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 ${ \bigotimes}_{ i\in I}^{ \rm ut} A_i$.
Categories: math.RA math.OA