Bi-free Infinite Divisibility James Mingo (Queen’s University at Kingston) joint work with Jerry Gu (Queen’s) and Hao-Wei Huang (Kaohsiung) (on the arXiv) Free Probability and the Large N Limit, V Berkeley, March 22, 2016 1 / 15
Kaohsiung Harbour 2 / 15
classical infinite divisibility (de Finetti, Kolmogorov, cin, 1928-1937 ( ∗ ) ) L´ evy, Hinˇ ◮ X real random variable, ◮ ϕ ( t ) = E ( e itX ) characteristic function ◮ X is infinitely divisible if for all n there exist X 1 , . . . , X n independent and identically distributed such that D X ∼ X 1 + · · · + X n ◮ X is infinitely divisible ⇔ ∃ α ∈ R and σ pos. measure s.t. � 1 + x 2 � � itx e itx − 1 − log ( ϕ ( t )) = α t + d σ ( t ) 1 + x 2 x 2 such an X is ‘manifestly’ infinite divisible ( ∗ ) according to Steutel & van Harn, 2004 3 / 15
free version, Bercovici-Voiculescu (1993) Compactly supported case ◮ R ( z ) = κ 1 + κ 2 z + κ 3 z 2 + · · · is the R -transform of a compactly supported measure on R ◮ X is freely infinitely divisible ⇔ ∃ and σ pos. measure s.t. � z R ( z ) = κ 1 + 1 − tz d σ ( t ) ⇔ R can be extended to C + and maps C + to C + ⇔ other equivalences . . . such an X is ‘manifestly’ infinite divisible 4 / 15
Triangular Arrays (Nica-Speicher, 2006) ◮ suppose that for each N , a N ,1 , . . . , a N , N ∈ ( A N , ϕ N ) free and identically dist. dist ◮ a N ,1 + · · · + a N , N −→ b ∈ ( A , ϕ ) N N ϕ N ( a n ⇔ lim N ,1 ) exists (and = κ n ( b , . . . , b ) if limit exists) ( use moment-cumulant formula and find leading order terms ) this condition implies ◮ { κ n } n ( cumulants of b ) are conditionally positive, which means �� � α n a n � ∗ � � α m a m �� � α m α n κ m + n = lim N N ϕ N � 0 m , n � 1 m n ◮ which implies that ∃ a finite pos. measure σ s.t. � t n d σ ( t ) κ n + 2 = 5 / 15
conditionally positive sequences ◮ if the cumulants of b satisfy � t n d σ ( t ) κ n + 2 = ◮ then the R -transform of b can be written � � κ n + 1 z n = κ 1 + z κ n + 2 z n = κ 1 + ( tz ) n d σ ( t ) � � R ( z ) = n � 0 n � 0 n � 0 � z = κ 1 + 1 − tz d σ ( t ) ◮ also such a { κ n } n produces an inner product on C 0 [ X ] = poly. variable X without constant term, let H be the corresponding Hilbert space and F ( H ) the full Fock space over H 6 / 15
Fock space ◮ H a Hilbert space, ◮ ξ ∈ H , ℓ ( ξ ) = left creation operator and ◮ ℓ ( ξ ) ∗ = left annihilation operator ◮ T ∈ B ( H ) , Λ ( T ) Ω = 0, Λ ( T )( ξ 1 ⊗ · · · ⊗ ξ n ) = T ( ξ 1 ) ⊗ · · · ⊗ ξ n ◮ for Y 1 = ℓ ( ξ ) , Y 2 = ℓ ( η ) ∗ , Y 3 = Λ ( T ∗ ) , Y 4 = α I then the only non-vanishing cumulant of κ n ( Y i 1 , . . . , Y i n ) is κ n ( ℓ ( η ) ∗ , Λ ( T 1 ) , . . . , Λ ( T n − 2 ) , ℓ ( ξ )) = � T 1 · · · T n − 1 ξ , η � ( use limit theorem from 2 pages back ) 7 / 15
