Rings of fractions of \(B(H)\) (Q1346674)

Let \(A\) be a ring with a unit 1. A subset \(S\) of \(A\) is said to be a denominator set if \(1\in S\), \(0\not\in S\). Moreover, if \(sa=0\) for some \(s\in S\), then \(at=0\) for some \(t\in S\). We also have if \(s\in S\) and \(a\in A\), then there is \(t\in S\) and \(b\in A\) such that \(sb= at\). If \(A\) is a ring and \(S\) is a denominator set then the following equivalence relation is defined on \(A\times S\). We write \((a,s)\equiv (b,t)\) if there is \(c,d\in A\) such that \(ac= bd\) and \(sc= td\in S\). The quotient obtained is denoted by \(A[S^{-1} ]\) and is called the ring of fractions of \(A\) with respect to \(S\). Addition and multiplication are defined appropriately. It is shown that if a subset \(S\) of bounded operators is a denominator set, then \(S\subset F\), where \(F\) is the set of all Fredholm operators. If \(S\) is a denominator set of \(B(H)\) there are two cases either \(S\) consists of invertible elements in which case \(B(H) [S^{-1} ]\cong B(H)\) otherwise the ring of fractions is isomorphic to \(B(H)/ J\) where \(J\) is the ideal of finite rank operators.