Operator algebras for multivariable dynamics
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Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally compact Hausdorff space with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> proper continuous self maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript i Baseline colon upper X right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma _i:X \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i less-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>i</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \le i \le n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . To this we associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A left-parenthesis upper X comma tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}(X,\tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the semicrossed product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C 0 left-parenthesis upper X right-parenthesis times Subscript tau Baseline double-struck upper F Subscript n Superscript plus"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo> × </mml:mo> <mml:mi> τ </mml:mi> </mml:msub> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {C}_0(X)\times _\tau \mathbb {F}_n^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We develop the necessary dilation theory for both models. In particular, we exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra. We introduce a new concept of conjugacy for multidimensional systems, called piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from the algebraic structure of either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A left-parenthesis upper X comma sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal A( X , \sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C 0 left-parenthesis upper X right-parenthesis times Subscript sigma Baseline double-struck upper F Subscript n Superscript plus"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo> × </mml:mo> <mml:mi> σ </mml:mi> </mml:msub> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {C}_0(X)\times _\sigma \mathbb {F}_n^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Various classification results follow as a consequence. For example, if
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| Category | Codex | Gemma |
|---|---|---|
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| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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