# Tagged Questions

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### An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
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### A question about tensor product

If $A$ is an algebra, $M_{n}(A)$ denotes the algebra of all $n\times n$ matrices with entries in $A$. The operations are defined just as for scalar matrices. If $A$ is a *-algebra, so is $M_{n}(A)$, ...
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### A question about range projection in von Neumann algebra.

I am reading a book about C*-algebra. And I meet with a problem. Recall the range projection of an operator $a\in B(H)$ is the projection on the closure of $\{a(\eta):\eta\in H\}$(Here, $H$ is a ...
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### What is the approximate units for an ideal?

In the Blackadar's book Operator algebras: theory of C*-algebras and von Neumann algebras, p103, there is "approximate units for J", here J is an ideal of C*-algebra A, but I do not know why an ideal ...
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### Cyclic vectors of an irreducible representation of a C*-algebra

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi)$ an irreducible representation of $\mathcal{A}$. I want to prove the statement: all $\xi \in H$ are cyclic or $\pi(\mathcal{A})=\{0\}$ and ...
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### Positive elements in $C^*$-algebras

I'm trying to prove the following, and I'm not sure if the proof is correct? If $A,B$ are $C^*$-algerbas, and $f$ is a $*$-homomorphism from $A$ onto $B$ then $f(A_+)=B_+$.Proof: let $a\in A_+$ then ...
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### Cube root in $C^{*}$-algebra.

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
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### Polar decomposition of invertible elements in a unital C$^{*}$-algebra.

If $A$ is a unital C$^{*}$-algebra and $a$ is invertible, then $a = u|a|$ for a unique unitary element $u$ of $A$. If $\| a \| = \| a^{-1} \| = 1$, what can you say about $|a|$? I ...
I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* ...
I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course). I think $K(H)$, the compact operators over a Hilbert space ...