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A $C^*$-algebra $A$ is isometric to an $L_1$-space (even as Banach space!) iff it is one dimensional. Assume $A$ is isometric to $L_1$ space and $\operatorname{dim}(A)>1$, then $A$ is weakly sequentially complete. By result of Sakai (proposition 2), this is possible only if $A$ is finite dimensional. By classification theorem for $C^*$ algebras we know ...


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In what follows, we assume that $ G $ is a locally compact Hausdorff group that does not have to be abelian. You can certainly define an involution $ ^{*} $ on $ {L^{1}}(G) $ by $$ \forall f \in {L^{1}}(G), ~ \forall x \in G: \quad {f^{*}}(x) \stackrel{\text{df}}{=} \overline{f(x^{-1})} \cdot \Delta(x^{-1}), $$ where $ \Delta $ denotes the modular ...


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This definition is from "Banach Algebras and Automatic Continuity- H. G. Dales". Definition 1.4.5 Let $A$ be an ideal in an algebra $B$. Then $B$ is left faithful over $A$ if $\{ b\in B : bA = 0\} = 0$, right faithful over $A$ if $\{b\in B : Ab = 0\} = 0$, and faithful over $A$ if $\{b \in B : bA=Ab=0\}=0$. An algebra $A$ is [left], [right] faithful if it ...


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The assumption is that the map $X\to X_{kj}$ is continuous for each $k,j$. This means that there exists a constant $c$ such that $\|X_{kj}\|\leq c\|X\|$ for all $X\in M_n(A)$. Consider, on $M_n(A)$, the norm $$ \|X\|_1=\sum_{k,j}\|X_{kj}\|. $$ Note that $(M_n(A),\|\cdot\|_1)$ is a Banach space. The above shows that $\|X\|_1\leq cn^2\|X\|$, so the identity ...


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Yes, it follows from the following general theorem (see Dixmier "$C^*$-algebras and representations", 2.10.2): Every irreducible representation $\rho$ of a $C^*$-subalgebra $C$ of a $C^*$-algebra $A$ can be continued to an irreducible representation $\pi$ of $A$ in a possibly larger Hilbert space. You can also apply the Lemma 2.10.1 from Dixmier directly. ...



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