i i Xi=1 The compact operators (H) (H) are the closure of the finite rank K ⇢B operators.

Fact (H)= (H) if dim H < . Otherwise (H) is a maximal proper closed K B 1 K ideal in (H). B

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 13 / 52 More Examples

Fact 2 (` )= 1 Mn(C) where K i=1 S A 0 Mn(C) A Mn+1(C). 3 7! 002 ✓ ◆ Example

The CAR algebra is i1=1 M2n (C)where

S A 0 M2n (C) A M2n+1 (C). 3 7! 0 A 2 ✓ ◆

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 14 / 52 Some Operator Theory

Definition An operator T (H)issaidtobe: 2B I self adjoint if T ⇤ = T . ⇠((T ⇠,⇠) R)) )8 2 I positive if self adjoint and ⇠((T ⇠,⇠) 0). 8 I normal if T ⇤T = TT ⇤.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 15 / 52 Some Operator Theory

Definition An operator T (H)issaidtobe: 2B I self adjoint if T ⇤ = T . ⇠((T ⇠,⇠) R)) )8 2 I positive if self adjoint and ⇠((T ⇠,⇠) 0). 8 I normal if T ⇤T = TT ⇤.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 15 / 52 Some Operator Theory

Definition An operator T (H)issaidtobe: 2B I self adjoint if T ⇤ = T . ⇠((T ⇠,⇠) R)) )8 2 I positive if self adjoint and ⇠((T ⇠,⇠) 0). 8 I normal if T ⇤T = TT ⇤.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 15 / 52 The Spectrum

Definition For T (H), the spectrum 2B (T ):= C : T I GL(H) . { 2 62 } Fact (T ) is nonempty compact.

Fact If (T ) then either: 2 I ⇠ = 0(T ⇠ = ⇠) 9 6 I ✏>0 ⇠( ⇠ =1 T ⇠ ⇠ <✏) 8 9 k k ^k k I (T I )(H) = H. 6

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 16 / 52 The Spectrum

Definition For T (H), the spectrum 2B (T ):= C : T I GL(H) . { 2 62 } Fact (T ) is nonempty compact.

Fact If (T ) then either: 2 I ⇠ = 0(T ⇠ = ⇠) 9 6 I ✏>0 ⇠( ⇠ =1 T ⇠ ⇠ <✏) 8 9 k k ^k k I (T I )(H) = H. 6

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 16 / 52 The Spectrum

Definition For T (H), the spectrum 2B (T ):= C : T I GL(H) . { 2 62 } Fact (T ) is nonempty compact.

Fact If (T ) then either: 2 I ⇠ = 0(T ⇠ = ⇠) 9 6 I ✏>0 ⇠( ⇠ =1 T ⇠ ⇠ <✏) 8 9 k k ^k k I (T I )(H) = H. 6

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 16 / 52 The Spectrum

Fact If T (H) is normal and (T ),thenthereexists(⇠ ), ⇠ =1s.t. 2B 2 n k nk T ⇠ ⇠ 0. k n nk! Proof. T normal T ⇠ ⇠ = T ⇠ ⇠¯ . )k k k ⇤ k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 17 / 52 The Spectrum

Definition T (H)thespectralradius (T )ismax : (T ) . 2B | | {| | 2 } Fact > T then T I GL(H). | | k k 2 Proof. Use power series to invert T I .

Corollary (T ) T | | k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 18 / 52 Spectral Radius

Theorem If T (H), T 0,then T = (T ) 2B k k | | Proof. Wlog T = 1. Choose (⇠ ), ⇠ = 1, T ⇠ 1. Define k k n k nk k nk! (⇠,⌘)T := (T ⇠,⌘) positive semidefinite. We have 1 = limn(⇠n, T ⇠n)T and lim (⇠ ,⇠ ) , lim (T ⇠ , T ⇠ ) 1whence⇠ = T ⇠ + ⌘ where n n n T n n n T  n n n T ⌘ 0(Cauchy-Schwarz).Butthen ⌘ 0 by optimality of norm k nk! k nk! estimate. Thus T ⇠ ⇠ 0and1 (T ). k n nk! 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 19 / 52 Spectral Mapping

p(z) complex -polynomial, ⇤ p(z)=(z a ) (z a )(¯z b ) (¯z b ). 1 ··· m 1 ··· n T (H), normal, 2B

p(T ):=(T a I ) (T a I )(T ⇤ b I ) (T ⇤ b I ) 1 ··· j 1 ··· k

Theorem (Spectral Mapping I) If T (H), T normal, then p((T )) = (p(T )). 2B Proof. p(T )⇠ 0 either lim inf T ⇠ a ⇠ = 0 for some i or k nk! , k n i nk lim inf T ⇠ b ⇠ = 0 for some j. k ⇤ n j nk

