groupoid of partially invertible elements w-algebras
TRANSCRIPT
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IntroductionBanach manifold structures
Varna, 8-13 June 2012
Groupoidof
partially invertible elementsof W ∗-algebras
Aneta Sli»ewska
Institut of MathematicsUniversity in Biaªystok
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
REFERENCES:
1 K. Mackenzie. General Theory of Lie Groupoids and Lie
Algebroids, Cambridge University Press,2005.
2 A. Odzijewicz, A. Sli»ewska. Groupoids and inverse semigroups
associated to W ∗-algebra, arXiv:1110.6305.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
BASIC DEFINITIONS
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Groupoid over the base set B is a set G with actions:
(i) source map s : G → B and target map t : G → B
(ii) product m : G(2) → G
m(g, h) =: gh,
dened on the set of composable pairs
G(2) := (g, h) ∈ G × G : s(g) = t(h),
(iii) iniective identity section ε : B → G,(iv) inverse map ι : G → G,
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
which satisfy the following conditions:
s(gh) = s(h), t(gh) = t(g), (1)
k(gh) = (kg)h, (2)
ε(t(g))g = g = gε(s(g)), (3)
ι(g)g = ε(s(g)), gι(g) = ε(t(g)), (4)
where g, k, h ∈ G.Notation: G ⇒ B.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
GROUPOID G(M)
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Left support l(x) ∈ L(M) (right support r(x) ∈ L(M)) ofx ∈M is the least projection in M, such that
l(x)x = x (resp. x r(x) = x). (5)
If x ∈M is selfadjoint, then support s(x)
s(x) := l(x) = r(x).
Polar decomposition for x ∈M
x = u|x|, (6)
where u ∈M is partial isometry and |x| :=√x∗x ∈M+. Then
l(x) = s(|x∗|) = uu∗, r(x) = s(|x|) = u∗u.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Let G(pMp) - the group of all invertible elements in
W ∗-subalgebra pMp ⊂M.
We dene the set G(M) of partially invertible elements in M
G(M) := x ∈M; |x| ∈ G(pMp), where p = s(|x|)
Remark: G(M) M.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Theorem
The set G(M) with
1 the source and target maps s, t : G(M)→ L(M)
s(x) := r(x), t(x) := l(x),
2 the product dened as the product in M on the set
G(M)(2) := (x, y) ∈ G(M)× G(M); s(x) = t(y),
3 the identity section ε : L(M) → G(M) as the identity,
4 the inverse map ι : G(M)→ G(M) dened by
ι(x) := |x|−1u∗,
is the groupoid over L(M).
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
GROUPOID REPRESENTATIONS
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
The set G(E) of linear isomorphisms enm : Em → En between the
bres of a vector bundle π : E→M has the groupoid structure
over M .
It is called the structural groupoid of the bundle π : E→M .
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Let G be a groupoid over B. Representation of G ⇒ B on a
vector bundle π : E→M is a groupoid morphism of G into the
structural groupoid G(E) of this bundle:
G G(E)
B M
?? ??
-
-
s t t′s′
φ
ϕ(7)
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
One has the bundle π :MR(M)→ L(M), where
MR(M) := (y, p) ∈M× L(M) : p r(y) = r(y)
and π := pr2. The bre π−1(p) over p ∈ L(M) is isomorphic to
the right W ∗-ideal (=M-modul) pM of M generated by p.
Fact
For x ∈ G(M)qp the left action
Lx : pM → qM
is the isomorphism of the right W ∗-ideals.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Theorem
1 The structural groupoid G(MR(M)) of the bundle
π :MR(M)→ L(M) of the right W ∗-ideals is isomorphic to
G(M).
2 The structural groupoid G(ML(M)) of the bundle
π :ML(M)→ L(M) of the left W ∗-ideals is isomorphic to
G(M).