circle closed ◮ if { t n } n is conditionally positive and H the Hilbert space obtained from C 0 [ X ] we let X be the operator of left multiplication on H (bounded because of growth assumptions on { t n } n ), also we let b = ℓ ( X ) + ℓ ( X ) ∗ + Λ ( X ) + t 1 ∈ B ( F ( H )) then κ n ( b , . . . , b ) = t n so { t n } n is the cumulant sequence of a bounded self-adjoint operator ◮ operators of the form ℓ ( X ) + ℓ ( X ) ∗ + Λ ( X ) + t 1 are ‘manifestly’ freely infinitely divisible ( ∗ ) : � X ⊕ 0 ⊕ · · · 0 � X ⊕ 0 ⊕ · · · 0 + Λ ( X ⊕ 0 ⊕ · · · 0 ) + t 1 � ∗ � √ √ ℓ + ℓ N N N . . . � 0 ⊕ 0 ⊕ · · · X � 0 ⊕ 0 ⊕ · · · X + Λ ( 0 ⊕ 0 ⊕ · · · X ) + t 1 � ∗ � √ + ℓ √ ℓ N N N ( ∗ ) because Hilbert space is infinitely divisible 8 / 15
bi-freeness ( slightly simplified ) ◮ X i = C ξ i ⊕ ˚ X i vector spaces with distinguished subspace of co-dimension 1 ⊕ ˚ ◮ ( X , ˚ X , ξ ) = ∗ i ( X i , ˚ � � X i 1 ⊗ · · · ⊗ ˚ X i , ξ i ) = C ξ ⊕ X i n n � 1 i 1 � ··· � i n ◮ l r , r i : L ( X i ) −→ L ( X ) “left” and “right” actions ◮ ( A i , B i ) ⊂ L ( X i ) , a pair of faces, act on X via l i and r i ◮ � · ξ , ξ � gives a state on the pairs ( l i ( A i ) , r i ( B i )) ◮ the pairs of faces (algebras) are bi-free by construction ◮ ∃ ? a description of bi-freeness without explicit use of free products, a challenge no cumulantologist can resist 9 / 15
bi-free cumulants (Mastnak-Nica) ◮ given χ : [ n ] → { l , r } let χ − 1 ( l ) = { i 1 < · · · < i p } and χ − 1 ( r ) = { j 1 < · · · < j n − p } ◮ usual non-crossing partitions are with respect to the order ( 1, 2, 3, . . . , n ) ◮ NC χ ( n ) are non-crossing with respect to ( i 1 , . . . , i p , j n − p , . . . , j i ) � κ χ ◮ ϕ ( a 1 · · · a n ) = π ( a 1 , . . . , a n ) π ∈ NC χ ( n ) ( moment-cumulant formula ) ◮ bi-freeness ⇔ vanishing of mixed bi-free cumulants ( Charlesworth, Nelson & Skoufranis ) 10 / 15
bi-variate case: [ a , b ] = 0 ◮ suppose a and b are commuting self-adjoint operators in a C ∗ -algebra with a state ϕ ◮ get µ ∈ M ( R 2 ) a compactly supported probability measure ◮ given χ : [ n ] → { l , r } let c 1 , . . . , c n be defined by c i = a if χ i = l and c i = b if χ i = r ◮ κ χ n ( c 1 , . . . , c n ) only depends on # ( χ − 1 ( l )) and # ( χ − 1 ( r )) ◮ κ m , n ( a , b ) means m occurrences of a and n occurrences of b � κ m , n z m w n , G ( z , w ) = ϕ (( z − a ) − 1 ( w − b ) − 1 ) R a , b ( z , w ) = m , n � 0 m + n � 1 zw ◮ R a , b ( z , w ) = zR a ( z ) + wR b ( w ) + 1 − G ( K a ( z ) , K b ( w )) ◮ µ 1 ⊞ ⊞ µ 2 is the distribution of the pair ( a 1 + a 2 , b 1 + b 2 ) where ( a 1 , b 1 ) and ( a 2 , b 2 ) are bi-free 11 / 15
