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I already asked two questions about the Moyal $\star$-product here and here but I think I'll have a lot more similar questions, so I'm wondering if anyone can help me with finding some good resources.

I'm interested in the properties of the $\star$-product based operations and objects, such as (obviously) multiplication, division, $\star$-exponents, $\star$-logarithms and $\star$-elementary functions in general. I'm also interested in $\star$-based calculus and objects such as $\star$-determinants. In other words, I'm looking for results, theorems and techniques related to anything that one would ordinarily find in analysis, but with $\star$-products instead of usual multiplication.

I found several papers/notes by Akira Yoshioka and they were useful, but I need something that could help me do the dirty work and make some calculations easier. There's obviously a ton of papers and books about the $\star$-product in the context of deformation quantization. That's all great and helpful, but it doesn't really go beyond the abstract level into some of the calculations.

If you need me to clarify something, please do tell, I'm not exactly sure how exactly to say what I need, hopefully someone will understand.

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    $\begingroup$ The Moyal product is equivalent to a matrix product for matrices of infinite size, see arXiv:math-ph/0001039; so any resource on infinite-dimensional linear algebra should be effective. $\endgroup$
    – Carlo Beenakker
    Commented Aug 1, 2018 at 9:47
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    $\begingroup$ I understand this is self-promotional, but we wrote our basic text-book Concise Treatise on Quantum Mechanics in Phase Space for that very purpose, including the exercises... The idea is to maximize the number of exponential arguments and Bopp-shift away. Or else to use the Baker integral-shift representation and reduce integrals. $\endgroup$
    – Cosmas Zachos
    Commented Apr 27, 2019 at 13:33
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    $\begingroup$ @CosmasZachos Nothing wrong with shameless self-promotion if it's helpful! :) Thanks! $\endgroup$
    – lel
    Commented Apr 28, 2019 at 0:31
  • $\begingroup$ @CosmasZachos I have asked the following MO question based on the comments. Could you take a look? mathoverflow.net/questions/459365/… $\endgroup$
    – Diego Santos
    Commented Dec 5, 2023 at 5:53
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    $\begingroup$ Answered it. You misquoted, misunderstood, and misapplied the Bopp Shift! $\endgroup$
    – Cosmas Zachos
    Commented Dec 5, 2023 at 15:09

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