## The connectedness of the group of automorphisms of $L^ 1(0,1)$

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- by F. Ghahramani PDF
- Trans. Amer. Math. Soc.
**302**(1987), 647-659 Request permission

## Abstract:

For each of the radical Banach algebras ${L^1}(0,1)$ and ${L^1}(w)$ an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of ${L^1}(0,1)$ and ${L^1}(w)$ endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if $||| \cdot ||{|_p}$ denotes the norm of $B({L^p}(0,1)$, ${L^1}(0,1))$, $1 < p \leq \infty$, then the group of automorphisms of ${L^1}(0,1)$ topologized by $||| \cdot ||{|_p}$ is arcwise connected. It is shown that every automorphism $\theta$ of ${L^1}(0,1)$ is of the form $\theta = {e^{\lambda d}}{\operatorname {lim}}{e^{qn}}({\text {BSO}})$, where each ${q_n}$ is a quasinilpotent derivation. It is shown that the group of principal automorphisms of ${l^1}(w)$ under operator norm topology is arcwise connected, and every automorphism has the form ${e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$, where $\alpha \in {\mathbf {R}}$, $\lambda > 0$, and $D$ is a derivation, and where ${({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$ denotes the extension by continuity of ${e^{\lambda d}}{e^D}{e^{ - \lambda d}}$ from a dense subalgebra of ${l^1}(w)$ to ${l^1}(w)$.## References

- John M. Bachar, Philip C. Curtis Jr., H. Garth Dales, and Marc P. Thomas (eds.),
*Radical Banach algebras and automatic continuity*, Lecture Notes in Mathematics, vol. 975, Springer-Verlag, Berlin-New York, 1983. MR**697577** - W. F. Donoghue Jr.,
*The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation*, Pacific J. Math.**7**(1957), 1031–1035. MR**92124**, DOI 10.2140/pjm.1957.7.1031 - F. Ghahramani,
*Homomorphisms and derivations on weighted convolution algebras*, J. London Math. Soc. (2)**21**(1980), no. 1, 149–161. MR**576191**, DOI 10.1112/jlms/s2-21.1.149 - F. Ghahramani,
*Isomorphisms between radical weighted convolution algebras*, Proc. Edinburgh Math. Soc. (2)**26**(1983), no. 3, 343–351. MR**722565**, DOI 10.1017/S0013091500004417 - Sandy Grabiner,
*Weighted convolution algebras as analogues of Banach algebras of power series*, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 282–289. MR**697590** - Sandy Grabiner,
*Derivations and automorphisms of Banach algebras of power series*, Memoirs of the American Mathematical Society, No. 146, American Mathematical Society, Providence, R.I., 1974. MR**0415321** - Frederick P. Greenleaf,
*Norm decreasing homomorphisms of group algebras*, Pacific J. Math.**15**(1965), 1187–1219. MR**194911**, DOI 10.2140/pjm.1965.15.1187 - Edwin Hewitt and Kenneth A. Ross,
*Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations*, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0156915** - Einar Hille,
*Functional Analysis and Semi-Groups*, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, New York, 1948. MR**0025077** - Nicholas P. Jewell and Allan M. Sinclair,
*Epimorphisms and derivations on $L^1(0,1)$ are continuous*, Bull. London Math. Soc.**8**(1976), no. 2, 135–139. MR**402507**, DOI 10.1112/blms/8.2.135 - B. E. Johnson,
*An introduction to the theory of centralizers*, Proc. London Math. Soc. (3)**14**(1964), 299–320. MR**159233**, DOI 10.1112/plms/s3-14.2.299 - Herbert Kamowitz and Stephen Scheinberg,
*Derivations and automorphisms of $L^{1}\,(0,\,1)$*, Trans. Amer. Math. Soc.**135**(1969), 415–427. MR**233210**, DOI 10.1090/S0002-9947-1969-0233210-4 - Stephen Scheinberg,
*Automorphisms of commutative Banach algebras*, Problems in analysis (papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 319–323. MR**0352989**

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**302**(1987), 647-659 - MSC: Primary 46J35; Secondary 43A20
- DOI: https://doi.org/10.1090/S0002-9947-1987-0891639-6
- MathSciNet review: 891639