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 20 / 52 Spectral Mapping

p (x) uniformly convergent on (T )tof , T = T . { n } ⇤ Theorem (Spectral Mapping II)

(f (T )⇠,⌘):=limn(pn(T )⇠,⌘) exists and f ((T )) = (f (T )).

Proof. Uniformly over pairs ⇠,⌘ in unit ball (p (T )⇠,⌘) is Cauchy by spectral { n } mapping, so defines a (bounded) bilinear form, thus an operator f (T ). Then p (T ) f (T ) 0. k n k!

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 21 / 52 Corollary T (H),thenT 0 i↵ S(T = S S) i↵ S(T = S2 S = S ). 2B 9 ⇤ 9 ^ ⇤

Spectral Theorem

T (H), T = T , C (T ) smallest norm closed, real algebra containing 2B ⇤ R⇤ T and I . Theorem (Spectral Theorem I) There is an isometric isomorphism

C ((T )) C ⇤(T ) R $ R

sending id(T ) to T .

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 22 / 52 Spectral Theorem

T (H), T = T , C (T ) smallest norm closed, real algebra containing 2B ⇤ R⇤ T and I . Theorem (Spectral Theorem I) There is an isometric isomorphism

C ((T )) C ⇤(T ) R $ R

sending id(T ) to T .

Corollary T (H),thenT 0 i↵ S(T = S S) i↵ S(T = S2 S = S ). 2B 9 ⇤ 9 ^ ⇤

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 22 / 52 Spectral Theorem

T (H), T = T , C (T ) smallest norm closed, real algebra containing 2B ⇤ R⇤ T and I . Theorem (Spectral Theorem I) There is an isometric isomorphism

C ((T )) C ⇤(T ) R $ R

sending id(T ) to T .

Corollary T (H),thenT 0 i↵ S(T = S S) i↵ S(T = S2 S = S ). 2B 9 ⇤ 9 ^ ⇤

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 22 / 52 Spectral Theorem

Corollary

Every element of a C⇤-algebra is a linear combination of four positive elements.

Corollary

In a unital C⇤-algebra the set of positive elements A+ is a complete cone

Definition x, y A , x y if y x 0. 2 +  Fact If x A ,thenx x 1. 2 + k k·

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 23 / 52 Spectral Theorem

T (H), T normal, C (T )smallestC-algebra containing T , I . 2B ⇤ ⇤ Theorem (Spectral Theorem II) There is an isometric isomorphism

C((T )) C ⇤(T ) $

sending id(T ) to T .

I How to bootstrap to abelian C⇤-algebras? I Need a complete set of “eigenvectors”.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 24 / 52 Spectral Theorem

T (H), T normal, C (T )smallestC-algebra containing T , I . 2B ⇤ ⇤ Theorem (Spectral Theorem II) There is an isometric isomorphism

C((T )) C ⇤(T ) $

sending id(T ) to T .

I How to bootstrap to abelian C⇤-algebras? I Need a complete set of “eigenvectors”.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 24 / 52 I Let (A) be the set of states. (A) is nonempty, convex, weak* S S compact.

Fact A is positive i↵ (1) = . 2 ⇤ k k

States

A (H)unitalC-subalgebra ⇢B ⇤ Definition A is positive if a 0 (a) 0andastateif 0and(1) = 1. 2 ⇤ ) (x):=(x⇠,⇠), ⇠ = 1 is a state on (H). Also lim (x⇠ ,⇠ ), ⇠ = 1. k k B ! n n k nk

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 25 / 52 Fact A is positive i↵ (1) = . 2 ⇤ k k

States

A (H)unitalC-subalgebra ⇢B ⇤ Definition A is positive if a 0 (a) 0andastateif 0and(1) = 1. 2 ⇤ ) (x):=(x⇠,⇠), ⇠ = 1 is a state on (H). Also lim (x⇠ ,⇠ ), ⇠ = 1. k k B ! n n k nk I Let (A) be the set of states. (A) is nonempty, convex, weak* S S compact.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 25 / 52 States

A (H)unitalC-subalgebra ⇢B ⇤ Definition A is positive if a 0 (a) 0andastateif 0and(1) = 1. 2 ⇤ ) (x):=(x⇠,⇠), ⇠ = 1 is a state on (H). Also lim (x⇠ ,⇠ ), ⇠ = 1. k k B ! n n k nk I Let (A) be the set of states. (A) is nonempty, convex, weak* S S compact.

