Boundaries of reduced C*C^{*}-algebras of discrete groups
Bibliographic record
Abstract
Abstract For a discrete group G , we consider the minimal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> C^{*} -subalgebra of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℓ</m:mi> <m:mi>∞</m:mi> </m:msup> </m:math> \ell^{\infty} ( G ) that arises as the image of a unital positive G -equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra C ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>F</m:mi> </m:msub> <m:mo></m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> \partial_{F}G ) of continuous functions on Furstenberg’s universal G -boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>F</m:mi> </m:msub> <m:mo></m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {\partial_{F}G} . This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G -action on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>F</m:mi> </m:msub> <m:mo></m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> {\partial_{F}G} is amenable, and use this fact to prove Ozawa’s conjecture that if G is exact, then there is an embedding of the reduced <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> C^{*} -algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mi>r</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{C}^{*}_{r}(G)} of G into a nuclear <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> C^{*} -algebra which is contained in the injective envelope of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mi>r</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{C}^{*}_{r}(G)} . The algebra C ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>F</m:mi> </m:msub> <m:mo></m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> \partial_{F}G ) arises as an injective envelope in the sense of Hamana, which implies rigidity results for certain G -equivariant maps. We prove a generalization of a rigidity result of Ozawa for G -equivariant maps between spaces of functions on the hyperbolic boundary of a hyperbolic group. Our result applies to hyperbolic groups, but also to groups that are not hyperbolic or even relatively hyperbolic, including certain mapping class groups. It is a longstanding open problem to determine which groups are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> C^{*} -simple, in the sense that the algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>C</m:mi> <m:mi>r</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo></m:mo> <m:mrow>
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".