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1]. For ???? ≥ 2, deﬁne ???? ∑ Δ???? = ????1 ⊗ ????1 + (???????? − ????????−1 ) ⊗(???????? − ????????−1 ). ????=2 We then have the following identities: ????(Δ???? ) = ???????? for all ????; ???? ⋅ Δ???? = Δ???? ⋅ ???? (1) for all ???? and all ???? ∈ ????. (2) The identity (1) can be shown by direct calculation, using property (i). The identity (2) is true for ???? = ???????? (???? arbitrary); this is another direct calculation using (i), which is most easily done by treating the cases ???? ≤ ???? and ???? > ???? separately. Hence, by linearity and continuity (using property (ii)), this identity holds for all ???? ∈ ????, as claimed.

It turns out that there is a continuous algebra homomorphism ???? : ???? → ℬ(ℋ) whose range is closed, so that ????(????) is a singly generated, biﬂat operator algebra. Moreover, given such an embedding ????, one can construct an embedding of ???? as a closed subalgebra of a ﬁnite, Type I von Neumann algebra. (The basic idea is as follows. If (???????? ) denotes the standard unit basis of ???? = ℓ1 , let ???????? = lin(????1 , . . 8]). This approach is somewhat indirect, and does not seem to give an explicit description of an embedding.

This approach is somewhat indirect, and does not seem to give an explicit description of an embedding. It is therefore desirable to have an explicit construction of an embedding of ???? as a closed subalgebra of a product of matrix algebras. This can be done with the following construction, which was shown to me by M. de la Salle [7], and is included here with his kind permission. The presentation here is paraphrased slightly from his original wording. It seems likely that similar embeddings were known previously, but I was unable to ﬁnd an explicit description in the literature.