Fact A is positive i↵ (1) = . 2 ⇤ k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 25 / 52 States

Fact (Cauchy–Schwarz) A , 0,then 2 ⇤ 1/2 1/2 (y ⇤x) (x⇤x) (y ⇤y) . | | Corollary 0, (1) = 0,then 0. ⌘ Fact (State extension) Any state (A) extends to a state ( (H)). 2S 0 2S B Proof. Hahn–Banach

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 26 / 52 Pure States

Definition Astate (A)ispureif is an extreme in (A). Let (A) denote the 2S S P pure states.

Fact (Krein–Milman) The convex hull of (A) is (A). P S Fact a A, a =0,thereisapurestatesuchthat(a) =0 2 6 6 Proof. If H is a complex Hilbert space, then (T ⇠,⇠) = 0, ⇠ H T 0. 8 2 ) ⌘

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 27 / 52 States

Let A = `1(N). What is (A)? P I n(f ):=f (n).

Fact

In general, , X N, µ(X ):=(X ) defines a finitely additive 2S ⇢ probability measure. Conversely for any such measure µ, f fdµ is a 7! state. R

I (`1(N)) µ(X ) 0, 1 , X N 2P $ 2{ } 8 ⇢ I (`1(N)) ultrafilters on N !! P ${ }

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 28 / 52 States

Let A = `1(N). What is (A)? P I n(f ):=f (n).

Fact

In general, , X N, µ(X ):=(X ) defines a finitely additive 2S ⇢ probability measure. Conversely for any such measure µ, f fdµ is a 7! state. R

I (`1(N)) µ(X ) 0, 1 , X N 2P $ 2{ } 8 ⇢ I (`1(N)) ultrafilters on N !! P ${ }

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 28 / 52 States

Definition , A , is (a) (a)foralla 0. 2 ⇤   Fact 0 , (A),then = c for some c [0, 1].   2P · 2 Proof. Wlog  := (1) (0, 1). 2 1 1 =  [ ]+(1 ) [(1 ) ( )]. · ·

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 29 / 52 Characters

Definition Acharacteris a unital -homomorphism : A C ⇤ !

I a 0, (a)=(x x)=(x)(x) 0, so (A). ⇤ 2S I (A). 2P I A = C(X ), X compact T , x X , (f ):=f (x)character. 2 8 2 x I Any character on C(X ) is of the form x . (Reisz representation theorem)

I May be no characters, e.g., Mn(C), n 2.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 30 / 52 Characters

Definition Acharacteris a unital -homomorphism : A C ⇤ !

I a 0, (a)=(x x)=(x)(x) 0, so (A). ⇤ 2S I (A). 2P I A = C(X ), X compact T , x X , (f ):=f (x)character. 2 8 2 x I Any character on C(X ) is of the form x . (Reisz representation theorem)

I May be no characters, e.g., Mn(C), n 2.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 30 / 52 Characters

Definition Acharacteris a unital -homomorphism : A C ⇤ !

I a 0, (a)=(x x)=(x)(x) 0, so (A). ⇤ 2S I (A). 2P I A = C(X ), X compact T , x X , (f ):=f (x)character. 2 8 2 x I Any character on C(X ) is of the form x . (Reisz representation theorem)

I May be no characters, e.g., Mn(C), n 2.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 30 / 52 Characters

Definition Acharacteris a unital -homomorphism : A C ⇤ !

I a 0, (a)=(x x)=(x)(x) 0, so (A). ⇤ 2S I (A). 2P I A = C(X ), X compact T , x X , (f ):=f (x)character. 2 8 2 x I Any character on C(X ) is of the form x . (Reisz representation theorem)

I May be no characters, e.g., Mn(C), n 2.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 30 / 52 Characters

Definition Acharacteris a unital -homomorphism : A C ⇤ !

