## A transitivity theorem for algebras of elementary operators

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- by Bojan Magajna
- Proc. Amer. Math. Soc.
**118**(1993), 119-127 - DOI: https://doi.org/10.1090/S0002-9939-1993-1158004-6
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## Abstract:

Let $\mathcal {A}$ be a ${C^{\ast }}$-algebra and $\mathcal {E}$ the algebra of all elementary operators on $\mathcal {A}$, and let $\vec a = ({a_1}, \ldots ,{a_n}),\;\vec b = ({b_1}, \ldots ,{b_n}) \in {\mathcal {A}^n}$. It is proved that $\vec b$ is contained in the closure of the set $\{ (E{a_1}, \ldots ,E{a_n}):E \in \mathcal {E}\}$ if and only if each complex linear combination $\sum \nolimits _{j = 1}^n {{\lambda _j}} {b_j}$ is contained in the closed two-sided ideal generated by $\sum \nolimits _{j = 1}^n {{\lambda _j}} {a_j}$. In particular, a bounded linear operator on $\mathcal {A}$ preserves all closed two-sided ideals if and only if it is in the strong closure of $\mathcal {E}$.## References

- Constantin Apostol and Lawrence Fialkow,
*Structural properties of elementary operators*, Canad. J. Math.**38**(1986), no.Β 6, 1485β1524. MR**873420**, DOI 10.4153/CJM-1986-072-6 - Lawrence A. Fialkow,
*The range inclusion problem for elementary operators*, Michigan Math. J.**34**(1987), no.Β 3, 451β459. MR**911817**, DOI 10.1307/mmj/1029003624 - C. K. Fong and A. R. Sourour,
*On the operator identity $\sum \,A_{k}XB_{k}\equiv 0$*, Canadian J. Math.**31**(1979), no.Β 4, 845β857. MR**540912**, DOI 10.4153/CJM-1979-080-x - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. I*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR**719020** - David R. Larson and Ahmed R. Sourour,
*Local derivations and local automorphisms of ${\scr B}(X)$*, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp.Β 187β194. MR**1077437**, DOI 10.1090/pspum/051.2/1077437 - Bojan Magajna,
*A system of operator equations*, Canad. Math. Bull.**30**(1987), no.Β 2, 200β209. MR**889539**, DOI 10.4153/CMB-1987-029-2 - Martin Mathieu,
*Elementary operators on prime $C^*$-algebras. I*, Math. Ann.**284**(1989), no.Β 2, 223β244. MR**1000108**, DOI 10.1007/BF01442873
β, - Martin Mathieu,
*Rings of quotients of ultraprime Banach algebras, with applications to elementary operators*, Conference on Automatic Continuity and Banach Algebras (Canberra, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 21, Austral. Nat. Univ., Canberra, 1989, pp.Β 297β317. MR**1022011** - Gerard J. Murphy,
*$C^*$-algebras and operator theory*, Academic Press, Inc., Boston, MA, 1990. MR**1074574** - Richard S. Pierce,
*Associative algebras*, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR**674652** - Masamichi Takesaki,
*Theory of operator algebras. I*, Springer-Verlag, New York-Heidelberg, 1979. MR**548728**

*Applications of ultraprime Banach algebras in the theory of elementary operators*, Thesis, TΓΌbingen, 1986.

## Bibliographic Information

- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**118**(1993), 119-127 - MSC: Primary 46L05; Secondary 47B48, 47D25
- DOI: https://doi.org/10.1090/S0002-9939-1993-1158004-6
- MathSciNet review: 1158004