scholarly journals Sphere and projective space of a C*-algebra with a faithful state

2019 ◽  
Vol 52 (1) ◽  
pp. 410-427
Author(s):  
Andrea C. Antunez
Keyword(s):  
Initial Data ◽  
Projective Space ◽  
Banach Spaces ◽  
Lie Group ◽  
Unit Sphere ◽  
Homogeneous Spaces ◽  
Inner Product ◽  
C Algebra ◽  
Faithful State ◽  
Banach Lie Group

AbstractLet 𝒜 be a unital C*-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊ϕ = {x ∈ 𝒜 : ϕ(x*x) = 1} and the projective space ℙϕ = 𝕊ϕ/𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰ϕ(𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙϕ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

Highly Symmetric Homogeneous Spaces

1974 ◽  
Vol 26 (02) ◽  
pp. 291-293 ◽  
Author(s):  
L. N. Mann
Keyword(s):  
Projective Space ◽  
Lie Group ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Real Projective Space ◽  
Standard Sphere ◽  
Connected Lie Group ◽  
Trivial Subgroup

We consider effective homogeneous spacesM=G/HwhereGis a compact connected Lie group,His a closed subgroup andGacts effectively onM(i.e.,Hcontains no non-trivial subgroup normal inG). It is known that dimG≦m2/2 +m/2 wherem= dimMand that if dimG=m2/2 +m/2, thenMis diffeomorphic to the standard sphereSmor the standard real projective spaceRPm[1]. In addition it has been shown that for fixedmthere are gaps in the possible dimensions forGbelow the maximal bound [4; 5].

Concurrent and parallel chords of spheres in normed linear spaces

2010 ◽  
Vol 47 (4) ◽  
pp. 505-512
Author(s):  
Horst Martini ◽  
Senlin Wu
Keyword(s):  
Banach Spaces ◽  
Unit Sphere ◽  
Normed Linear Space ◽  
Inner Product ◽  
Product Spaces ◽  
Euclidean Case ◽  
Linear Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

Homogeneous spaces yielding solutions of the k[S] –hierarchy and its strict version

2021 ◽  
pp. 315-336
Author(s):  
Gerard F. Helminck ◽  
Jeffrey A. Weenink
Keyword(s):  
Lie Group ◽  
Commutative Algebra ◽  
Homogeneous Spaces ◽  
Shift Operator ◽  
The Matrix ◽  
Lax Equations ◽  
Banach Lie Group ◽  
Dimension One

The k[S] -hierarchy and its strict version are two deformations of the commutative algebra k[S], k=R or C; in the N×N-matrices, where S is the matrix of the shift operator. In this paper we show first of all that both deformations correspond to conjugating k[S] with elements from an appropriate group. The dressing matrix of the deformation is unique in the case of the k[S]-hierarchy and it is determined up to a multiple of the identity in the strict case. This uniqueness enables one to prove directly the equivalence of the Lax form of the k[S]-hierarchy with a set of Sato-Wilson equations. The analogue of the Sato-Wilson equations for the strict k[S]-hierarchy always implies the Lax equations of this hierarchy. Both systems are equivalent if the setting one works in, is Cauchy solvable in dimension one. Finally we present a Banach Lie group G(S_2), two subgroups P_+ (H) and U_+ (H) of G(S_2), with U_+ (H)⊂P_+ (H), such that one can construct from the homogeneous spaces G(S_2 )/P_+ (H) resp. G(S_2)/U_+ (H) solutions of respectively the k[S]-hierarchy and its strict version.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

scholarly journals The spherical maximal function on the free two-step nilpotent Lie group

2006 ◽  
Vol 99 (1) ◽  
pp. 99 ◽  
Author(s):  
Véronique Fischer
Keyword(s):  
Lie Group ◽  
Maximal Function ◽  
Unit Sphere ◽  
Nilpotent Lie Group ◽  
Homogeneous Unit

We consider here the free two step nilpotent Lie group, provided with the homogeneous Korányi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Double Covers and Metastable Immersions of Spheres

1974 ◽  
Vol 26 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Robert Wells
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Projective Space ◽  
Unit Sphere ◽  
Real Line ◽  
Double Cover ◽  
Sphere Bundle ◽  
Canonical Line Bundle ◽  
Projective Space Bundle ◽  
Double Covers

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

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