I a 0, (a)=(x x)=(x)(x) 0, so (A). ⇤ 2S I (A). 2P I A = C(X ), X compact T , x X , (f ):=f (x)character. 2 8 2 x I Any character on C(X ) is of the form x . (Reisz representation theorem)

I May be no characters, e.g., Mn(C), n 2.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 30 / 52 Abelian C⇤-Algebras

Theorem

If A is a unital, abelian C⇤-algebra, then any pure state is a character.

Proof. (A), a, b 0, a , b 1. 2P k k k k 1/2 1/2 I x 0 (x):=(xb)=(b xb ) 0 ) b 1/2 1/2 1/2 1/2 I (x)=(xb)=(x bx ) b (x 1x ) (x) 0 b k k  )  b  I (A) =  (x) 2P ) b · I (b)= (1) =  (1) =  b · I (ab)=(a)(b)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 31 / 52 Abelian C⇤-Algebras

Theorem

If A is a unital, abelian C⇤-algebra, then any pure state is a character.

Proof. (A), a, b 0, a , b 1. 2P k k k k 1/2 1/2 I x 0 (x):=(xb)=(b xb ) 0 ) b 1/2 1/2 1/2 1/2 I (x)=(xb)=(x bx ) b (x 1x ) (x) 0 b k k  )  b  I (A) =  (x) 2P ) b · I (b)= (1) =  (1) =  b · I (ab)=(a)(b)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 31 / 52 Abelian C⇤-Algebras

Theorem

If A is a unital, abelian C⇤-algebra, then any pure state is a character.

Proof. (A), a, b 0, a , b 1. 2P k k k k 1/2 1/2 I x 0 (x):=(xb)=(b xb ) 0 ) b 1/2 1/2 1/2 1/2 I (x)=(xb)=(x bx ) b (x 1x ) (x) 0 b k k  )  b  I (A) =  (x) 2P ) b · I (b)= (1) =  (1) =  b · I (ab)=(a)(b)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 31 / 52 Abelian C⇤-Algebras

Theorem

If A is a unital, abelian C⇤-algebra, then any pure state is a character.

Proof. (A), a, b 0, a , b 1. 2P k k k k 1/2 1/2 I x 0 (x):=(xb)=(b xb ) 0 ) b 1/2 1/2 1/2 1/2 I (x)=(xb)=(x bx ) b (x 1x ) (x) 0 b k k  )  b  I (A) =  (x) 2P ) b · I (b)= (1) =  (1) =  b · I (ab)=(a)(b)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 31 / 52 Abelian C⇤-Algebras

Theorem

If A is a unital, abelian C⇤-algebra, then any pure state is a character.

Proof. (A), a, b 0, a , b 1. 2P k k k k 1/2 1/2 I x 0 (x):=(xb)=(b xb ) 0 ) b 1/2 1/2 1/2 1/2 I (x)=(xb)=(x bx ) b (x 1x ) (x) 0 b k k  )  b  I (A) =  (x) 2P ) b · I (b)= (1) =  (1) =  b · I (ab)=(a)(b)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 31 / 52 Gel’fand Spectrum

Definition

Let A be a unital, abelian C⇤-algebra. The Gel’fand spectrum is

⌃ := (A)= all characters of A A P { } ⌃ is a weak* closed subset of (A), whence is compact, Hausdor↵itself. A S Fact (Gel’fand–Mazur) ⌃ maximal proper closed ideals of A A ${ }

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 32 / 52 Gel’fand Transform

Definition :A C(⌃ ),(a)():=(a), Gel’fand transform. ! A Fact Easy to check that is a unital contractive -homomorphism. ⇤ Fact is an isometry.

Proof. A,abelian -algebra every a A is normal. Hence there is a state s.t. ⇤ ) 2 (a)= a . k k

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 33 / 52 Gel’fand Transform

Theorem is surjective.

Proof. Image of is a -subalgebra that separates points. The image is closed ⇤ since (a) determines the values of (a⇠,⇠), ⇠ H,whencethebilinear 8 2 form (a⇠,⌘). an isometry, thus (an) converges uniformly implies an converges uniformly and limn (an)=(limn an). Stone–Weierstrass finishes the job.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 34 / 52 Gel’fand transform

Fact ((a)()=) (a) 9 , 2 Proof.

