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1

You only need to use that The sum and product of two such operators correspond to the sum and product of the corresponding functions. That said, for any operator $A$, the square of the operator $\eta_\mathscr I(A)$ equals to $\eta_\mathscr I^2(A)$ -- whatever it will mean --, but as a real (or complex) function, we have $\eta_\mathscr I^2=\eta_\mathscr ...


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1) In that page the author is not claiming that $\eta_{\mathscr{I}}(A)$ exists (yet), but is rather discussing what properties it should have before constructing it. 2) Note that $\eta_{\mathscr{I}}(A)$ is an operator, not a function. The idea of functional calculus is that the map $f\longmapsto f(A)$ should be a $*$-homomorphism, i.e. it should preserve ...


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In a $C^\ast$ algebra, this is easy. You have $\Vert a \Vert = \Vert a^\ast \Vert$ and (by definition of a $C^\ast$ algebra) you have $$ \Vert a^\ast \Vert^2 = \Vert (a^\ast)^\ast a^\ast \Vert = \Vert a \cdot a^\ast\Vert, $$ so that taking $b = \frac{a^\ast}{\Vert a\Vert}$ yields your claim, since the estimate $\Vert ab \Vert \leq \Vert a \Vert \Vert b ...


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Yes, your argument is the standard way of proving it.


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Because you are trying to prove (first) that $\phi$ is unbounded on $A^+$. You do this because you want to use that your map is positive, so it makes sense to work on the positive part of $A$. The summands are positive. Because $\|\phi(p)\|$ would be an upper bound for the natural numbers.


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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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