Dual 𝐴*-algebras of the first kind

Johnson, David L.; Lahr, Charles D.

doi:10.1090/S0002-9939-1979-0524307-1

Dual $A^{\ast }$-algebras of the first kind
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by David L. Johnson and Charles D. Lahr

Proc. Amer. Math. Soc. 74 (1979), 311-314

DOI: https://doi.org/10.1090/S0002-9939-1979-0524307-1

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Abstract:

Let A be an ${A^\ast }$-algebra of the first kind. It is proved that A has property P2 of Máté if and only if ${A^2}$ is dense in A if and only if A possesses an (operator-bounded) approximate identity. Further, it is shown that an ${A^\ast }$-algebra of the first kind having property P2 is a dual algebra if and only if it is a modular annihilator algebra. As applications, these results are used to strengthen certain theorems about Hilbert algebras.

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