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 $\bigotimes_{i\in I}^{\rm ut} A_i$.