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

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

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.

scholarly journals Decomposable projections related to the Fourier and flip automorphisms

2010 ◽  
Vol 107 (2) ◽  
pp. 174 ◽  
Author(s):  
S. Walters
Keyword(s):  
Fourier Transform ◽  
Orthogonal Projection ◽  
Analogous Result ◽  
Irrational Rotation ◽  
Topological Invariants ◽  
Orthogonal Projections ◽  
C Algebra ◽  
Orthogonal Decompositions ◽  
The Fourier Transform

In this paper we classify Fourier invariant projections $g$ in the irrational rotation $C^*$-algebra that can be decomposed in the form 26741 g = f + \sigma(f) + \sigma^2(f) + \sigma^3(f) 26741 for some Fourier orthogonal projection $f$, where $\sigma$ is the Fourier transform automorphism. The analogous result is shown for the flip automorphism as well as the existence of flip-orthogonal projections. Both classifications are achieved by means of topological invariants (given by unbounded traces) and the canonical trace. We also show (in both the flip and Fourier cases) that invariant projections $h$ are subprojections of orthogonal decompositions $g$ for some projection $f$ such that $\tau(f) = \tau(h)$ (where $\tau$ is the canonical trace).

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