On invertibility in non-selfadjoint operator algebras

Let $\mathcal {L}$ be a complete commutative subspace lattice on a Hilbert space. When $\mathcal {L}$ is purely atomic, we give a necessary and sufficient condition for $\sigma (T)= \sigma _{\mathcal {L}}(T)$ for every $T$ in $alg\mathcal {L}$, where $\sigma _{\mathcal {L}}(T)$ and $\sigma (T)$ denote the spectrum of $T$ in $alg\mathcal {L}$ and $B(H)$ respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in $alg\mathcal {L}$.