On genuine infinite algebraic tensor products

Abstract

In this paper, we study genuine infinite tensor products of some algebraic structures. By a genuine infinite tensor product of vector spaces, we mean a vector space $\bigotimes\_{i\in I} X\_i$ whose linear maps coincide with multilinear maps on an infinite family ${X\_i}{i\in I}$ of vector spaces. After establishing its existence, we give a direct sum decomposition of $\bigotimes{i\in I} X\_i$ over a set $\Omega\_{I;X}$, through which we obtain a more concrete description and some properties of $\bigotimes\_{i\in I} X\_i$. If ${A\_i}{i\in I}$ is a family of unital $^\*$-algebras, we define, through a subgroup $\Omega^{\rm ut}{I;A}\subseteq \Omega\_{I;A}$, an interesting subalgebra $\bigotimes\_{i\in I}^{\rm ut} A\_i$. When all $A\_i$ are $C^$-algebras or group algebras, it is the linear span of the tensor products of unitary elements of $A\_i$. Moreover, it is shown that $\bigotimes\_{i\in I}^{\rm ut} \mathbb{C}$ is the group algebra of $\Omega^{\rm ut}{I;\mathbb{C}}$. In general, $\bigotimes{i\in I}^{\rm ut} A\_i$ can be identified with the algebraic crossed product of a cocycle twisted action of $\Omega^{\rm ut}{I;A}$. On the other hand, if ${H\_i}{i\in I}$ is a family of inner product spaces, we define a Hilbert $C^(\Omega^{\rm ut}{I;\mathbb{C}})$-module $\overline{\bigotimes}^{\rm mod}{i\in I} H\_i$, which is the completion of a subspace $\bigotimes\_{i\in I}^{\rm unit} H\_i$ of $\bigotimes\_{i\in I} H\_i$. If $\chi\_{\Omega^{\rm ut}{I;\mathbb{C}}}$ is the canonical tracial state on $C^\*(\Omega^{\rm ut}{I;\mathbb{C}})$, then $\overline{\bigotimes}^{\rm mod}{i\in I} H\_i\otimes{\chi\_{\Omega^{\rm ut}{I;\mathbb{C}}}}\mathbb{C}$ coincides with the Hilbert space $\overline{\bigotimes}^{\phi\_1}{i\in I} H\_i$ given by a very elementary algebraic construction and is a natural dilation of the infinite direct product $\prod {\otimes}{i\in I} H\_i$ as defined by J. von Neumann. We will show that the canonical representation of $\bigotimes{i\in I}^{\rm ut} \mathcal{L}(H\_i)$ on $\overline\bigotimes^{\phi\_1}{i\in I} H\_i$ is injective (note that the canonical representation of $\bigotimes{i\in I}^{\rm ut} \mathcal{L}(H\_i)$ on $\prod {\otimes}{i\in I} H\_i$ is not injective). We will also show that if ${A\_i}{i\in I}$ is a family of unital Hilbert algebras, then so is $\bigotimes\_{i\in I}^{\rm ut} A\_i$.