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 ⊗iεIXi whose linear maps coincide with multilinear maps on an infinite family {Xi}iεI of vector spaces. After establishing its existence, we give a direct sum decomposition of⊗iεIXi over a set ΩI;X, through which we obtain a more concrete description and some properties of ⊗iεIXi. If {A i}iεI is a family of unital C*-algebras, we define, through a subgroup ΩutI;A ⊗ ΩI;A, an interesting subalgebra ⊗utiεIAi. When all Ai are C*-algebras or group algebras, it is the linear span of the tensor products of unitary elements of Ai. Moreover, it is shown that ⊗utiεIC is the group algebra of ΩutI;C. In general, ⊗utiεIA i can be identified with the algebraic crossed product of a cocycle twisted action of ΩutI;A. On the other hand, if {Hi}iεI is a family of inner product spaces, we define a Hilbert C*(ΩutI;C)-module ⊗modiεI Hi, which is the completion of a subspace ⊗unitiεI Hi of ⊗iεIHi. If χΩut I;C is the canonical tracial state on C*(ΩutI;C), then ⊗ modiεI Hi ⊗χ Ωut I;C C coincides with the Hilbert space ⊗ Φ1iεIHi given by a very elementary algebraic construction and is a natural dilation of the infinite direct product ⊗⊗iεI Hi as defined by J. von Neumann. We will show that the canonical representation of ⊗utiεIL(Hi) on ⊗Φ1iεIHi is injective (note that the canonical representation of ⊗utiεIL(Hi) on ⊗⊗iεI Hi is not injective). We will also show that if {Ai}iεI is a family of unital Hilbert algebras, then so is ⊗utiεIA i. © European Mathematical Society.