3 The structural groupoid G(A(M)) of the bundle
π : A(M)→ L(M) of W ∗-subalgebras is isomorphic to G(M),where A(M) := (y, p) ∈M× L(M) : y ∈ pMp.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Theorem
1 The structural groupoid G(MR(M)) of the bundle
π :MR(M)→ L(M) of the right W ∗-ideals is isomorphic to
G(M).
2 The structural groupoid G(ML(M)) of the bundle
π :ML(M)→ L(M) of the left W ∗-ideals is isomorphic to
G(M).
3 The structural groupoid G(A(M)) of the bundle
π : A(M)→ L(M) of W ∗-subalgebras is isomorphic to G(M),where A(M) := (y, p) ∈M× L(M) : y ∈ pMp.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
GroupoidW∗-algebraSupports and polar decompositionGroupoid of partially invertible elementsGroupoid representations
Theorem
1 The structural groupoid G(MR(M)) of the bundle
π :MR(M)→ L(M) of the right W ∗-ideals is isomorphic to
G(M).
2 The structural groupoid G(ML(M)) of the bundle
π :ML(M)→ L(M) of the left W ∗-ideals is isomorphic to
G(M).
3 The structural groupoid G(A(M)) of the bundle
π : A(M)→ L(M) of W ∗-subalgebras is isomorphic to G(M),where A(M) := (y, p) ∈M× L(M) : y ∈ pMp.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
DIFFERENTIAL STRUCTURES
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
We consider the following locally convex topologies, for which hold:
σ−topology ≺ s−topology ≺ s∗−topology ≺ uniform topology
where for ω ∈M+∗
1 σ-topology is dened by a family of semi-norms
‖x‖σ := |〈x, ω〉|,2 s-topology is dened by a family of semi-norms
‖x‖ω :=√〈x∗x, ω〉; x ∈M ;
3 s∗-topology is dened by a family of semi-norms
‖·‖ω , ‖·‖∗ω : ω ∈M+
∗ where ‖x‖∗ω :=
√〈xx∗, ω〉; x ∈M.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Theorem
For an innite dimmentional W ∗-algebra M the groupoid G(M) is
not topological with respect to above topologies of M.
Example: Let p ∈ L(M) and xn ∈ G(M) as
xn = p+1
n(1− p), n ∈ N.
Then
s(xn) = t(xn) = 1 and s(p) = t(p) = p.
The uniform limit limn→∞ xn = p, so the source and target
maps are not continuous.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
L(M) as a BANACH MANIFOLD
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
For p ∈ L(M) let us dene the set
Πp := q ∈ L(M) : M = qM⊕ (1− p)M
then q ∧ (1− p) = 0, q ∨ (1− p) = 1and q = x− y ∈ qM⊕ (1− p)M.
Dene the maps
σp : Πp → qMp, ϕp : Πp → (1− p)Mp
by
σp(q) := x, ϕp(q) := y.
The map ϕp is the bijection of Πp onto the Banach space
(1− p)Mp.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
In order to nd the transitions maps
ϕp ϕ−1p′ : ϕp′(Πp ∩Πp′)→ ϕp(Πp ∩Πp′)
in the case Πp ∩Πp′ 6= ∅, let us take for q ∈ Πp ∩Πp′ the following
splittings
M = qM⊕ (1− p)M = pM⊕ (1− p)MM = qM⊕ (1− p′)M = p′M⊕ (1− p′)M.
(8)
The splittings (9) lead to the corresponding decompositions of pand p′
p = x− y p = a+ bp′ = x′ − y′ 1− p = c+ d
(9)
where x ∈ qMp, y ∈ (1− p)Mp, x′ ∈ qMp′, y′ ∈ (1− p′)Mp′,a ∈ p′Mp, b ∈ (1− p′)Mp, c ∈ p′M(1− p) and
d ∈ (1− p′)M(1− p).Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Combining equations from (10) we obtain
q = ι(x′) + y′ι(x′) (10)
q = (a+ cy)ι(x) + (b+ dy)ι(x). (11)
Comparing (11) and (12) we nd that
ι(x′) = (a+ cy)ι(x) (12)
y′ι(x′) = (b+ dy)ι(x). (13)
After substitution (13) into (14) and noting that t(a+ cy) 6 p′ wenally get the formula
y′ = (ϕp′ ϕ−1p )(y) = (b+ dy)ι(a+ cy).
