Characterization of Spectral Flow Magdalena Georgescu 42nd Canadian - PowerPoint PPT Presentation

Introduction Uniqueness of spectral flow Characterization of Spectral Flow Magdalena Georgescu 42nd Canadian Annual Symposium on Operator Algebras and Their Applications June 23, 2014

Introduction Uniqueness of spectral flow Outline • Example • Definition of spectral flow and context • Characterization of spectral flow in a type I factor • Characterization of spectral flow in a type II factor

Introduction Uniqueness of spectral flow Disclaimer

Introduction Uniqueness of spectral flow Example Hilbert space H = L 2 ( T ) , fix basis { h n := 1 2 π e int } n ∈ Z . √ Consider B ( L 2 ( T )) . • self-adjoint unbounded operator D = 1 d dt (so D maps h n to n · h n ) i • unitary operator u the adjoint of the bilateral shift (maps h n to h n + 1 ). Consider the path t �→ D + t · 1.

Introduction Uniqueness of spectral flow Example (cont’d) On the previous slide, we defined D = 1 d dt (so h n �→ n · h n for n ∈ Z ) and i denoted by u the adjoint of the bilateral shift ( h n �→ h n + 1 for n ∈ Z ). • • • • • • Let D 0 = D and D t = D 0 + t · 1. 1 1 Then D 1 = u ∗ Du ; • • • • 0 0 in general, D t takes h n to ( n + t ) h n . • • • • -1 -1 From the picture, spectral flow( { D t } ) = 1. • • -2 -2 D 0 D 1 spectral flow in B ( H ) : defined for paths of self-adjoint Fredholm operators (either bounded or unbounded).

Introduction Uniqueness of spectral flow Spectral flow in von Neumann algebras: mise-en-sc` ene B ( H ) N with a semifinite, faithful, normal trace τ compact operators K ( H ) τ -compact operators, K N (the norm closed ideal generated by finite trace projections) generalized Calkin algebra N / K N Calkin algebra Fredholm operators Breuer-Fredholm operators (operators which are invertible modulo the τ -compacts)

Introduction Uniqueness of spectral flow Definitions of Spectral Flow: Bounded Operators Definition (Phillips, 1997) Suppose { F t } is a path of self-adjoint Breuer-Fredholm operators. Let P t = χ [ 0 , ∞ ) ( F t ) . Then π ( P t ) is continuous, so we can find indices i 0 , i 1 ,... i n such that � π ( P t 1 ) − π ( P t 2 ) � < 1 for all t 1 , t 2 ∈ [ i k , i k + 1 ] . This ensures that P t ik P t ik + 1 is a Breuer-Fredholm operator when considered as an operator between P t ik + 1 H and P t ik H and we can define sf ( { F t } ) = ∑ ind ( P t ik P t ik + 1 ) .

Introduction Uniqueness of spectral flow Definitions of Spectral Flow: Unbounded Operators • gap continuous unbounded operators The Cayley map imaginary axis i D �→ ( D − i )( D + i ) − 1 − 1 allows us to change a gap- 0 continuous path of unbounded op- − 1 1 real axis 1 erators to a path of unitary opera- − i tors. Definition (Wahl, 2008) Apply a normalization function Ξ to D t (warning: Ξ( D t ) is bounded, but t �→ Ξ( D t ) is not continuous), and let U t = e π i (Ξ( D t )+ 1 ) . Define � 1 1 0 τ ( U − 1 d sf ( { D t } ) = winding number ( { U t } ) = dt ( U t − 1 )) dt . t 2 π i

Introduction Uniqueness of spectral flow Context D - unbounded self-adjoint Breuer-Fredholm operator with ( 1 + D 2 ) − 1 ∈ K N and u ∈ N - unitary such that [ D , u ] is bounded Let P = χ [ 0 , ∞ ] ( D ) (the projection onto the non-negative spectral subspace of D ). The PuP is a Breuer-Fredholm operator and ind ( PuP ) = sf ( D , uDu ∗ ) . This is connected to the pairing between (odd) K-theory and K-homology. In certain conditions, there are integral formulas for spectral flow. Proving that such a formula calculates spectral flow is a non-trivial proposition, though worth the effort, as having the integral formula allows for different kinds of algebraic manipulation (e.g. the proof of the Local Index Theorem given by Carey, Phillips, Rennie and Sukochev, 2006).

Introduction Uniqueness of spectral flow Properties of spectral flow Concatenation: ξ ρ sf ( ρ ∗ ξ ) = sf ( ρ )+ sf ( ξ ) Homotopy: ρ sf ( ρ ) = sf ( ξ ) ξ NOTE: can change the homotopy requirement so that ρ and ξ do not have the same endpoints, but the endpoints are invertible operators and remain invertible throughout the homotopy.

Introduction Uniqueness of spectral flow Characterization of Spectral Flow (type I ∞ factor) CF sa - unbounded self-adjoint Fredholm operators (necessarily closed and densely-defined) Theorem (Lesch, 2005) Let µ : Ω( CF sa , ( CF sa ) × ) → Z be a map which satisfies the concatenation and homotopy property (as suggested by the previous slide). Suppose in addition that the following property holds: ’Normalization’ property: Fix T 0 ∈ ( F sa , ∗ ) × with σ ( T 0 ) = {± 1 } . Suppose that there exists a rank one projection P ∈ B ( H ) such that ( 1 − P ) T 0 ( 1 − P ) ∈ B ( P ⊥ H ) is invertible and such that µ ( { t ⊕ P ⊥ T 0 P ⊥ } t ∈ [ − 1 2 ] ) = 1 . 2 , 1 Then µ = sf . Overview of proof: Use the gaps in the spectrum to break up the path in such a way that the ’action’ is happening on a finite-dimensional corner. Appeal to the result for finite-dimensional matrices to get the result.

