Two conditional expectations in unbounded operator algebras (O * -algebras) are discussed. One is a vector conditional expectation defined by a linear map of an O * -algebra into the Hilbert space on which the O * -algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear map Ᏹ of an O * -algebra ᏹ onto a given O * -subalgebra ᏺ of ᏹ. Here the domain D(Ᏹ) of Ᏹ does not equal to ᏹ in general, and so such a conditional expectation is called unbounded.
Abstract. The notion of weights on (topological) * -algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C * -algebra.
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