(GL(A)) = GL(C(⌃A))

Corollary T normal invertible in (H) i↵invertible in C (T ). B ⇤ Corollary (Gel’fand–Naimark) A an abelian C -algebra, then A is -isomorphic to C(⌃ ). ⇤ ⇤ A

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 35 / 52 Useful Facts

Fact A unital, then every self-adjoint a A is an average of two unitaries 2 (u u =1=uu ). A is the span of (A), unitary group of A. ⇤ ⇤ U Fact A -homomorphism of C -algebras in contractive. ⇤ ⇤ Proof. Gel’fand–Naimark

Fact The image of a C -algebra under a -homomorphism is a C -algebra, i.e., ⇤ ⇤ ⇤ images are closed.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 36 / 52 Representations of C⇤-Algebras

Definition A representation is a -homomorphism ⇡ : A (K). ⇤ !B Definition I Arep’n⇡ is faithful if injective (=isometric).

I Arep’n⇡ is cyclic if ⇡(A)⇠ dense in K for some ⇠ K. 2

I ⇡ ⇡ : A (K K ) 1 2 !B 1 2 I ⇡ ⇡ : A (K K ) 1 ⌦ 2 !B 1 ⌦ 2

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 37 / 52 Gel’fand–Naimark–Segal Construction

Theorem (GNS Construction) A aunitalC-algebra. For every (A) there is: ⇤ 2S I Acyclicrep’n⇡ : A (K ) !B I A distinguished cyclic vector ⇠ I (⇡ (a)⇠ ,⇠ )=(a), a A 8 2

I ⇡ : A (K), ⇠ K, ⇠ = 1, then (⇡(a)⇠,⇠)isastate. !B 2 k k I ⇡ = ⇡ ⇡ (irreducible) i↵ (A). 6⇠ 0 2P

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 38 / 52 Proof of GNS

Proof.

I (x, y) := (y ⇤x) positive semidefinite on A. I complete to Hilbert space K , A a aˆ K dense. 3 7! 2 I ⇡(a)bˆ := ab,well-definedbyC-S. 1/2 1/2 I ⇡(a)bˆ = (b⇤a⇤ab) a⇤a (b⇤1b)= a b . k k c k k k k·k k I ⇠ = 1.ˆ

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 39 / 52 Universal Representation

Definition ⇡ : A (H ) is the universal representation of A where u !B u

Hu := H,⇡u := ⇡ (A) (A) 2MS 2MS

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 40 / 52 Ultraproducts of Representations

I A : i I , I adirectedset. { i 2 } I ⇡ : i I , ⇡ : A (H ). { i 2 } i i !B i I ultrafilter on I . U

A := A / (a ):lim a =0 i i { i k i k } I U YU Y Fact

The direct product representation I descends to a representation Q ⇡ : Ai (Hi ) (H ) U ! B ⇢B U YU YU which is faithful i↵ ⇡ is -almost always faithful. i U

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 41 / 52 Building C⇤-Algebras from Relations

I G = x : i I set of variables. “generators” { i 2 } I R = p : j J set of noncommuting polynomials in x ’s and x ’s { i 2 } i i⇤ “relations”

Definition A aC-algebra models (G R), A =(G R)ifthereisamapx T A ⇤ | | | i ! i 2 s.t. T , T generates A and p (T )=0forallj J. { i i⇤} j 2 Definition We say (G R) consistent if it admits a model. |

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 42 / 52 Building C⇤-Algebras from Relations

Theorem (Compactness) (G R) is consistent i↵every subcollection of finitely many generators and | relations is consistent.

Proof. F finite subcollection of generators and relations. directed set of all F finite subcollections. ultrafilter on . U F If A = F ,then F | A =(G R). F | | YU

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 43 / 52 Universal Models

Theorem If (G R) is consistent, then there is a unique C -algebra C (G R) s.t. | ⇤ ⇤ | I C (G R) =(G R). ⇤ | | | I if B =(G R) then there is a -epimorphism ⇡ : C (G R) B. | | ⇤ ⇤ | ! Proof. K isomorphism classes of models of (G R), then | (k) xi (T )k K ( Hk ) 7! i 2 2B k K M2 generates C (G R). ⇤ |

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 44 / 52 Group C⇤-Algebras

Theorem Generators and relations defining a group are consistent.