Introduction Uniqueness of spectral flow Characterization of Spectral Flow (type II factor) Setting: N is a factor (i.e. the center is trivial) Theorem N - type II factor Ω( BF sa , BF × sa ) - paths of (bounded) Breuer-Fredholm self-adjoint operators with invertible endpoints Suppose µ : Ω( BF sa , BF × sa ) → R is a map which satisfies the following three properties • homotopy: if ξ , ρ : Ω( BF sa , BF × sa ) and ξ , ρ are homotopic (with endpoints not necessarily fixed, but remaining invertible) then µ ( ξ ) = µ ( ρ ) . • concatenation: if ξ , ρ ∈ Ω( BF sa , BF × sa ) with ρ ( 1 ) = ξ ( 0 ) then µ ( ρ ∗ ξ ) = µ ( ρ )+ µ ( ξ ) . • normalization: there exists a finite-trace non-zero projection P 0 ∈ N such that if Q , R are projections with Q ≤ P 0 and R ≤ 1 − Q then µ ( { t ⊕ 1 ⊕− 1 } t ∈ [ − 1 , 1 ] ) = τ ( Q ) . � �� � ∈ Q H ⊕ R H ⊕ ( Q + R ) ⊥ H Then µ calculates spectral flow for paths in Ω( BF sa , BF × sa ) .

Introduction Uniqueness of spectral flow Cayley map revisited imaginary axis i − 1 Recall the Cayley map 0 κ : D �→ ( D − i )( D + i ) − 1 . − 1 1 real axis 1 − i Applying the Cayley map to unbounded self-adjoint Breuer-Fredholm operators, we get unitaries U such that 1 + U is Breuer-Fredholm, and 1 is not an eigenvalue of U . Denote by U κ the unitaries in the image of the Cayley transform (applied to the unbounded self-adjoint Breuer-Fredholm operators), and U + 1 the unitaries in U κ which do not have − 1 in the spectrum (ie. corresponding to κ the self-adjoint invertible operators)

Introduction Uniqueness of spectral flow Lemma Suppose ρ ∈ Ω( U κ , U κ + 1 ) is such that {− i , i } �∈ σ ( ρ ( t )) for any t ∈ [ 0 , 1 ] . If µ satisfies the concatenation, homotopy and normalization properties then µ ( ρ ) = sf ( ρ ) .

Introduction Uniqueness of spectral flow Lemma Suppose ρ ∈ Ω( U κ , U κ + 1 ) is such that {− i , i } �∈ σ ( ρ ( t )) for any t ∈ [ 0 , 1 ] . If µ satisfies the concatenation, homotopy and normalization properties then µ ( ρ ) = sf ( ρ ) . Sketch of proof: 2 ] ( ρ ( t )) is continuous, which means that P t = U t P 0 U ∗ • P t = χ [ π t for some path 2 → 3 π of unitaries { U t } imaginary axis real axis � � A t 0 • we can use { U t } to get a homotopy to some path { } (with respect 0 B t to the decomposition P 0 H ⊕ P ⊥ 0 H ); moreover, − 1 �∈ σ ( B t ) . � � A t 0 • construct a second homotopy to { } . 0 B 0 � � � � A t 0 A t 0 Conclude that we must have µ ( ) = sf ( ) (using the 0 B 0 0 B 0 description of spectral flow for bounded operators in P 0 N P 0 ), and hence µ ( ρ ) = sf ( ρ ) .

Introduction Uniqueness of spectral flow Introducing gaps at ± i U r 1 U 1 U 0 � � X t V t with − 1 �∈ σ ( Y t ) , On each of the subpaths, can write the operators as W t Y t and the X t corner finite-trace. We add the requirement that σ ( X t ) and σ ( X t − V t ( Y t + 1 ) − 1 W t ) should be contained in an arc of length π 4 around -1. At each division point, we can add little extrusions (as indicated by the dotted line) to get paths with endpoints in U κ + 1 . A technical lemma now gives us the homotopy which allows us to get a gap in the spectrum at ± i along each of these new paths.

Introduction Uniqueness of spectral flow Technical Lemma � � X V (with respect to some decompostion of H ) is unitary and If U = W Y − 1 �∈ σ ( Y ) then, for any fixed s ∈ [ 0 , 1 ] , √ � � X − sV ( sY + 1 ) − 1 W 1 − s 2 V ( sY + 1 ) − 1 √ Z s = 1 − s 2 ( sY + 1 ) − 1 W ( Y + s )( sY + 1 ) − 1 is also unitary. Moreover, the following hold: • − 1 �∈ σ ( U ) ⇒ − 1 �∈ σ ( Z s ) . • if s � = 1 then 1 �∈ σ ( U ) ⇒ 1 �∈ σ ( Z s ) . • if 1 is not an eigenvalue of U then 1 is not an eigenvalue of Z s *except* in the case when s = 1. Note that (for s = 1) we have � � X − V ( Y + 1 ) − 1 W 0 Z 1 = . 0 1