JournalsrmiVol. 29, No. 1pp. 329–356

On genuine infinite algebraic tensor products

  • Chi-Keung Ng

    Nankai University, Tianjin, China
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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 iIXi\bigotimes_{i\in I} X_i whose linear maps coincide with multilinear maps on an infinite family {Xi}iI\{X_i\}_{i\in I} of vector spaces. After establishing its existence, we give a direct sum decomposition of iIXi\bigotimes_{i\in I} X_i over a set ΩI;X\Omega_{I;X}, through which we obtain a more concrete description and some properties of iIXi\bigotimes_{i\in I} X_i. If {Ai}iI\{A_i\}_{i\in I} is a family of unital ^*-algebras, we define, through a subgroup ΩI;AutΩI;A\Omega^{\rm ut}_{I;A}\subseteq \Omega_{I;A}, an interesting subalgebra iIutAi\bigotimes_{i\in I}^{\rm ut} A_i. When all AiA_i are CC^*-algebras or group algebras, it is the linear span of the tensor products of unitary elements of AiA_i. Moreover, it is shown that iIutC\bigotimes_{i\in I}^{\rm ut} \mathbb{C} is the group algebra of ΩI;Cut\Omega^{\rm ut}_{I;\mathbb{C}}. In general, iIutAi\bigotimes_{i\in I}^{\rm ut} A_i can be identified with the algebraic crossed product of a cocycle twisted action of ΩI;Aut\Omega^{\rm ut}_{I;A}. On the other hand, if {Hi}iI\{H_i\}_{i\in I} is a family of inner product spaces, we define a Hilbert C(ΩI;Cut)C^*(\Omega^{\rm ut}_{I;\mathbb{C}})-module iImodHi\overline{\bigotimes}^{\rm mod}_{i\in I} H_i, which is the completion of a subspace iIunitHi\bigotimes_{i\in I}^{\rm unit} H_i of iIHi\bigotimes_{i\in I} H_i. If χΩI;Cut\chi_{\Omega^{\rm ut}_{I;\mathbb{C}}} is the canonical tracial state on C(ΩI;Cut)C^*(\Omega^{\rm ut}_{I;\mathbb{C}}), then iImodHiχΩI;CutC\overline{\bigotimes}^{\rm mod}_{i\in I} H_i\otimes_{\chi_{\Omega^{\rm ut}_{I;\mathbb{C}}}}\mathbb{C} coincides with the Hilbert space iIϕ1Hi\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 iIHi\prod {\otimes}_{i\in I} H_i as defined by J. von Neumann. We will show that the canonical representation of iIutL(Hi)\bigotimes_{i\in I}^{\rm ut} \mathcal{L}(H_i) on iIϕ1Hi\overline\bigotimes^{\phi_1}_{i\in I} H_i is injective (note that the canonical representation of iIutL(Hi)\bigotimes_{i\in I}^{\rm ut} \mathcal{L}(H_i) on iIHi\prod {\otimes}_{i\in I} H_i is not injective). We will also show that if {Ai}iI\{A_i\}_{i\in I} is a family of unital Hilbert algebras, then so is iIutAi\bigotimes_{i\in I}^{\rm ut} 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