On the homotopy type of the regular group of a realW *-algebra

Schröder, Herbert

doi:10.1007/bf01195808

Cited by 11 publications

(5 citation statements)

References 6 publications

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“…Here i * denotes the homomorphism induced by the inclusion i : H E ֒→ U M . We can then use results by Handelmann [15] and Schröeder [27] on computing the homotopy group of the unitary group of a von Neumann algebra. Case (1.)…”

Section: Corolary 24mentioning

confidence: 99%

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Orbits of conditional expectations

Argerami¹,

Stojanoff²

2001

Illinois J. Math.

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Let N ⊆ M be von Neumann algebras and E : M → N a faithful normal conditional expectation. In this work it is shown that the similarity orbit S(E) of E by the natural action of the invertible group of G M of M has a natural complex analytic structure and the map given by this action: G M → S(E) is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a Reductive Structure. These results were known previously under the conditions of finite index and N ′ ∩ M ⊆ N , which are removed in this work. Conversely, if the orbit S(E) has an Homogeneous Reductive Structure for every expectation defined on M , then M is finite. For every algebra M and every expectation E, a covering space of the unitary orbit U (E) is constructed in terms of the connected component of 1 in the normalizer of E. Moreover, this covering space is the universal covering in any of the following cases: 1) M is a finite factor and Ind(E) < ∞; 2) M is properly infinite and E is any expectation; 3) E is the conditional expectation onto the centralizer of a state. Therefore, in those cases, the fundamental group of U (E) can be characterized as the Weyl group of E.

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Section: Corolary 24mentioning

confidence: 99%

“…): Since M is a II 1 factor and Ind(E) < ∞ it is known (see [25]) that N = E(M) is also of type II 1 and dim Z(N) < ∞. Let us recall the following results (see [5], [15] and [27]):…”

Section: Corolary 24mentioning

confidence: 99%

Orbits of conditional expectations

Argerami¹,

Stojanoff²

2001

Illinois J. Math.

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“…In case b), i.e. L A (X) finite, Schröder [23] proved that π 2 (L A (X)) = 0 . In this case π 1 (U (1−e)LA(X)(1−e) ) ≃ Z(A) sa (1 − e) , i.e.…”

Section: Projective Space Of a Selfdual Modulementioning

confidence: 99%

“…In case b), i.e. L A (X) finite, Schröder [23] proved that π 2 (L A (X)) = 0 . In If A is a factor, then L A (X) is either finite or properly infinite.…”

Section: Projective Space Of a Selfdual Modulementioning

confidence: 99%

PROJECTIVE SPACE OF A C^*-MODULE

Andruchow¹,

Corach

Stojanoff

2001

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Let X be a right Hilbert C * -module over A. We study the geometry and the topology of the projective space P(X) of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere Sp(X) and the natural fibration Sp(X) → P(X), where Sp(X) = {x ∈ X: x, x = p}, for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra L A (X) of adjointable operators of X. The homotopy theory of these spaces is examined.

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“…Larotonda [CoLa1] that the map π i (GL n−1 (A)) → π i (GL n (A)) between the homotopy groups is surjective for n ≥ Bsr(A)+i+1 and injective for n ≥ Bsr(A) + i + 2. For other results of this type we refer to [Ri2], [Th], [Sc1], [Sc2], [Zh1], [Zh2].…”

Section: Computing Homotopy Groupsmentioning

confidence: 99%

Stable ranks, 𝐾-groups and Witt groups of some Banach and 𝐶*-algebras

Badea¹

1999

Function Spaces

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We show that certain dense and spectral invariant subalgebras of a C * -algebra have the same bilateral Bass stable rank. This is a partial answer for (a version of) an open problem raised by R.G. Swan. Then, for certain Banach algebras, we indicate when the homotopy groups π i (GL n (A)) stabilize for large n. This is an improvement of a result due to G. Corach and A. Larotonda. Using some results due to M. Karoubi, we show the isomorphism of the Witt group of a symmetric Banach algebra with the K 0

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On the homotopy type of the regular group of a realW *-algebra

Cited by 11 publications

References 6 publications

Orbits of conditional expectations

Orbits of conditional expectations

PROJECTIVE SPACE OF A C^*-MODULE

Stable ranks, 𝐾-groups and Witt groups of some Banach and 𝐶*-algebras

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On the homotopy type of the regular group of a realW *-algebra

Cited by 11 publications

References 6 publications

Orbits of conditional expectations

Orbits of conditional expectations

PROJECTIVE SPACE OF A C*-MODULE

Stable ranks, 𝐾-groups and Witt groups of some Banach and 𝐶*-algebras

Contact Info

Product

Resources

About

PROJECTIVE SPACE OF A C^*-MODULE