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 whose linear maps coincide with multilinear maps on an infinite family of vector spaces. After establishing its existence, we give a direct sum decomposition of over a set , through which we obtain a more concrete description and some properties of . If is a family of unital -algebras, we define, through a subgroup , an interesting subalgebra . When all are -algebras or group algebras, it is the linear span of the tensor products of unitary elements of . Moreover, it is shown that is the group algebra of . In general, can be identified with the algebraic crossed product of a cocycle twisted action of . On the other hand, if is a family of inner product spaces, we define a Hilbert -module , which is the completion of a subspace of . If is the canonical tracial state on , then coincides with the Hilbert space given by a very elementary algebraic construction and is a natural dilation of the infinite direct product as defined by J. von Neumann. We will show that the canonical representation of on is injective (note that the canonical representation of on is not injective). We will also show that if is a family of unital Hilbert algebras, then so is .
Cite this article
Chi-Keung Ng, On genuine infinite algebraic tensor products. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 329–356DOI 10.4171/RMI/722