# On genuine infinite algebraic tensor products

### Chi-Keung Ng

Nankai University, Tianjin, China

## 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 $⨂_{i∈I}X_{i}$ whose linear maps coincide with multilinear maps on an infinite family ${X_{i}}_{i∈I}$ of vector spaces. After establishing its existence, we give a direct sum decomposition of $⨂_{i∈I}X_{i}$ over a set $Ω_{I;X}$, through which we obtain a more concrete description and some properties of $⨂_{i∈I}X_{i}$. If ${A_{i}}_{i∈I}$ is a family of unital $_{∗}$-algebras, we define, through a subgroup $Ω_{I;A}⊆Ω_{I;A}$, an interesting subalgebra $⨂_{i∈I}A_{i}$. 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 $⨂_{i∈I}C$ is the group algebra of $Ω_{I;C}$. In general, $⨂_{i∈I}A_{i}$ can be identified with the algebraic crossed product of a cocycle twisted action of $Ω_{I;A}$. On the other hand, if ${H_{i}}_{i∈I}$ is a family of inner product spaces, we define a Hilbert $C_{∗}(Ω_{I;C})$-module $⨂ _{i∈I}H_{i}$, which is the completion of a subspace $⨂_{i∈I}H_{i}$ of $⨂_{i∈I}H_{i}$. If $χ_{Ω_{I;C}}$ is the canonical tracial state on $C_{∗}(Ω_{I;C})$, then $⨂ _{i∈I}H_{i}⊗_{χ_{Ω}}C$ coincides with the Hilbert space $⨂ _{i∈I}H_{i}$ given by a very elementary algebraic construction and is a natural dilation of the infinite direct product $∏⊗_{i∈I}H_{i}$ as defined by J. von Neumann. We will show that the canonical representation of $⨂_{i∈I}L(H_{i})$ on $⨂ _{i∈I}H_{i}$ is injective (note that the canonical representation of $⨂_{i∈I}L(H_{i})$ on $∏⊗_{i∈I}H_{i}$ is not injective). We will also show that if ${A_{i}}_{i∈I}$ is a family of unital Hilbert algebras, then so is $⨂_{i∈I}A_{i}$.

## Cite this article

Chi-Keung Ng, On genuine infinite algebraic tensor products. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 329–356

DOI 10.4171/RMI/722