What is the best place to learn about the mathematical foundations of quantum mechanics? I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind it from a mathematical point of view: the definition of a state, mean value of operators, representation of states using the spectral theorem and so on. In special, I'd like to see some discussions on the relation between states (in Dirac notation) and wave functions. Most references I know (from the mathematical point of view) discuss the foundations only using wave functions and $L^{2}$ spaces and no connection to the actual general picture is provided.
EDIT: Maybe I should express myself a little better to avoid confusion. I know some books on quantum mechanics from a mathematical point of view e.g. Gustafson and Sigal's book. However, these references usually avoid the axioms or discuss them very briefly and the main object of study become wave functions living on $L^{2}$ spaces, where the Schrödinger equantion is solved for a bunch of different potentials and so on. I'd like to have some nice presentation on the axioms itself and how quantum mechanics arises from them in a more systematic way.
 A: The best introductory book that I know is L. D. Faddeev and O. Ya. Yakubovskii,
Lectures on quantum mechanics for mathematics students, translated by the AMS in 2009.
A: Von Neumann and Dirac are hard to beat, but if you want a more recent perspective you might take a look at Mathematical Foundations of Quantum Mechanics: An Advanced Short Course by Valter Moretti.

This is a review of the formulation of Quantum Mechanics, and quantum
theories in general, from a mathematically advanced viewpoint,
essentially based on the orthomodular lattice of elementary
propositions, discussing some fundamental ideas, mathematical tools
and theorems also related to the representation of physical
symmetries. The final step consists of an elementary introduction to
the C$^*$ algebraic formulation of quantum theories.

A: I think Strocchi's book An introduction to the Mathematical structure of Quantum Mechanics  may do the job.
In particular section 1.3 explains what you want (states, observables, expectations, $C^*$-algebras and representations). The discussion is carried out in more detail in the first two chapters.
A: Just like studying bioinformatics programming is a good way for a non-biologist to get a basic knowledge of the fundamentals of molecular biology, studying quantum computing is a good way to get a basic knowledge of the fundamentals of quantum mechanics. The book Mathematics of Quantum Computing: An Introduction by Wolfgang Sherer (Springer 2019) gives a good introduction to quantum mechanics for computer scientists. It is less advanced than some of the other recommendations and has a different focus, but it might be helpful (especially if your motivation for asking the question is a desire to understand quantum computing).
A: The question is a little unclear --- you want something axiomatic but not rigorous? Anyway,  if you don't care about rigor and you like Dirac deltas, I don't think there's any better place to start than Dirac's Principles of Quantum Mechanics. Then if you want to understand the connection to $L^2$ spaces and the spectral theorem, I'd recommend Mathematical Foundations of Quantum Mechanics by von Neumann. The notation is out of date but the exposition is excellent.
A: Another possibility might be Mackey's Mathematical Foundations of Quantum Mechanics (Mackey obviously being an eminent researcher in functional analysis and noncommutative geometry).
A positive review in Bulletin of the American Mathematical Society can be found here if you need more information on the content of the book.
