• DOI:10.1007/s11005-022-01514-5
• Corpus ID: 244714988

Truncated geometry on the circle

@article{Hekkelman2022TruncatedGO,
  title={Truncated geometry on the circle},
  author={E. Hekkelman},
  journal={Letters in Mathematical Physics},
  year={2022}
}

• E. Hekkelman
• Published 27 November 2021
• Art
• Letters in Mathematical Physics

<jats:p>In this letter, we prove that the pure state space on the <jats:inline-formula><jats:alternatives><jats:tex-math>$$n \times n$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>n</mml:mi>
<mml:mo>×</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> complex Toeplitz matrices converges in the Gromov–Hausdorff sense to the state space on <jats:inline-formula><jats:alternatives><jats:tex-math>$$C(S^1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mo>(</mml:mo>
<mml:msup>
<mml:mi>S</mml:mi>
<mml:mn>1</mml:mn>
</mml:msup>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> as <jats:italic>n</jats:italic> grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on <jats:inline-formula><jats:alternatives><jats:tex-math>$$S^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>S</mml:mi>
<mml:mn>1</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> with density functions <jats:inline-formula><jats:alternatives><jats:tex-math>$$c \prod _{j=1}^n (1-\cos (t-\theta _j))$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>c</mml:mi>
<mml:msubsup>
<mml:mo>∏</mml:mo>
<mml:mrow>
<mml:mi>j</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>-</mml:mo>
<mml:mo>cos</mml:mo>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>-</mml:mo>
<mml:msub>
<mml:mi>θ</mml:mi>
<mml:mi>j</mml:mi>
</mml:msub>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> is dense in the set of all positive Borel measures on <jats:inline-formula><jats:alternatives><jats:tex-math>$$S^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>S</mml:mi>
<mml:mn>1</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> in the weak<jats:inline-formula><jats:alternatives><jats:tex-math>$$^*$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow />
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> topology.</jats:p>