scholarly journals Homogeneity of the Pure State Space of a Separable C*-Algebra

2003 ◽  
Vol 46 (3) ◽  
pp. 365-372 ◽  
Author(s):  
Akitaka Kishimoto ◽  
Narutaka Ozawa ◽  
Shôichirô Sakai
Keyword(s):  
Automorphism Group ◽  
State Space ◽  
Pure State ◽  
C Algebra ◽  
Inner Automorphisms

AbstractWe prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple C*-algebras. The first result of this kind was shown by Powers for the UHF algbras some 30 years ago.

scholarly journals HOMOGENEITY OF THE PURE STATE SPACE FOR SEPARABLE C*-ALGEBRAS

2001 ◽  
Vol 12 (07) ◽  
pp. 813-845 ◽  
Author(s):  
HAJIME FUTAMURA ◽  
NOBUHIRO KATAOKA ◽  
AKITAKA KISHIMOTO
Keyword(s):  
Automorphism Group ◽  
State Space ◽  
Large Class ◽  
Pure State ◽  
C Algebras ◽  
Inner Automorphisms ◽  
Af Algebras

We prove that the pure state space is homogeneous under the action of the automorphism group (or a certain smaller group of approximately inner automorphisms) for a fairly large class of simple separable nuclear C*-algebras, including the approximately homogeneous C*-algebras and the class of purely infinite C*-algebras which has been recently classified by Kirchberg and Phillips. This extends the known results for UHF algebras and AF algebras by Powers and Bratteli.

scholarly journals Homogeneity of the Pure State Space of the Cuntz Algebra

2000 ◽  
Vol 171 (2) ◽  
pp. 331-345 ◽  
Author(s):  
Ola Bratteli ◽  
Akitaka Kishimoto
Keyword(s):  
State Space ◽  
Pure State ◽  
Cuntz Algebra

On the automorphism group of the integral group ring of the infinite dihedral group

1988 ◽  
Vol 108 (1-2) ◽  
pp. 1-10
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Group Ring ◽  
Polynomial Ring ◽  
Dihedral Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Index 2 ◽  
Skolem Theorem ◽  
Inner Automorphisms

SynopsisLet ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

scholarly journals The automorphism group of the irrational rotation C*-algebra

1993 ◽  
Vol 155 (1) ◽  
pp. 3-26 ◽  
Author(s):  
George A. Elliott ◽  
Mikael Rørdam
Keyword(s):  
Automorphism Group ◽  
Irrational Rotation ◽  
C Algebra

scholarly journals UHF FLOWS AND THE FLIP AUTOMORPHISM

2001 ◽  
Vol 13 (09) ◽  
pp. 1163-1181 ◽  
Author(s):  
A. KISHIMOTO
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Adjoint Action ◽  
Restricted Class ◽  
C Algebra ◽  
Type Formula ◽  
Car Algebra ◽  
Suitable Sense ◽  
Cocycle Conjugacy

A UHF algebra is a C*-algebra A of the type [Formula: see text] for some sequence (ni) with ni≥2, where Mn is the algebra of n×n matrices, while a UHF flow α is a flow (or a one-parameter automorphism group) on the UHF algebra A obtained as [Formula: see text], where [Formula: see text] for some [Formula: see text]. This is the simplest kind of flows on the UHF algebra we could think of; yet there seem to have been no attempts to characterize the cocycle conjugacy class of UHF flows so that we might conclude, e.g., that the non-trivial quasi-free flows on the CAR algebra are beyond that class. We give here one attempt, which is still short of what we have desired, using the flip automorphism of A⊗A. Our characterization for a somewhat restricted class of flows (approximately inner and absorbing a universal UHF flow) says that the flow α is cocycle conjugate to a UHF flow if and only if the flip is approximated by the adjoint action of unitaries which are almost invariant under α⊗α. Another tantalizing problem is whether we can conclude that a flow is cocycle conjugate to a UHF flow if it is close to a UHF flow in a suitable sense. We give a solution to this, as a corollary, for the above-mentioned restricted class of flows. We will also discuss several kinds of flows to clarify the situation.

scholarly journals SOME FEATURES OF THE STATE-SPACE TRAJECTORIES FOLLOWED BY ROBUST ENTANGLED FOUR-QUBIT STATES DURING DECOHERENCE

2010 ◽  
Vol 08 (03) ◽  
pp. 505-515 ◽  
Author(s):  
A. P. MAJTEY ◽  
A. BORRAS ◽  
A. R. PLASTINO ◽  
M. CASAS ◽  
A. PLASTINO
Keyword(s):  
State Space ◽  
Recent Work ◽  
Mixed State ◽  
Pure State ◽  
The State ◽  
Time Dependent ◽  
Space Trajectories ◽  
Geometrical Features ◽  
Initial States ◽  
Qubit States

In a recent work (Borras et al., Phys. Rev. A79 (2009) 022108), we have determined, for various decoherence channels, four-qubit initial states exhibiting the most robust possible entanglement. Here, we explore some geometrical features of the trajectories in state space generated by the decoherence process, connecting the initially robust pure state with the completely decohered mixed state obtained at the end of the evolution. We characterize these trajectories by recourse to the distance between the concomitant time-dependent mixed state and different reference states.

ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

2008 ◽  
Vol 18 (02) ◽  
pp. 209-226 ◽  
Author(s):  
VITALY ROMAN'KOV
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Lie Algebra ◽  
Free Subgroup ◽  
Finite Set ◽  
Free Metabelian Lie Algebra ◽  
Inner Automorphisms ◽  
Infinite Set ◽  
Wild Automorphisms

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3 of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3 of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3 cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3 which generates a free subgroup F∞ modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.

scholarly journals Limits of pure states, II

1992 ◽  
Vol 35 (2) ◽  
pp. 227-231
Author(s):  
R. J. Archbold ◽  
A. M. Zaki
Keyword(s):  
State Space ◽  
Pure State ◽  
Pure States

We answer a question raised in an earlier paper concerning the pure state space of a separable C*-algebra.

scholarly journals Limits of pure states

1989 ◽  
Vol 32 (2) ◽  
pp. 249-254 ◽  
Author(s):  
R. J. Archbold
Keyword(s):  
State Space ◽  
Pure State ◽  
Transition Probabilities ◽  
Weak Limit ◽  
General Convex ◽  
Pure States

In [7, Section 5], Glimm showed that if φ and ψ are inequivalent pure states of a liminal C*-algebra A such that the Gelfand-Naimark-Segal (GNS) representations πφ and πψ cannot be separated by disjoint open subsets of the spectrum  then ½ (φ+ψ) is a weak*-limit of pure states. We extend this to arbitrary C*-algebras (and more general convex combinations) by means of what we hope will be regarded as a transparent proof based on the notion of transition probabilities. As an application, we show that if J is a proper primal ideal of a separable C*-algebra A then there exists a state φ in (the pure state space) such that J=ker πφ (Theorem 3). The significance of this is discussed below after the introduction of further notation and terminology.

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.

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