Strongly smooth paths of idempotents

Andruchow, Esteban

doi:10.1016/j.jmaa.2010.08.010

Cited by 3 publications

(10 citation statements)

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“…That is, α(t) = e tz x is the solution of (2) with α(0) = x, i.e. Γ t = e tz | M is the propagator of this equation, and the proof follows using Theorem 2.6 of [2] Remark 5.8. Suppose that e 0 , e 1 as above satisfy the condition e 0 ∧ e ⊥ 1 = 0 = e ⊥ 0 ∧ e 1 (weaker that e 0 − e 1 < 1), then there exists a unique geodesic δ(t) = e tz e 0 e −tz joining e 0 and e 1 .…”

Section: Hopf-rinow Theorem In Finite Factorsmentioning

confidence: 99%

Geodesics of projections in von Neumann algebras

Andruchow¹

2020

Preprint

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Let A be a von Neumann algebra and P A the manifold of projections in A. There is a natural linear connection in P A , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n . In this paper we show that two projections p, q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A), if and only ifwhere ∼ stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q ⊥ = p ⊥ ∧ q = 0. If A is a finite factor, any pair of projections in the same connected component of P A (i.e., with the same trace) can be joined by a minimal geodesic.We explore certain relations with Jones' index theory for subfactors. For instance, it is shown that if N ⊂ M are II 1 factors with finite index [M : N ] = t −1 , then the geodesic distance d(e N , e M ) between the induced projections e N and e M is d(e N , e M ) = arccos(t 1/2 ).

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Section: Hopf-rinow Theorem In Finite Factorsmentioning

confidence: 99%

Geodesics of projections in von Neumann algebras

Andruchow¹

2020

Preprint

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“…Theorem 4.4. Assume Hypothesis (1). Then for each t ∈ I, the map θ t := G t | B0 : B 0 → B t is a multiplicative * -isomorphism.…”

Section: For Eachmentioning

confidence: 99%

“…We refer the reader to [1] for the proofs of these facts, which though perfomed in K(H), are formally identical in our situation.…”

mentioning

confidence: 92%

“…It was shown in [1] in a different context, that the unitary propagator u t of this equation, (defined by u t ξ 0 = µ(t)), verifies…”

mentioning

confidence: 93%

“…This paper is a sequel to [1], where a similar matter is treated with more strict hypothesis. In [1] we considered a stronger smoothness condition, namely, that for each a ∈ A, the map I ∋ t → E t (a) ∈ A is continuously differentiable (in norm). The current regularity assumption on E t implies the existence of the bounded derivative operator, for each t ∈ I and a ∈ A…”

Section: Introductionmentioning

confidence: 99%

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Smooth Paths of Conditional Expectations

Andruchow

Larotonda

2011

Int. J. Math.

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Let A be a von Neumann algebra with a finite trace τ , represented in H = L 2 (A, τ ), and let Bt ⊂ A be sub-algebras, for t in an interval I (0 ∈ I). Let Et : A → Bt be the unique τ -preserving conditional expectation. We say that the path t → Et is smooth if for every a ∈ A and ξ ∈ H, the mapIf this operator verifies the additional boundedness condition,for any closed bounded sub-interval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras Bt are * -isomorphic. More precisely, there exists a curve Gt : A → A, t ∈ I of unital, * -preserving linear isomorphisms which intertwine the expectations,The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps B0 onto Bt. We show that this restriction is a multiplicative isomorphism. 1 * 2010 MSC. Primary 46L10; Secondary 58B10, 47D06. 1 Therefore a curve of possibly unbounded symmetric operators dE t is defined in H, with common domain A ⊂ H. We shall make the following assumption on dE:

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Geodesics of projections in von Neumann algebras

Andruchow

2021

Proc. Amer. Math. Soc.

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Let A {\mathcal {A}} be a von Neumann algebra and P A {\mathcal {P}}_{\mathcal {A}} the manifold of projections in A {\mathcal {A}} . There is a natural linear connection in P A {\mathcal {P}}_{\mathcal {A}} , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n \mathbb {C}^n . In this paper we show that two projections p , q p,q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A {\mathcal {A}} ), if and only if p ∧ q ⊥ ∼ p ⊥ ∧ q , \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} where ∼ \sim stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q ⊥ = p ⊥ ∧ q = 0 p\wedge q^\perp = p^\perp \wedge q=0 . If A {\mathcal {A}} is a finite factor, any pair of projections in the same connected component of P A {\mathcal {P}}_{\mathcal {A}} (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ⊂ M {\mathcal {N}}\subset {\mathcal {M}} are II 1 _1 factors with finite index [ M : N ] = t − 1 [{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1} , then the geodesic distance d ( e N , e M ) d(e_{\mathcal {N}},e_{\mathcal {M}}) between the induced projections e N e_{\mathcal {N}} and e M e_{\mathcal {M}} is d ( e N , e M ) = arccos ⁡ ( t 1 / 2 ) d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2}) .

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Strongly smooth paths of idempotents

Cited by 3 publications

References 8 publications

Geodesics of projections in von Neumann algebras

Geodesics of projections in von Neumann algebras

Smooth Paths of Conditional Expectations

Geodesics of projections in von Neumann algebras

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