The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be closed under taking adjoints. Every operator algebra I have played with up until this point has been a C*-algebra, and so I am currently a bit uncomfortable with the object I am working with.
In seeking out references for some basic theory of non-self adjoint operator algebras, I have come across papers over particular results but nothing I would label as a collection of essentials. Thus my question:
- Is there a book or paper that goes over the basics of non-self adjoint operator algebras?
- If not, what are the fundamental results that I should seek out?
Thanks in advance.
I don't know enough to give you a very authoritative answer. But, as far as I can tell, there is no "general theory" of non-selfadjoint subalgebras of $B(H)$ the way that there is a theory for c$^*$-algebras or von Neumann algebras. There is a rather complete classification of nest algebras, and some generalizations.
The original source for non-selfadjoint subalgebras of $B(H)$ is Arveson's 1969 Acta paper. With this and other references, you will find that usually the study occurs under suitable hypotheses. Further work was done, to this day, by Davidson and his (many) students, Effros, Muhly, and many others.
If you look at (some of) these references you'll see that, as in single operator theory, the study is always done in some smallish subset where working properties can be deduced.