| Autor: |
Goldstein, Dmitry, Krupnik, Naum |
| Zdroj: |
Integral Equations & Operator Theory; May2000, Vol. 37 Issue 1, p20-31, 12p |
| Abstrakt: |
It is known [KRS] that for each finitely generated Banach algebra $$\mathcal{A}$$ there exists a number N such that for each n>N the matrix algebras $$M_n (\mathcal{A})$$ can be generated by three idempotents. In this paper we show that the same statement is true for direct sums $$\tilde {\cal A} = M_{n_1 } ({\cal A}) \oplus M_{n_2 } ({\cal A}) \oplus \ldots \oplus M_{n_p } ({\cal A})$$ and $$\tilde {\cal B} = M_{n_1 } ({\cal B}) \oplus M_{n_2 } ({\cal B}) \oplus \ldots \oplus M_{n_p } ({\cal B}) (n_j > 1)$$ , where $$\mathcal{B}$$ is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras $$M_n (\mathcal{A})$$ because the number N we obtain here improves known estimates (see for example [R]). We show that the algebra $${\tilde {\cal A}}$$ can be generated by two idempotents if and only if n j=2 for each j and $$\mathcal{A}$$ is singly generated. Also we give an example of a free singly generated algebra $$\mathcal{B}$$ for which $$M_2 (\mathcal{B})$$ can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra $$\mathcal{B}$$ . [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
| Externí odkaz: |
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