Proof. g G,defineu (`2G) 2 g 2U u ⇠(h):=⇠(gh),⇠ `2G. g 2

We define the reduced group C⇤-algebra

2 C ⇤(G):=C ⇤(u , g G) (` G) r g 2 ⇢B

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 45 / 52 Group C⇤-Algebras

C ⇤(G)theuniversalC⇤-algebra given by group G generators and relations. Fact Any unitary rep’n ⇡ : G (H) extends to a rep’n ⇡ : C (G) (H). !U ⇤ !B Fact Cr⇤(Z) ⇠= C(T).

Proof. Fourier transform

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 46 / 52 Group C⇤-Algebras

Fact 2 C ⇤(Z) = C ⇤(1, u 1=1⇤ =1 , u⇤u =1=uu⇤) = C ⇤(Z) ⇠ | ⇠ r

I C ⇤(G) ⇠= Cr⇤(G)i↵G is amenable. (Fell) Fact F free group on countably many generators. C ⇤(F ) is the universal 1 1 separable C⇤-algebra.

I Cr⇤(F )issimple.(Powers) 1 I C ⇤(F F ) = C ⇤(F ) C ⇤(F ) (Hu Hu)?? (Kirchberg, 1 ⇥ 1 ⇠ 1 ⌦ 1 ⇢B ⌦ Connes)

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 47 / 52 More Universal Algebras

Example C (x x = x ) = C[0, 1) ⇤ | ⇤ ⇠ Theorem (Coburn) C (1, v 1=1 =12, v v = 1) -isomorphic to the Toeplitz algebra C (S), ⇤ | ⇤ ⇤ ⇤ ⇤ S : `2N `2N shift. ! Example G = 1, v ,...,v , R = 1=1 =12, v v =1, v v =1 The { 1 n} { ⇤ i⇤ i i i i⇤ } Cuntz algebra n is C (G R). O ⇤ | P

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 48 / 52 Cuntz Algebras

Theorem (Cuntz) Let S ,...,S (`2N) isometries such that S S =0i = j and 1 n 2B i⇤ j 6 S S = I .ThenC (S ,...,S ) = . i i i⇤ ⇤ 1 n ⇠ On P Theorem (Cuntz) , n 2,isseparable,simple,purelyinfinite. (“simple + purely infinite” On a A , a =0, x(x ax = 1).) ,8 2 + 6 9 ⇤

I Cuntz algebras can be generalized to Cuntz–Kreiger algebras, etc.

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 49 / 52 Stable Relations

Definition (G R)=(x p )isstableif for any >0,thereexists✏>0suchthatfor | i | j any C -algebra A and any map x a such that p (a) <✏for all j, ⇤ i 7! i k j k there exist (bi )inA s.t. sup ai bi i k k such that pj (b)=0forallj.

Fact This is equivalent to: ⇡ : C ⇤(G R) Ai ⇡i : C ⇤(G R) Ai , -almost always. | ! U )9 | ! U Q

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 50 / 52 Stable Relations

Theorem (p p = p = p2) is stable. | ⇤ Proof. Wlog p = p,soC (p)isabelian.p p2 (p) [0,✏)(✏, 1]. ⇤ ⇤ ⇠ ) ⇢ p id .Definef = 0 on [0,✏) (p), f = 1 on (✏, 1] (p). f $ (p) \ \ corresponds to an element q C (p)s.t.q = q = q2 and 2 ⇤ ⇤ p q = id f . k k k k1

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 51 / 52 Stable Relations

Theorem (1, u 1=1 =12, u u =1=uu ) is stable. | ⇤ ⇤ ⇤ Proof. Wlog 1 is 1 A .1 u u implies u u invertible. Let v = u(u u) 1/2. 2 ⇠ ⇤ ⇤ ⇤ 1/2 1/2 u 1 v ⇤v =(u⇤u) u⇤u(u⇤u) = u⇤u(u ) =1. ⇤ This implies that (vv )2 = vv =: p.Butp 1, p 1impliesthatp = 1. ⇤ ⇤  ⇠

T. Sinclair (Purdue University) Intro to C⇤-algebras September 20, 2017 52 / 52