Some consequences of left invertibility

Hochwald, Scott H.; Morrel, Bernard B.

doi:10.1090/S0002-9939-1987-0883410-1

Some consequences of left invertibility
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by Scott H. Hochwald and Bernard B. Morrel

Proc. Amer. Math. Soc. 100 (1987), 109-110

DOI: https://doi.org/10.1090/S0002-9939-1987-0883410-1

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Abstract:

Let $A$ be a noncommutative Banach algebra with identity $e$. Let $L$ be a multiplicative semigroup of left-invertible elements of $A$ which properly contains the invertible elements of $A$. Then there does not exist a function $g:L \to A$ such that $g(ab) = g(b)g(a)$ and $g(a)a = e$ for all elements $a$ and $b$ of $L$. This paper contains an elementary proof of this result, and thereby answers a question posed by G. R. Allan.

References

G. R. Allan, Holomorphic left-inverse functions, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 7–14. MR 769494, DOI 10.1090/conm/032/769494