Positive element

of an algebra $ A $ with an involution $ {} ^ {*} $

An element $ x $ of $ A $ of the form $ x = y ^ {*} y $, where $ y \in A $. The set $ P( A) $ of positive elements in a Banach $ * $- algebra $ A $ contains the set $ Q( A) $ of squares of the Hermitian elements, which in turn contains the set $ P _ {0} ( A) ^ {+} $ of all Hermitian elements with positive spectrum (cf. Spectrum of an element), but in general it does not contain the set $ A ^ {+} $ of all Hermitian elements with non-negative spectrum. The condition $ P( A) \subset A ^ {+} $ defines the class of completely-symmetric (or Hermitian) Banach $ * $- algebras. For a $ * $- algebra to be completely symmetric it is necessary and sufficient that all Hermitian elements in it have real spectrum. The equality $ P( A) = A ^ {+} $ holds if and only if $ A $ is a $ C ^ {*} $- algebra. In that case $ P( A) $ is a reproducing cone (cf. Semi-ordered space) in the space of all Hermitian elements of the algebra $ A $.

References

[1]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[2]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[3]	D.N. Raikov, Dokl. Akad. Nauk. SSSR , 54 : 5 (1946) pp. 391–394
[4]	V. Pták, "On the spectral radius in Banach algebras with involution" Bull. London Math. Soc. , 2 (1970) pp. 327–334
[5]	T.W. Palmer, "Hermitian Banach -algebras" Bull. Amer. Math. Soc. , 78 (1972) pp. 522–524