# Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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### Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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### Binormal operator - equivalent definitions?

I have seen two different definitions of a binormal operator A. A is unitarily equivalent to a block 2x2 matrix of commuting normal matrices. AA* commutes with A*A. I am hoping these definitions ...
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### How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that $\overline{\... 1answer 69 views ### Does$C^*(G) \cong C^*(H)$imply that$\mathbb{C}G \cong \mathbb{C}H$? I wonder whether the underlying complex group algebra of a group$C^*$-algebra is unique? I.e. if$G$and$H$are discrete groups such that$C^*(G) \cong C^*(H)$(or$C^*_r(G) \cong C^*_r(H)$) as$C^*$... 0answers 75 views ### Difference between Schmidt decomposition and singular value decomposition Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let$O$be an operator acting on the Hilbert space$\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$.$\...
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I cannot understand that a simple GCR C*algebra is *-isomorphic to the set of all compact operators on some Hilbert space. Please tell me how to show this.
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### Multiplicity free representation contain irreducible representation (for type I representation)?

While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central ...
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### Idempotents which are not Murray-von Neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Murray-von Neumann equivalent to $e^{*}$?
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### States in a $C^*$-algebra bounded?

A functional $\phi$ on a $C^*$-algebra $A$ with unit element, i.e. $\phi: A \rightarrow \mathbb{C}$ is called a state if $\phi(T^*T) \ge 0$ for all $T \in A$ and $\phi( \operatorname{id}) = 1.$ Now, I ...
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### the ideal structure of group $C^*$-algebras

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ? If G to be the group of integers $Z$ , then $C^*$($Z$)=C($T$). so because ideal structure of C(...
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### Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?
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### Minimum of the Schatten 1-norm

Given two operators or non-zero matrices $A$ and $B$, where $A\neq B$, tr$(A)=1$ and tr$(B)=1$ and tr$(A-B)=0$, what is a lower bound of the Schatten p-norm ($p=1$) $\|A-B\|_1$? Any helpful references?...
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### Is the bilinear map $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
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### Independence of choice of faithful representation in reduced $C^*$ crossed product

In the definition of the reduced $C^*$ crossed product associated with an action of a discrete group $G$ on a $C^*$-algebra $A$, one can begin with any faithful representation of $A$ on a Hilbert ...
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### Existence of uniform multiplicity projection in abelian Von Neumann algebras.

I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...
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### postliminal $C^*$-algebra

A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ‎...
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### Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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### Homotopy of bounded homomorphisms between Banach algebras

Let $A$ and $B$ be Banach algebras. Say that two bounded homomorphisms $\phi_0$ and $\phi_1$ from $A$ to $B$ are homotopic if there is a path $(\phi_t)_{t\in[0,1]}$ of bounded homomorphisms from $A$ ...
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### Second countable property for abelian von neumann algebras

I am looking at Murphy's book "$C^*$algebras and operator theory", in the section on abelian Von Neumann algebras (end of chapter 4). There, it is explained that any (unit containing) abelian Von ...
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### reduced density matrix for the given composite system

Given the composite system of two qubits $$|\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle)$$ with the density matrix of the composite ...
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### subnormal operator

I know that ‎$‎‎u\in B(H)$ ‎is a‎ ‎normal ‎operator if ‎‎$‎‎uu^*=u^*u$‎. I ‎know ‎that ‎if ‎‎$‎u‎$‎‎ ‎is ‎subnormal ‎‎‎‎then ‎‎‎ ‎‎$‎‎uu^*‎\neq ‎u^*u$ ‎(like unilateral shift operator). ‎‎ My ...
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### $p$-complete boundedness of homomorphisms between $L^p$ operator algebras

Let $A$ and $B$ be non-unital Banach subalgebras of $B(L^p(X,\mu))$ where $p\in[1,\infty)$. We unitize $A$ (and similarly for $B$) by considering $A^+=A+\mathbb{C}I\subset B(L^p(X,\mu))$, and we norm ...
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### Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
### Classification of representations of compact $C^*$ algebras for single operators.
I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed ...