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Register a new userApproximating the group algebra of the lamplighter by infinite matrix products
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In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $mathcal{N}(G)$. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) rtimes_T mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.
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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 $mu$ 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 $mu$, namely that the rank of a characteristic function of a clopen subset $U subseteq X$ must equal the measure of $U$.
From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $mathfrak{L}$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $mathfrak{L}/mathbb{F}L_{0, 0}^0$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $mathcal{W}_3$, $A_omega^delta$, $A_{omega}$ and the 3-$W_{infty}$ algebra can be embedded in $mathfrak{L}$.
We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.
We classify all Rota-Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.
Given a grading $Gamma: A=oplus_{gin G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra).
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