By Ivan G. Todorov, Lyudmila Turowska

This quantity includes the lawsuits of the convention on Operator thought and its functions held in Gothenburg, Sweden, April 26-29, 2011. The convention was once held in honour of Professor Victor Shulman at the get together of his sixty fifth birthday. The papers integrated within the quantity cover a huge number of themes, between them the idea of operator beliefs, linear preservers, C*-algebras, invariant subspaces, non-commutative harmonic research, and quantum teams, and reflect contemporary advancements in those parts. The e-book contains both original learn papers and top of the range survey articles, all of which were carefully refereed.

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In particular, for ???? ∈ ???? and ???? ∈ ???? we deﬁne Θ????,???? : ???? → ???? such that Θ????,???? (????) = ???? ⟨????, ????⟩???? , for all ???? ∈ ????. It is easy to check that Θ????,???? ∈ ℒ(????, ???? ) with Θ∗????,???? = Θ????,???? . We denote by ????(????, ???? ) the closed linear space of ℒ(????, ???? ) spanned by {Θ????,???? : ???? ∈ ????, ???? ∈ ???? }. If ???? = ???? then ????(????, ????) ≡ ????(????) is a closed ideal of the C∗ -algebra ℒ(????, ????) ≡ ℒ(????). In a dual way we call ???? a left Hilbert ????-module if it is complete with respect to the norm induced by an inner-product left ????-module [⋅, ⋅]???? .

These have since been pursued in various directions by several diﬀerent groups of authors; see [20] for an overview of some of the results to date. In this article, we are only concerned with one such variant, which we now brieﬂy describe. Given a Banach algebra ???? and a Banach ????-bimodule ???? and ???? ∈ ????, we denote by ad ???? the inner derivation ???? → ???? ⋅ ???? − ???? ⋅ ????. 1. A Banach algebra ???? is boundedly approximately contractible if for each Banach ????-bimodule ???? and each continuous derivation ???? : ???? → ????, there exists a net (???????? ) ⊂ ????, not necessarily bounded, such that the net (ad ???????? ) is norm bounded (as a subset of ℬ(????, ????)) and converges in the strong operator topology of ℬ(????, ????) to ????.

Alaminos, J. R. Villena it follows that ???? ∑ ????= ???? (????????1 ⊗ ????????2 ⊗ ????????3 ) ????1 ,????2 ,????3 =1 ∑ = ????????????1 ????????−1 ????2 ???? (????????1 ⊗ ????????2 ⊗ ????????3 ) ∩????????1 =????????????1 ????????−1 ????3 ∩????????2 =∅ ∑ + ???? (????????1 ⊗ ????????2 ⊗ ????????3 ). −1 ????????????1 ????????−1 ???? ∩????????1 ∕=∅ or ????????????1 ???????????? ∩????????2 ∕=∅ 2 3 Assume that ∩ ????????1 ∕= ∅ and let ????0 ∈ ????????????1 , ????0 ∈ ????????????2 with ????0 ????0−1 ∈ ????????1 . If ???? ∈ ????????????1 and ???? ∈ ????????????2 , then ????????????1 ????????−1 ????2 ????????−1 = (????????0−1 )(????????0−1 )(????0 ????0−1 ) ∈ ????????/2 ????????/2 ????????1 ⊂ ????????1 +???? .