Explanation for anti-automorphism

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https://books.google.si/books?id=NZHkBwAAQBAJ&pg=PA142&lpg=PA142&dq=complementary+bases+for+dual+space&source=bl&ots=fWIjy8AHzt&sig=t9hUYoov-mcPEsSMWGn71YLrABg&hl=ru&sa=X&ved=2ahUKEwjK9uCa9KjbAhVLPFAKHfkJANIQ6AEwAXoECAEQMQ#v=onepage&q=complementary%20bases%20for%20dual%20space&f=false

In this book on page 148, there is anti-automorphism defined, and then it says we can turn left vector space into right vector space using and anti-automorphism.

My question is:

$\alpha \rightarrow \alpha^A$ that is defined there as an inverse of an automorphism. For what there stands the power of A and could someone please explain this notation better?