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Let $H, K$ be Hilbert spaces. Let $A\subseteq B(H)$ be a nonselfadjoint closed subalgebra such that the identity map is in $A$. Let $C_A$ denote the $C^*$-algebra generated by $A$.

Q1: (this question may be obvious to the experts) Let $\pi:A\to B(K)$ be a continuous homomorphism, which is a *-homomorphism restricted to $\{x\in A: x^*\in A\}$. Is $\pi$ completely contractive?

Q2: If $\textrm{socle}(A)=\{0\}$, is it true that $\textrm{socle}(C_A)=\{0\}$?

Q3: Let $C_{env}^*(A)$ denote the $C^*$-envelope of $A$. If $\textrm{socle}(A)=\{0\}$, is it true that $\textrm{socle}(C_{env}^*(A))=\{0\}$?

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    $\begingroup$ Both are false. As to Q1, any nontrivial $A$ such that $A\cap A^*=\mathbb{C}1$ is a counterexample, by considering the conjugation by an invertible operator. It makes more sense to ask if every contractive homomorphism is completely contractive. Pisier calls such operator algebras to be "full" and presents examples and counterexamples in his book [Introduction to Operator Space Theory, Section 26]. As to Q2, the unital operator algebra $A$ generated by the unilateral shift is a counterexample. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 23, 2022 at 13:40
  • $\begingroup$ @NarutakaOZAWA Thank you for your kind reply and for the precise reference to the section in Pisier's book. It's not so long ago that I started to learn the operator space theory from the basics, starting from the books of Blecher & le Merdy, Effros & Ruan, Pisier, Paulsen, along with next-step books to read on my Jabref shelf. Thanks for bearing with me and your kind response. $\endgroup$
    – Onur Oktay
    Commented Feb 23, 2022 at 14:43
  • $\begingroup$ Professor @NarutakaOZAWA, as opposed to the $C^{\ast}$-algebra generated by the unilateral shift $u$, its $C^{\ast}$-envelope is $C(\mathbb{R}/\mathbb{Z})$ that has trivial socle. Is it generally true that the socle of the $C^{\ast}$-envelope of an operator algebra $A$ is $\{0\}$ if $socle(A)=\{0\}$? If not, the socle of the $C^{\ast}$-envelope is (perhaps?) a boundary ideal. Which property of $A$ suffices the socle of the $C^{\ast}$-envelope to be equal to the Shilov boundary ideal? Thank you for your time in advance. $\endgroup$
    – Onur Oktay
    Commented Feb 23, 2022 at 20:03
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    $\begingroup$ Triviality of the socle of $A$ does not imply the same of the C*-envelope. An example can be derived from Parrott's example. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 24, 2022 at 13:11
  • $\begingroup$ Professor Ozawa, thanks again for your kind and helpful reply. $\endgroup$
    – Onur Oktay
    Commented Feb 24, 2022 at 13:23

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