# allowing `discontinuous functions' into a C* algebra

There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any useful comments welcomed.

What is wanted is the noncommutative analogue of taking an open subset of a compact Hausdorff topological space, and taking the continuous complex functions on that open set, with the topology of uniform convergence on compact subsets of the open set. (Noting that such functions are often unbounded.)

The NC setup - let $A$ be a unital separable C* algebra, with 2-sided ideal $J$. We do not want to take $A/J$, but instead take a countably many seminorm construction based on certain states on $A$. In the commutative case they would be states with measures supported away from the closed set where the elements of $J$ vanish. In the noncommutative set up, it seems likely that such states can be constructed from an approximate unit of $A/J$. An appropriate modification of $a\mapsto \phi(u_n a u_n)$ where $u_n\in A/J$ is the approximate unit and $\phi$ a state of $A/J$, with suitable cut offs and sums with $1/2^m$ weights might work. Then such states would give countably many seminorms to give an algebra larger than $A/J$ (and likely not separable).

The background: How to define meromorphic functions in noncommutative complex geometry? The starting ideal $J$ would correspond to where the poles (or divisor) would have to be. Of course the preceeding construction (if it worked) would generate really horrible functions in general, but the big problem is how to get unbounded functions at all - once they exist it should be possible to choose nicer ones.

G.R. Allan defined "GB*-algebras" as locally convex $\ast$-algebras $A$ such that

1. $1+xx^\ast$ is invertible for all $x\in A$
2. the set $\mathcal{B}$ of absolutely convex, closed, bounded submonoids of $(A,1,\cdot)$ has a largest element $B$.
3. Some form of completeness is satisfied.

In the commutative case this corresponds via a Gelfand-type theorem algebras of all continuous functions on a locally compact space $X$ with uniform convergence on compact subsets (and $B$ being the unit ball of $C_b(X)$).

Another commutative example you will recognize is the algebra of all measurable functions on a measure space $(X,\mathfrak{A},\mu)$ with convergence in measure. $B$ is the unit ball of $L^\infty(\mu)$ in this case.

The noncommutative case is a bit harder, because unbounded functions of course should correspond to unbounded operators which may take fiddeling with the domains of these operators, before you get the right notion of "representation" etc.

But there is literature on this subject.

• Very interesting... I shall have to look further into this, thanks for pointing it out. Jun 27, 2015 at 23:52