Differential properties of some dense subalgebras of C*-algebras

Abstract

The paper studies some classes of dense *-subalgebras B of C*-algebras A whose properties are close to the properties of the algebras of differentiable functions. In terms of a set of norms on B it defines -subalgebras of A and establishes that they are locally normal Q*-subalgebras. If x = x* ∈ B and f(t) is a function on Sp(x), some sufficient conditions are given for f(x) to belong to B. For p = 1, in particular, it is shown that -subalgebras are closed under C∞-calculus. If δ is a closed derivation of A, the algebras D(δp) are -subalgebras of A. In the case when δ is a generator of a one-parameter semigroup of automorphisms of A, it is proved that, in fact, D(δp) are -subalgebras. The paper also characterizes those Banach *-algebras which are isomorphic to subalgebras of C*-algebras.