Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

2005 ◽  
Vol 53 (1) ◽  
pp. 33-50 ◽  
Author(s):  
Carlos E. Durán ◽  
Luis E. Mata-Lorenzo ◽  
Lázaro Recht
Keyword(s):  
Unitary Group ◽  
Homogeneous Spaces ◽  
Metric Geometry ◽  
C Algebra ◽  
Algebra Part

scholarly journals Metric geometry in homogeneous spaces of the unitary group of a C∗-algebra: Part I—minimal curves

2004 ◽  
Vol 184 (2) ◽  
pp. 342-366 ◽  
Author(s):  
Carlos E. Durán ◽  
Luis E. Mata-Lorenzo ◽  
Lázaro Recht
Keyword(s):  
Unitary Group ◽  
Homogeneous Spaces ◽  
Metric Geometry ◽  
C Algebra ◽  
Minimal Curves ◽  
Algebra Part

THE TRACE SPACE INVARIANT AND UNITARY GROUP OF C*-ALGEBRA

2003 ◽  
Vol 24 (01) ◽  
pp. 115-122
Author(s):  
XIAOCHUN FANG
Keyword(s):  
Unitary Group ◽  
C Algebra ◽  
Trace Space

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.

scholarly journals GENERALIZED JENSEN’S EQUATIONS IN BANACH MODULES OVER A C∗-ALGEBRA AND ITS UNITARY GROUP

2003 ◽  
Vol 7 (4) ◽  
pp. 641-655 ◽  
Author(s):  
Deok-Hoon Boo ◽  
Sei-Qwon Oh ◽  
Chun-Gil Park ◽  
Jae-Myung Park
Keyword(s):  
Unitary Group ◽  
C Algebra ◽  
Banach Modules

scholarly journals Klein's programme and quantum mechanics

2015 ◽  
Vol 12 (06) ◽  
pp. 1560006
Author(s):  
Jesús Clemente-Gallardo ◽  
Giuseppe Marmo
Keyword(s):  
Quantum Mechanics ◽  
Linear Group ◽  
Unitary Group ◽  
General Linear Group ◽  
Open Quantum Systems ◽  
Compact Subgroup ◽  
Quantum Systems ◽  
General Linear ◽  
C Algebra ◽  
Open Quantum

We review the geometrical formulation of quantum mechanics to identify, according to Klein's programme, the corresponding group of transformations. For closed systems, it is the unitary group. For open quantum systems, the semigroup of Kraus maps contains, as a maximal subgroup, the general linear group. The same group emerges as the exponentiation of the C*-algebra associated with the quantum system, when thought of as a Lie algebra. Thus, open quantum systems seem to identify the general linear group as associated with quantum mechanics and moreover suggest to extend the Klein programme also to groupoids. The usual unitary group emerges as a maximal compact subgroup of the general linear group.

scholarly journals Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

2019 ◽  
Vol 31 (6) ◽  
pp. 1567-1605 ◽  
Author(s):  
Gabriel Larotonda
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Linear Operators ◽  
Finsler Metric ◽  
Invariant Metrics ◽  
Metric Geometry ◽  
Infinite Dimensional ◽  
Left Invariant Metrics ◽  
Geodesic Structure ◽  
Infinite Dimensional Lie Groups

AbstractWe study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

scholarly journals UNITARIES IN A SIMPLE C*-ALGEBRA OF TRACIAL RANK ONE

2010 ◽  
Vol 21 (10) ◽  
pp. 1267-1281 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
State Space ◽  
Unitary Group ◽  
Affine Function ◽  
Connected Component ◽  
Finite Spectrum ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Tracial State ◽  
Rank One ◽  
Tracial Rank

Let A be a unital separable simple infinite dimensional C*-algebra with tracial rank not more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ϵ > 0, there exists a self-adjoint element h ∈ As.a such that [Formula: see text] We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U0(A). Denote by [Formula: see text] the affine function on T(A) defined by [Formula: see text]. We show that u can be approximated by unitaries in A with finite spectrum if and only if u ∈ CU(A) and [Formula: see text] for all n ≥ 1. Examples are given for which there are unitaries in CU(A) which cannot be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.

scholarly journals Quantum Orbit Method in the Presence of Symmetries

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 724
Author(s):  
Nicola Ciccoli
Keyword(s):  
Lie Groups ◽  
Poisson Structure ◽  
Homogeneous Spaces ◽  
Orbit Method ◽  
C Algebra ◽  
Poisson Lie Groups ◽  
Poisson Homogeneous Spaces

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Sign in / Sign up

Export Citation Format

Share Document