# Tagged Questions

54 views

### Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
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### C*-algebra representations

Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ ...
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### Where is commutativity of $b$ needed?

I have a question about the following proof: If $e^{ia}-e^{i\lambda}=(a-\lambda)be^{i\lambda}$ and $(a-\lambda)$ is not invertible then $(a-\lambda)x$ is not invertible for all $x$. Why "since $b$ ...
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### Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
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### Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
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### Module homomorphism

Let $A$ be a Banach algebra with norm $\|.\|_A$ and $X$ be a Banach space with norm $\|.\|_X$. If there exists a operation $.:A\times X\to X$ such that for any $a,b\in A$ and $x,y\in X$ we have ...
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### Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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### application of c*algebras

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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### Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
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### Showing the $C^*$ identity

I'm working through a proof in Dixmier's book on $C^*$-algebras and I'm stuck on part of a proof. I'm given a Banach algebra $\mathcal{A}$ which has norm $\lVert\cdot\rVert$ and a semi-norm ...
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### Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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### Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
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### A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
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### Diagonalizing operator over $L^2(\mathbb{T})$

I've been asked to diagonalise an operator on $L^2(\mathbb{T})$, given by $Tf(z) = f(z^{-1}$). I know that I'm expected to find a $U$ such that $TU = UM_f$, where $M_f$ is the multiplication operator, ...
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### Positive elements in a C*-algebra [closed]

Prove that if $a$ is an element in a $C^*$-algebra $A$, then $a$ is positive if and only if $f(a) \geq 0$ for every state $f$ on $A$.
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### Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $||xx^{\#}||=||x||^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
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### Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge （or monotonous converge） to $1$ in $A^+$, does $\{x_n\}$ must be the ...
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### Approximation of certain continuous functions by analytic functions

Let $f\in C(S^{1},M_{n}(\mathbb{C}))$ be a unitary. Does there exist an analytic unitary function $g$ from $S^{1}$ to $M_{n}(\mathbb{C})$ that approximates $f$?
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### $\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C}))$ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
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### Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
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### Does *-operator be automatically continous

In the C*-algebras, does the * -operator be automatically continous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continous. Am I right? ...
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### Unitary Equivalence of Two Irreducible $*$-Representations of a GCR $C^{*}$-Algebra that Have the Same Kernel.

In general, if two irreducible $*$-representations of a $C^{*}$-algebra $A$ have the same kernel, then we can say that they are approximately unitarily equivalent. When $A$ is GCR, how can we ...
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### C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
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### A question about positive elements in $C^*$ algebras

Let $A$ be a $C^*$-algebra If $a\in A$ is positive, is it true that for any $0<\alpha<\frac{1}{2}$ we have $$\left(a+\frac{1}{n}1\right)^{\frac{-1}{2}}a^{\frac{1}{2}-\alpha}$$is self adjoint?A ...
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### strictly positive elements in $C^*$-algebra

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to find the following:a)What are the strictly ...
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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 ...
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### Double centralizers in the Murphy book

I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake: ...
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### $(\lambda-a)^{-1}$ as limits of 'polynomials'

For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives $$\sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a)$$ for any unital $C^*$-subalgebra $\mathcal{B}$. ...
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### Spectrum of elements in $C^*$-subalgebras

Assume $\mathcal{A}$ is a $C^*$-algebra with unit $1$ and $\mathcal{B}\subset\mathcal{A}$ is a $C^*$-subalgebra (i.e. a closed $*$-subalgebra) such that $1\in\mathcal{B}$. It is said that under these ...
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### Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
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### When is a Banach Algebra stellar?

I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
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### Involutive and C* Banach Algebras.

I want to prove the next theorem: If $\pi: A \rightarrow B$ is a star homomorphism, meaning it's an algebra homomorphism which also satisfies: $\pi(x^*)=(\pi(x))^*$, where $A$ is an involutive Banach ...
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### Reflexive Banach algebras?

I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space $X$, we investigate its dual $X'$ ...
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### Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
Let $A$ be a normed unital algebra. Suppose that $C\subseteq A$ is a commutative subalgebra which is dense in $A$. I ask myself the following question: Under the above assumptions, is $A$ necessarily ...
Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...