Isometric multipliers and isometric isomorphisms of $l_{1}(S)$
HTML articles powered by AMS MathViewer
- by Charles D. Lahr PDF
- Proc. Amer. Math. Soc. 58 (1976), 104-108 Request permission
Abstract:
Let $S$ be a commutative semigroup and $\Omega (S)$) the multiplier semigroup of $S$. It is shown that $T$ is an isometric multiplier of ${l_1}(S)$ if and only if there exists an invertible element $\sigma \in \Omega (S)$ and a complex number $\lambda$ of unit modulus such that $T(\alpha ) = \lambda \sum \nolimits _{x \in S} {\alpha (x){\delta _{\sigma (x)}}}$ for each $\alpha = \sum \nolimits _{x \in S} {\alpha (x){\delta _x} \in {l_1}} (S)$. Also, if ${S_1}$ and ${S_2}$ are commutative semigroups, and $L$ is an isometric isomorphism of ${l_1}({S_1})$ into ${l_1}({S_2})$, then it is proved that there exist a semicharacter $\chi ,|\chi (x)| = 1$ for all $x \in {S_1}$, and an isomorphism $i$ of ${S_1}$ onto ${S_2}$ such that $L(\alpha ) = \sum {\chi (x)\alpha (x){\delta _{i(x)}}}$ for each $\alpha = \sum \nolimits _{x \in {S_1}} {\alpha (x){\delta _x}} \in {l_1}({S_1})$.References
- Frank T. Birtel, Isomorphisms and isometric multipliers, Proc. Amer. Math. Soc. 13 (1962), 204–210. MR 176345, DOI 10.1090/S0002-9939-1962-0176345-6 M. M. Day, Normed linear spaces, Springer-Verlag, New York, 1962. MR 26 #2847.
- Edwin Hewitt and Herbert S. Zuckerman, The $l_1$-algebra of a commutative semigroup, Trans. Amer. Math. Soc. 83 (1956), 70–97. MR 81908, DOI 10.1090/S0002-9947-1956-0081908-4
- Charles Dwight Lahr, Multipliers for certain convolution measure algebras, Trans. Amer. Math. Soc. 185 (1973), 165–181. MR 333587, DOI 10.1090/S0002-9947-1973-0333587-6
- Ronald Larsen, The multiplier problem, Lecture Notes in Mathematics, Vol. 105, Springer-Verlag, Berlin-New York, 1969. MR 0435737
- J. G. Wendel, On isometric isomorphism of group algebras, Pacific J. Math. 1 (1951), 305–311. MR 49910
- J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 58 (1976), 104-108
- MSC: Primary 43A22
- DOI: https://doi.org/10.1090/S0002-9939-1976-0415209-7
- MathSciNet review: 0415209