bi-free infinite divisibility ◮ if for every N we can find µ N such that µ = µ ⊞⊞ N then µ is N bi-freely infinitely divisible thm : t . f . a . e . 1. µ bi-freely infinitely divisible 2. { κ m , n } m , n are conditionally positive and conditionally bounded 2-sequences ( to be explained ) 3. R a , b has the integral representation � z w R a , b ( z , w ) = zR 1 ( z ) + wR 2 ( w ) + 1 − wt d ρ ( s , t ) 1 − zs z w � � with R 1 ( z ) = κ 1,0 + 1 − zs d ρ 1 ( s , t ) , R 2 ( w ) = κ 0,1 + 1 − wt d ρ 2 ( s , t ) ρ 1 and ρ 2 compactly supported, ρ a signed Borel measure with compact support and | ρ ( { 0, 0 } ) | 2 � ρ 1 ( { 0, 0 } ) ρ 2 ( { 0, 0 } ) , td ρ 1 ( s , t ) = sd ρ ( s , t ) , sd ρ 2 ( s , t ) = td ρ ( s , t ) 12 / 15
conditionally positive and cond. bounded ◮ C 0 [ x , y ] polynomials in commuting variables without constant term ◮ � x m 1 y n 1 , x m 2 y n 2 � = κ m 1 + m 2 , n 1 + n 2 is a positive semi-def. inner product ( conditionally positive ) ◮ ∃ L > 0 s.t. | � x m y n p , p � | � L m + n � p , p � ( conditionally bounded ) ◮ inner product on C 0 [ x , y ] gives Hilbert space H and two multiplication operators T 1 (by x ) and T 2 (by y ) with spectral measures E 1 and E 2 (note T 1 ( y ) = T 2 ( x ) ) ◮ ρ ([ c 1 , d 1 ] × [ c 2 , d 2 ]) := � E 1 ([ c 1 , d 1 ]) x , E 2 ([ c 2 , d 2 ]) y � z w � κ m , n z m w n ( ∗ ) � = 1 − wt d ρ ( s , t ) ( by calculation ) ◮ 1 − zs m , n � 1 ◮ θ ( 1 ) m , n = κ m + 2, n , θ ( 2 ) m , n = κ m , n + 2 give positive finite compactly supported measures ρ 1 and ρ 2 ( s m t n ) t d ρ 1 ( s , t ) = κ m + 2, n + 1 = ( s m t n ) s d ρ ( s , t ) (by ( ∗ ) ) ◮ � � 13 / 15
bi-partite infinitely divisible operators ◮ H a Hilbert space, F ( H ) the full Fock space over H ◮ f , g ∈ H , T 1 = T ∗ 1 , T 2 = T ∗ 2 ∈ B ( H ) ◮ a = ℓ ( f ) + ℓ ( f ) ∗ + Λ l ( T 1 ) + λ 1 ∈ B ( F ( H )) ◮ b = r ( g ) + r ( g ) ∗ + Λ r ( T 2 ) + λ 2 ∈ B ( F ( H )) ◮ a , b commute i ff [ T 1 , T 2 ] = 0, T 1 ( g ) = T 2 ( f ) , � f , g � ∈ R � f ⊕ 0 ⊕ · · · ⊕ 0 � f ⊕ 0 ⊕ · · · ⊕ 0 � ∗ � ◮ a N ,1 = ℓ √ √ + ℓ + Λ l ( T 1 ⊕ N N 0 ⊕ · · · ⊕ 0 ) + λ 1 N � g ⊕ 0 ⊕ · · · ⊕ 0 � g ⊕ 0 ⊕ · · · ⊕ 0 � ∗ � ◮ b N ,1 = r √ + r √ + Λ r ( T 2 ⊕ N N 0 ⊕ · · · ⊕ 0 ) + λ 2 N ◮ ( a , b ) bi-freely infinite divisible ◮ κ m , n ( a , b ) = � T m − 1 f , T n − 1 g � , κ m ,0 = � T m − 2 f , f � , κ 1,0 = λ 1 2 1 1 14 / 15
example: bi-free Poisson ◮ ( α , β ) ∈ R 2 , λ > 0 1 − λ δ ( 0,0 ) + λ ◮ µ N = � � N δ ( α , β ) N N µ ⊞⊞ N ◮ µ = lim is bi-freely infinite divisible N ◮ has bi-free cumulants κ m , n = λα m β n ( use limit theorem ) � κ m , n z m w n = � λ ( α z ) m ( β w ) n ◮ and R ( z , w ) = m , n � 0 m , n � 0 m + n � 1 m + n � 1 β 2 w α 2 z λα z β w � � � � = λ z α + + λ w β + + 1 − α z 1 − β w ( 1 − α z )( 1 − β w ) ρ 1 ( s , t ) = λ s 2 δ ( α , β ) , ρ 2 ( s , t ) = λ t 2 δ ( α , β ) , ρ ( s , t ) = λ st δ ( α , β ) ( ρ positive when αβ > 0) 15 / 15
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