The following is a proposition of Takesaki's Operator theory:
My question: How does he assume, considering the C*-subalgebra generated by k instead of $*$-Banach algebra B? Are we sure that the C*-algebra generated by k is a C*-subalgebra of B?
2026-03-27 04:38:11.1774586291
*-isomorphism of a C*-algebra into an involutive Banach algebra is norm increasing
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It appears to be a typo. It should say the C*-algebra generated by $h$ and the closed $*$-algebra generated by $k$, or equivalent. There is no assumptions that $k$ sits in a C*-algebra. For instance, later it is noted that $\|k\|\geq \|k\|_{\mathrm{sp}}$, but this would be equality if it were in a C*-algebra.