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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Multipliers of $ A\sp{\ast} $-algebras


Authors: David L. Johnson and Charles D. Lahr
Journal: Proc. Amer. Math. Soc. 74 (1979), 315-317
MSC: Primary 46H05; Secondary 46K15, 46L05
DOI: https://doi.org/10.1090/S0002-9939-1979-0524308-3
MathSciNet review: 524308
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Abstract: Let A be an $ {A^\ast}$-algebra of the first kind with $ {C^\ast}$-algebra completion $ \mathfrak{A}$. It is known that if A is dual then $ {A^2}$ is dense in A and the Banach algebras $ {M_L}(A)$ and $ {M_L}(\mathfrak{A})$ of left multipliers of A and $ \mathfrak{A}$ are algebra isomorphic. In this note it is proved that $ {M_L}(A)$ and $ {M_L}(\mathfrak{A})$ are topologically algebra isomorphic when A is an arbitrary $ {A^\ast}$-algebra of the first kind such that $ {A^2}$ is dense in A. As a consequence, it follows that every left multiplier of a replete Hilbert algebra A is automatically continuous.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0524308-3
Keywords: Multiplier, $ {A^\ast}$-algebra, $ {C^\ast}$-algebra, Hilbert algebra, automatic continuity
Article copyright: © Copyright 1979 American Mathematical Society