The Space of Traces of the Free Group and Free Products of Matrix Algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914712322572288 |
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| author | Orovitz, Joav Slutsky, Raz Vigdorovich, Itamar |
| author_facet | Orovitz, Joav Slutsky, Raz Vigdorovich, Itamar |
| contents | We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_10955 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Space of Traces of the Free Group and Free Products of Matrix Algebras Orovitz, Joav Slutsky, Raz Vigdorovich, Itamar Group Theory Operator Algebras 20F65, 46L05, 20E06, 46L09, 46L10, 81P45 We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces. |
| title | The Space of Traces of the Free Group and Free Products of Matrix Algebras |
| topic | Group Theory Operator Algebras 20F65, 46L05, 20E06, 46L09, 46L10, 81P45 |
| url | https://arxiv.org/abs/2308.10955 |