Again, a provisional answer from a non-expert: likely someone who is a real Jedi Master in Mathematical Physics/Operator Algebras will chime in.

In classical QM, one starts from a Hilbert space of states $H$, and builds from there by looking at special types of operators acting on $H$ (unitary for simmetries, and hermitians for observables). So, in a sense, operator algebras are right there from the start, though in classical QM it looks and feels as if the basic entities are (quantum) states, and the secondary ones are processes (operators).

But I think it is fair to say that the movement has been toward inverting the order, in a sense beginning with the algebra of abstract operators and then modeling the set of states using the infamous Gelfand duality. What I just sketched is a supermarket chat on Algebraic Quantum Field Theory (you can find a condensate here).

You may ask why: I am not sure, but to me it seems that the movement toward processes as opposed to states makes sense

1. mathematically (for instance it connects with Non-Commutative Geometry of Connes, where one works directly on non-commutative algebras as if they were the algebras of functions over a ghost non commutative space). The algebras are good enough to capture the topology and geometry of the ghost space, and also lend itself to more abstract machinery
2. physically. There is a growing awareness that QM/QFT is about processes/interactions, rather than a world in which systems exist by themselves. See for instance Rovelli's Relational Interpretation, to just quote one option.

Again, a provisional answer from a non-expert: likely someone who is a real Jedi Master in Mathematical Physics/Operator Algebras will chime in.

In classical QM, one starts from a Hilbert space of states $H$, and builds from there by looking at special types of operators acting on $H$ (unitary for simmetries, and hermitians for observables). So, in a sense, operator algebras are right there from the start, though in classical QM it looks and feels as if the basic entities are (quantum) states, and the secondary ones are processes (operators).

But I think it is fair to say that the movement has been toward inverting the order, in a sense beginning with the algebra of abstract operators and then modeling the set of states using the infamous Gelfand duality. What I just sketched is a supermarket chat on Algebraic Quantum Field Theory (you can find a condensate here).

You may ask why: I am not sure, but to me it seems that the movement toward processes as opposed to states makes sense

1. mathematically (for instance it connects with Non-Commutative Geometry of Connes, where one works directly on non-commutative algebras as if they were the algebras of functions over a ghost non commutative space). The algebras are good enough to capture the topology and geometry of the ghost space, and also lend itself to more abstract machinery
2. physically. There is a growing awareness that QM/QFT is about processes/interactions, rather than a world in which systems exist by themselves. See for instance Rovelli's Relational Interpretation, to just quote one option.

ADDENDUM: so, are C* algebras the newest tool for QFT? The answer is: which QFT do you have in mind? For instance, in Quantum Gravity the answer is definitely no. There folks play with all sorts of goodies, running from higher category theory, to the already mentioned non commutative geometry, to ... pretty much anything under the sun, and even a tad more.