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 ${⨂}_{i\in I}{X}_i$ 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 ${⨂}_{i\in I}{X}_i$ over a set ${\Omega }_{I;X}$, through which we obtain a more concrete description and some properties of ${⨂}_{i\in I}{X}_i$. If $\{A_i{\}}_{i\in I}$ is a family of unital ${}^{\ast }$-algebras, we define, through a subgroup ${\Omega }_{I;A}^{\mathrm{ut}}\subseteq {\Omega }_{I;A}$, an interesting subalgebra ${⨂}_{i\in I}^{\mathrm{ut}}{A}_i$. When all $A_i$ are $C^{\ast }$-algebras or group algebras, it is the linear span of the tensor products of unitary elements of $A_i$. Moreover, it is shown that ${⨂}_{i\in I}^{\mathrm{ut}}\mathbb{C}$ is the group algebra of ${\Omega }_{I;\mathbb{C}}^{\mathrm{ut}}$. In general, ${⨂}_{i\in I}^{\mathrm{ut}}{A}_i$ can be identified with the algebraic crossed product of a cocycle twisted action of ${\Omega }_{I;A}^{\mathrm{ut}}$. On the other hand, if $\{H_i{\}}_{i\in I}$ is a family of inner product spaces, we define a Hilbert $C^{\ast }({\Omega }_{I;\mathbb{C}}^{\mathrm{ut}})$-module ${\overline{⨂}}_{i\in I}^{\mathrm{mod}}{H}_i$, which is the completion of a subspace ${⨂}_{i\in I}^{\mathrm{unit}}{H}_i$ of ${⨂}_{i\in I}{H}_i$. If ${\chi }_{{\Omega }_{I;\mathbb{C}}^{\mathrm{ut}}}$ is the canonical tracial state on $C^{\ast }({\Omega }_{I;\mathbb{C}}^{\mathrm{ut}})$, then ${\overline{⨂}}_{i\in I}^{\mathrm{mod}}{H}_i{\otimes }_{{\chi }_{{\Omega }_{I;\mathbb{C}}^{\mathrm{ut}}}}\mathbb{C}$ coincides with the Hilbert space ${\overline{⨂}}_{i\in I}^{{\varphi }_1}{H}_i$ 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 ${⨂}_{i\in I}^{\mathrm{ut}}\mathcal{L}(H_i)$ on ${\overline{⨂}}_{i\in I}^{{\varphi }_1}{H}_i$ is injective (note that the canonical representation of ${⨂}_{i\in I}^{\mathrm{ut}}\mathcal{L}(H_i)$ 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 ${⨂}_{i\in I}^{\mathrm{ut}}{A}_i$.