I am trying to learn operator algebra theory (I am tempted to start with Douglas' "Banach Algebra Techniques in Operator Theory").
One aspect that I am curious about is whether there are significant applications of that theory. (Especially in applied mathematics or any area that heavily relies on applied mathematics).
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The Jones polynomial demonstrates, for instance, a deep connection between von Neumann algebras and three-dimensional topology. This invariant in knot theory was discovered by Jones (the original paper of 1985 is in free access) as a by-product of his ground-breaking work on subfactors: see the notion of Jones index for subfactors and the most striking Jones index theorem (Inventiones 1983: this paper is extremely well written and actually understandable even if you don't know much about von Neumann algebras) in the previous link. Here is also a note by Jones himself explaining connections with braid theory, representation theory (Temperley-Lieb algebra), statistical mechanics, and quantum field theory. Another by-product of this work is Jones 1990 Fields medal.
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Two more examples:
Operator algebras can be applied to wavelets, which are used in signal processing.
See this summary
http://www.math.tamu.edu/~larson/macauarticle.pdf
The Kadison-Singer Conjecture/Problem is an Operator algebra problem that has many equivalent formulations and application in various areas of pure math as well as electrical engineering (time-frequency analysis).
This answer can be considered as a continuation of the comment by Michel. Operator algebra techniques have been used in the latest developments in additivity conjecture. For example, you can see this paper. Note that, in quantum information, convexity plays a major role.
Recently Junge et al applied operator space theory in the problem concerning Bell inequality, another fundamental problem in the quantum mechanics (see this). This can give you more ideas regarding the Banach space techniques, which are used in an active field of research. You may like see the works of Paulsen ans Stormer regarding the structure of positive maps.
A large part of functional analysis was developed to understand quantum mechanics. Operator theory is heavily used in the mathematical formulation of quantum mechanics. There is a reasonably long bibliography. You can see the (reasonably) elementary book Mathematical methods in quantum mechanics by Gerald Teschl and (more advanced) Quantum mechanics in Hilbert space by Prugovecki. You can also see Quantum Stochastic calculus by K R Parthasarathy, for many beautiful applications.
I believe, you will soon meet the the books 'Operator algebras and quantum statistical mechanics' (vol I and II) by Bratteli and Robinson (if you have not already seen it). There is an axiomatic formulation of quantum field theory as well (Local quantum theory - Haag). But I am not an expert in these areas. For the same reason, I have not mentioned the works of Alain Connes and Vaughan Jones. I hope some others will write about their works. Similarly, I do not know its applications in other applied areas (like mathematical finance), though I believe there are many.
Enjoy and discover!!