Algebraic Methods in Functional Analysis: The Victor Shulman by Ivan G. Todorov, Lyudmila Turowska

follow site By Ivan G. Todorov, Lyudmila Turowska

This quantity contains the lawsuits of the convention on Operator conception and its functions held in Gothenburg, Sweden, April 26-29, 2011. The convention used to be held in honour of Professor Victor Shulman at the social gathering of his sixty fifth birthday. The papers integrated within the quantity hide a wide number of subject matters, between them the speculation of operator beliefs, linear preservers, C*-algebras, invariant subspaces, non-commutative harmonic research, and quantum teams, and mirror fresh advancements in those parts. The publication includes either unique examine papers and prime quality survey articles, all of which have been conscientiously refereed. ​

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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.

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