Abstract
Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L∞(μ) such that: 1)¦F¦= 1 a.e. relative to μ; 2) F μ ε A1; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L∞(μ), L1(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H∞ to a very broad class of uniform algebras.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 80–89, 1976.
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Kislyakov, S.V. Uniform algebras as Banach spaces. J Math Sci 16, 1102–1108 (1981). https://doi.org/10.1007/BF02427719
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DOI: https://doi.org/10.1007/BF02427719