I am a bit worried about the equivalence of the definition of local C* algebra in Blackadar (the K-theory book, in terms of functional calculus using functions holomorphic in a neighbourhood of the spectrum) and other definitions involving subalgebras. In particular I am concerned about whether Blackadar's definition allows taking a square root of a positive operator which may not be invertible!
Any light on where to find a discussion of these matters would be much appreciated!