Abstract
To an arbitrary involutive stereotype algebra A the continuous envelope operation assigns its nearest, in some sense, involutive stereotype algebra EnvC A so that homomorphisms to various C*-algebras separate the elements of Env C A but do not distinguish between the properties of A and those of Env C A.
If A is an involutive stereotype subalgebra in the algebra C(M) of continuous functions on a paracompact locally compact topological space M, then, for C(M) to be a continuous envelope of A, i.e., Env C A = C(M), it is necessary but not sufficient that A be dense in C(M). In this note we announce a necessary and sufficient condition for this: the involutive spectrum of A must coincide with M up to a weakening of the topology such that the system of compact subsets in M and the topology on each compact subset remains the same.
References
S. S. Akbarov, Fundam. Prikl. Matem., 14:1 (2008), 3–178; English transl.: J. Math. Sci., 162:4 (2009), 459–586; http://arxivorg/abs/0806.3205v3.
S. S. Akbarov, Diss. Math., 513:1 (2016), 1–188; http://arxivorg/abs/1110.2013v11.
S. S. Akbarov, Continuous and smooth envelopes of topological algebras, http://arxivorg/ abs/1303.2424v5.
S. S. Akbarov, J. Math. Sci., 113:2 (2003), 179–349.
J. Kuznetsova, J. Operator Theory, 69:2 (2013), 571–600; http://arxivorg/abs/ 0907.1409.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 50, No. 2, pp. 75–77, 2016
Original Russian Text Copyright © by S. S. Akbarov
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Akbarov, S.S. An algebra of continuous functions as a continuous envelope of its subalgebras. Funct Anal Its Appl 50, 143–145 (2016). https://doi.org/10.1007/s10688-016-0138-z
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DOI: https://doi.org/10.1007/s10688-016-0138-z