On genuine infinite algebraic tensor products (1112.3128v1)

Abstract: A genuine infinite tensor product of complex vector spaces is 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. 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 ${\bigotimes}{i\in I}^{\rm ut} A_i$. 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 $\Omega^{\rm ut}{I;A}$. 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 $\bar\bigotimes^{\rm mod}{i\in I} H_i$, 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 $\bar\bigotimes^{\rm mod}{i\in I} H_i\otimes{\chi_{\Omega^{\rm ut}{I;\mathbb{C}}}}\mathbb{C}$ 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 $\bar\bigotimes^{\phi_1}{i\in I} H_i$ is injective. We will also show that if ${A_i}{i\in I}$ is a family of unital Hilbert algebras, then so is ${\bigotimes}{i\in I}^{\rm ut} A_i$.