Decomposition of Thompson group representations arising from Cuntz algebras

Abstract For every integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 , we consider a family <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>π</m:mi> <m:mi>w</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>w</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:msup> </m:mrow> </m:msub> </m:math> \{\pi_{w}\}_{w\in\{0,1,\ldots,n-1\}^{\mathbb{N}}} of irreducible representations of the Cuntz algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">O</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> \mathcal{O}_{n} . All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>I</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (I,f) with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> <m:mo lspace="0.500em">⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>mod</m:mi> <m:mo>⁢</m:mo> <m:mn> 1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> f(x)=nx\ (\mathrm{mod}\ 1) . We consider the embeddings <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>V</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>V</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⊂</m:mo> <m:msub> <m:mi>V</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:msub> <m:mi mathvariant="script">O</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> V=V_{2}\subset V_{n}\subset\mathcal{O}_{n} of the Thompson group 𝑉 in the Higman–Thompson group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>V</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> V_{n} obtained by Birget and then from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>V</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> V_{n} into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">O</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> \mathcal{O}_{n} which was obtained independently by Birget and Nekrashevych. The restriction of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>π</m:mi