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 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.

The Structure of the Irrational Rotation C * -Algebra

1993 ◽  
Vol 138 (3) ◽  
pp. 477 ◽  
Author(s):  
George A. Elliott ◽  
David E. Evans
Keyword(s):  
Irrational Rotation ◽  
C Algebra

ON GENERALIZED UNIVERSAL IRRATIONAL ROTATION ALGEBRAS AND ASSOCIATED STRONGLY IRREDUCIBLE OPERATORS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350059
Author(s):  
JUNSHENG FANG ◽  
CHUNLAN JIANG ◽  
HUAXIN LIN ◽  
FENG XU
Keyword(s):  
Irrational Number ◽  
Positive Function ◽  
Irrational Rotation ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Order Isomorphism ◽  
Strongly Irreducible ◽  
Zero Points

We introduce a class of generalized universal irrational rotation C*-algebras Aθ, γ = C*(x, w) which is characterized by the relations w*w = ww* = 1, x*x = γ(w), xx* = γ(e-2πiθw), and xw = e-2πiθwx, where θ is an irrational number and γ(z) ∈ C(𝕋) is a positive function. We characterize tracial linear functionals, simplicity, and K-groups of Aθ, γ in terms of zero points of γ(z). We show that if Aθ, γ is simple then Aθ, γ is an A𝕋-algebra of real rank zero. We classify Aθ, γ in terms of θ and zero points of γ(z). Let Aθ = C*(u, v) be the universal irrational rotation C*-algebra with vu = e2πiθuv. Then C*(u + v) ≅ Aθ,|1+z|2. As an application, we show that C*(u + v) is a proper simple C*-subalgebra of Aθ which has a unique trace, K1(C*(u + v)) ≅ ℤ, and there is an order isomorphism of K0(C*(u + v)) onto ℤ + ℤθ. Moreover, C*(u + v) is a unital simple A𝕋-algebra of real rank zero. We also show that u + v is strongly irreducible relative to the hyperfinite type II 1 factor.

scholarly journals The K-inductive structure of the noncommutative Fourier transform

2019 ◽  
Vol 124 (2) ◽  
pp. 305-319
Author(s):  
Samuel G. Walters
Keyword(s):  
Fixed Point ◽  
Fourier Transform ◽  
Partition Of Unity ◽  
Irrational Rotation ◽  
C Algebra ◽  
Sigma G

The noncommutative Fourier transform $\sigma (U)=V^{-1}$, $\sigma (V)=U$ of the irrational rotation C*-algebra $A_\theta $ (generated by canonical unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta } UV$) is shown to have the following K-inductive structure (for a concrete class of irrational parameters, containing dense $G_\delta $'s). There are approximately central matrix projections $e_1$, $e_2$, $f$ that are σ-invariant and which form a partition of unity in $K_0$ of the fixed-point orbifold $A_\theta ^\sigma $, where $f$ has the form $f = g+\sigma (g) +\sigma ^2(g)+\sigma ^3(g)$, and where $g$ is an approximately central matrix projection as well.

scholarly journals An action of the Klein four-group on the irrational rotation C*-algebra

1997 ◽  
Vol 56 (1) ◽  
pp. 135-148
Author(s):  
P.J. Stacey
Keyword(s):  
Fixed Point ◽  
Irrational Rotation ◽  
C Algebra ◽  
Fixed Point Algebra

Explicit automorphisms of the irrational rotation algebra are constructed which are associated with the two 2 × 2 diagonal integer matrices of determinant −1. The fixed point algebra of the product of these two automorphisms is shown to be isomorphic to the fixed point algebra of the flip.

On a class of operators in the hyperfinite $\mathrm{II}_1$ factor

2017 ◽  
Vol 120 (2) ◽  
pp. 249
Author(s):  
Zhangsheng Zhu ◽  
Junsheng Fang ◽  
Rui Shi
Keyword(s):  
Measurable Function ◽  
Von Neumann Algebra ◽  
Irrational Number ◽  
Irrational Rotation ◽  
Identity Operator ◽  
Lebesgue Measurable Function ◽  
Von Neumann ◽  
C Algebra ◽  
Subspace Problem ◽  
Invariant Subspace Problem

Let $R$ be the hyperfinite $\mathrm {II}_1$ factor and let $u$, $v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta } uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.

Sign in / Sign up

Export Citation Format

Share Document