Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spacesLet U be a unitary transformation from Hilbert space H to Hilbert space K.How do you call a *-homomorphism f from B(H) to B(K), defined by f(a)=UaU−1?I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.

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Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spacesLet U be a unitary transformation from Hilbert space H to Hilbert space K.How do you call a *-homomorphism f from B(H) to B(K), defined by f(a)=UaU−1?I&#039;m interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.

attefrimibeocx · Accepted Answer

This homomorphism is often denoted Ad(U) in the case where&amp;nbsp;H=K. The map&amp;nbsp;Ad:U(H)→Aut(B(H))&amp;nbsp;is a surjective homomorphism from the unitary group of H to the ∗-automorphism group of B(H) with kernel&amp;nbsp;TI. For example, this notation is used in Davidson&#039;s C*-algebras by example and in Raeburn and Williams&#039;s Morita equivalence and continuous trace C*-algebras. The same notation is often used for the corresponding automorphism of the algebra of compact operators on H, even though in that case it is not an inner automorphism unless H is finite dimensional.I don&#039;t know whether the same notation is in common use for the case where&amp;nbsp;H≠K.

elvis0217t2x · Accepted Answer

If H=K and U is a unitary, then f is called an inner automorphism of B(H), because it is given by conjugation with a unitary element of your algebra B(H) (these are the &quot;obviously existing automorphism&quot;).The above naming doesn&#039;t nicely generalise to your more general situation and I don&#039;t think there&#039;s a common name for isomorphisms given by conjugation with a unitary transformation.

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spac

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