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Theorem
The family of maps
(Πp, ϕp) p ∈ L(M)
denes a smooth atlas on a L(M). This atlas is modeled by the
family of Banach spaces (1− p)Mp, for p ∈ L(M).
Fact: If p′ ∈ Op then (1− p)Mp ∼= (1− p′)Mp′.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 25: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/25.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
G(M) as a BANACH MANIFOLD
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 26: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/26.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
For projections p, p ∈ L(M) we dene the set
Ωpp := t−1(Πp) ∩ s−1(Πp)
and a map
ψpp : Ωpp → (1− p)Mp⊕ pMp⊕ (1− p)Mp
in the following way
ψpp(x) := (ϕp(t(x)), ι(σp(t(x)))xσp(s(x)), ϕp(s(x))) ,
where p = σp(q)− ϕp(q) ∈ qM⊕ (1− p)M.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 27: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/27.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Theorem
The family of maps
(Ωpp, ψpp) p, p ∈ L(M)
denes a smooth atlas on the groupoid G(M). The complex
Banach manifold structure of G(M) has type G, where G is the set
of Banach spaces
(1− p)Mp⊕ pMp⊕ (1− p)Mp
indexed by the pair of equivalent projections of L(M).
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 28: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/28.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
U(M) as a BANACH MANIFOLD
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 29: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/29.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
The groupoid U(M) is the set of xed points of the automorphism
J : G(M)→ G(M) dened by
J(x) := ι(x∗).
Expressing J : Ωpp → Ωpp in the coordinates
ψpp(x) = (y, z, y) ∈ (1− p)Mp⊕ ipMhp⊕ (1− p)Mp
we nd that (ψpp J ψ−1pp
)(y, z, y) =
=(y, ι(σp(ϕ
−1p (y))∗σp(ϕ
−1p (y)))ι(z∗)σp(ϕ
−1p (y))∗σp(ϕ
−1p (y)), y
),
where z ∈ G(M)pp ⊂ pMp. Since J2(x) = x for x ∈ U(M) one has
(DJ(x))2 = 1
for DJ(x) : TxG(M)→ TxG(M).Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Thus one obtains a splitting of the tangent space
TxG(M) = T+x G(M)⊕ T−x G(M) (14)
dened by the Banach space projections
P±(x) :=1
2(1±DJ(x)) . (15)
The Frechét derivative Dι(z) of the inversion map
ι : G(M)pp 3 z 7→ ι(z) ∈ G(M)pp at the point z is given by
Dι(z) Mz = −ι(z) Mz ι(z),
where Mz ∈ pMp.
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
![Page 31: Groupoid of partially invertible elements W-algebras](https://reader031.vdocuments.mx/reader031/viewer/2022012013/6158599d0005db49271dd08c/html5/thumbnails/31.jpg)
IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
Theorem
The groupoid U(M) of partial isometries has a natural structure of
the real Banach manifold of the type G, where the family G consist
of the real Banach spaces
(1− p)Mp⊕ pMhp⊕ (1− p)Mp
parameterized by the pairs (p, p) ∈ L(M)× L(M) of equivalent
projections.
Corollary
The groupoid G(M) is the complexication of U(M).
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra
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IntroductionBanach manifold structures
Complex Banach manifold structure on L(M)Complex Banach manifold structure on G(M)Banach manifold structure on U(M)
THANK YOU
Aneta Sli»ewska Groupoid of partially invertible elements of W∗-algebra