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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Linear maps preserving quasi-commutativity

### Tom 184 / 2008

Studia Mathematica 184 (2008), 191-204 MSC: 15A04, 15A27, 47B49. DOI: 10.4064/sm184-2-7

#### Streszczenie

Let $X$ and $Y$ be Banach spaces and ${\cal B}(X)$ and ${\cal B}(Y)$ the algebras of all bounded linear operators on $X$ and $Y$, respectively. We say that $A,B \in {\cal B}(X)$ quasi-commute if there exists a nonzero scalar $\omega$ such that $AB = \omega BA$. We characterize bijective linear maps $\phi : {\cal B}(X) \to {\cal B}(Y)$ preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

#### Autorzy

• Heydar RadjaviDepartment of Pure Mathematics
University of Waterloo
200 University Avenue West
e-mail
• Peter ŠemrlDepartment of Mathematics
University of Ljubljana