The banach-lie group of lie triple automorphisms of an H*-algebra

2010 ◽  
Vol 30 (4) ◽  
pp. 1219-1226 ◽  
Author(s):  
A.J. Calderón Martín ◽  
C. Martín González
Keyword(s):  
Lie Group ◽  
Banach Lie Group

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.

Infinite Dimensional DeWitt Supergroups and their Bodies

2014 ◽  
Vol 57 (2) ◽  
pp. 283-288 ◽  
Author(s):  
Ronald Fulp
Keyword(s):  
Banach Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Grassmann Algebra ◽  
The Body ◽  
Exponential Mapping ◽  
Infinite Dimensional ◽  
Group Operation ◽  
Banach Lie Group ◽  
Quotient Manifold

AbstractFor Dewitt super groups G modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group BG compatible with the group operation on G, then, generically, the kernel K of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra κ has the property that for each a ∊ κ ada has a zero spectrum. We also show that the exponential mapping from κ to K is surjective and that K is a quotient manifold of the Banach space κ via a lattice in κ.

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.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

The Centralizer

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Diffeomorphism Group ◽  
Group Diff ◽  
Local Solutions

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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