Let $M\subset B(H)$ be a von Neumann algebra, with cyclic separating vector $\xi$.
Then the modular conjugation operator $S$ is defined to be the closure of the operator $$S_{0}:M\xi\to M\xi\text{ defined by }S_{0}(x\xi) = x^{*}\xi$$
Then the modular operator is defined by $\Delta := S^{*}S$.
Now my question is this:
For a real number $t$, how do we make sense of the expression $\Delta^{it}$?
If we were dealing with bounded operators on a Hilbert space, we could use the Analytic, Continuous, or Borel Functional Calculus. But since $S$ need not be bounded, it seems we cannot expect $\Delta$ to be.
So what tools are available to assign meaning to $\Delta^{it}$?
Positive unbounded operators satisfy the Spectral Theorem, so Borel functional calculus applies to them.