Sylvester matrix rank functions on crossed products
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2019
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| _version_ | 1866910324554203136 |
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| author | Ara, Pere Claramunt, Joan |
| author_facet | Ara, Pere Claramunt, Joan |
| contents | In this paper we consider the algebraic crossed product $\mathcal A := C_K(X) \rtimes_T \mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field and $C_K(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on $\mathcal A$ by means of full ergodic $T$-invariant probability measures $μ$ on $X$. To do so, we present a general construction of an approximating sequence of $*$-subalgebras $\mathcal A_n$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $*$-algebra $\mathcal A$ into $\mathcal M_K$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on $\mathcal A$ by restricting the unique one defined on $\mathcal M_K$. This process gives a way to obtain a Sylvester matrix rank function on $\mathcal A$, unique with respect to a certain compatibility property concerning the measure $μ$, namely that the rank of a characteristic function of a clopen subset $U \subseteq X$ must equal the measure of $U$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1902_06476 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sylvester matrix rank functions on crossed products Ara, Pere Claramunt, Joan Rings and Algebras Dynamical Systems Operator Algebras Primary 16E50, Secondary 16S35, 37A05, 16D70 In this paper we consider the algebraic crossed product $\mathcal A := C_K(X) \rtimes_T \mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field and $C_K(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on $\mathcal A$ by means of full ergodic $T$-invariant probability measures $μ$ on $X$. To do so, we present a general construction of an approximating sequence of $*$-subalgebras $\mathcal A_n$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $*$-algebra $\mathcal A$ into $\mathcal M_K$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on $\mathcal A$ by restricting the unique one defined on $\mathcal M_K$. This process gives a way to obtain a Sylvester matrix rank function on $\mathcal A$, unique with respect to a certain compatibility property concerning the measure $μ$, namely that the rank of a characteristic function of a clopen subset $U \subseteq X$ must equal the measure of $U$. |
| title | Sylvester matrix rank functions on crossed products |
| topic | Rings and Algebras Dynamical Systems Operator Algebras Primary 16E50, Secondary 16S35, 37A05, 16D70 |
| url | https://arxiv.org/abs/1902.06476 |