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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Restriction of a *-homomorphism

Let $A$ be C*-algebra then we know that $M_n(A)$ is also a C*-algebra. Let $\rho:M_n(A)\rightarrow B(K)$ be a *-representation of $M_n(A)$ on some Hilbert space $K$. Then there exists a ...
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Positivity in matrix algebra

Let $A$ be a unital C*-algebra. Then we know that $M_n(A)$ is also a C*-algebra. Let $x=[x_{ij}]\in M_n(A)$. I want to prove that if for every state $\phi$ on $A$ and for every ...
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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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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 ...
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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 ...
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33 views

example of positive operators a,b, $a\le b$ but $b^2-a^2$ is not positive [on hold]

Give an example of a C*-algebra $\mathscr{A}$ and positive elements a,b in $\mathscr{A}$ such that $a\le b$ but $b^2-a^2\notin \mathscr{A}_+$, i.e. $b^2-a^2$ is not positive element in $\mathscr{A} $
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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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21 views

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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the c*-algebra generated by a closed ideal and a c*-subalgebra

If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ . Show that the c*-algebra generated by ...
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21 views

the c*-algebra generated by the Volterra operator

Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$. Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where ...
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A condition on quotient norms on quotient Banach algebras

Let $A$ be a non-unital Banach algebra, and let $A^+$ be the unitization of $A$ consisting of elements of the form $(a,z)$ where $a\in A$ and $z\in\mathbb{C}$ with multiplication ...
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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22 views

Closed ideals in $\mathbb B(H)$

Let $\mathbb{H}$ be a non-separable Hilbert space. If $\alpha$ is an countably many infinite cardinal number, let $I_{\alpha}=\{A\in \mathbb{B(H)}\:dim~ cl(ran A)\le \alpha\}.$ Show that ...
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30 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
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support of an operator on a Hilbert space [closed]

let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. let $A=\int\lambda \, ...
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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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24 views

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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liminal and postliminal $C^*$-algebras

A $‎‎C^*$‎‎‎-algebra ‎$‎‎A$ ‎is said to be ‎postliminal (liminal) ‎‎ if for every non-zero irreducible representation‎$‎‎(H,‎\varphi)$ of ‎$‎‎A$ ‎we have‎‎‎ ‎‎$‎‎K(H)‎\subseteq ‎\varphi‎(A)$ ...
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Polar decomposition for completely bounded linear maps

Let $M$ be a W*-algebra and $f$ be a norm one and normal functional on $M$. Polar decomposition says that, there is a unique positive linear functional, denoted by $|f|$, satisfying in: ...
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27 views

closed convex hull of projection

$1$:I know that if ‎$‎‎F$ is a ‎locally convex ‎compact ‎space ‎then ‎‎$‎‎‎\overline{co}(‎Ext (F))=F$‎ ($Ext$: means extreme point) $2$:I ‎know ‎that ‎if ‎‎$‎‎M$ ‎is a ‎Von ‎Neumann ‎algebra ‎then ...
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30 views

weakly convergent sequence of operator on $B(H)$

Let $H$ be a Hilbert space. Assume that $\{u_n\} \subseteq B(H)$ is W.O.T convergent.(‎ ‎$‎u_n‎\rightarrow u‎$‎‎ ‎in ‎W.O.T ‎topology ‎iff ‎‎$‎‎‎‎<‎u_n(x),y>\rightarrow‎ ‎‎‎<‎u_n(x),y> ‎ ...
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44 views

which von Neumann algebras have many sufficiently normal irreducible representations?

Let $M$ be a von Neumann algebra. We say that $M$ has many sufficiently normal irreducible representations, namely $\{\pi_i\}$, if $||a||=\sup ||\pi_i(s)||$. For example the second dual of a ...
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1answer
58 views

If an operator have only Real eigenvalues + symmetric then it's self-adjoint?

I know that if an operator is self-adjoint then has Real eigenvalues but I'm not sure about the converse i.e. if it has only Real eigenvalues and is symmetric then the operator is selfadjoint. Is that ...
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In a proof of the theorem about the abstract index group of a Banach algebra

The following is a proposition in the Banach Algebra Techniques in Operator Theory by Douglas: I don't quite understand the very last step of the proof. Let $\pi:G\to G/G_0$ be the cannonical ...
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*-isomorphism and spectrum

‎‎‎$A$ is a ‎‎‎‎$‎‎C^∗$-algebra and $P(A)$ is a set of projection of it. Assume that $A$ ‎admits a‎ ‎strictly ‎positive ‎element ‎‎‎‎‎$a$ ‎such ‎that ‎‎‎‎‎$‎‎‎‎σ(a)‎-\{‎0\}$ ‎is ‎discrete‎. I want to ...
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strictly positive elments $a$ when $‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete

If ‎$‎‎A$ is a ‎‎$‎‎C^*$-algebra ‎and it ‎admits a‎ ‎strictly ‎positive ‎element ‎‎$‎‎a$ ‎such ‎that ‎‎$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete‎ then‎ Q1:‎$‎‎A$ admits ‎an ‎approximate ‎unit ...
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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 $\tilde{A}=A+\mathbb{C}I\subset B(L^p(X,\mu))$, and we ...
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3answers
38 views

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 ...
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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 ...
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GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, ...
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Representing an operator in different bases

Say I have a random operator $\hat {A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represented in the basis $\mathbf {e} = \left \{ \hat {e}_1, \hat {e}_2\right \}$ How should ...
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Coherent states - operator algebra problem with physics motivation

Motivation: I have a mathematical problem motivated by quantum field theory in physics. It should be quite easy to prove, but for some reason I can't do it. Intro: Let there be operators $\hat{a_i}$ ...
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Change of basis for the matrix representation of an operator $L$

Suppose I have an operator, $L$, represented, in matrix form, in the orthonormal basis $\mathbf{e} = \left \{ \hat{e_1}, \hat{e_2} \right \}$, as $$L = \begin{pmatrix} 3 & \frac{3}{2} \\ ...
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Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
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Von Neumann algebras with uncountable sets of incompatible projections

Which von Neumann algebras acting on separable Hilbert space $H$ have uncountable antichains of projections? ("Antichain" meaning a set of projections any pair of which has no shared nonzero ...
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31 views

Is a unital $*$-homomorphism preserving a state is one-to-one?

Let $M$ be a von Neumann algebra and let $\varphi$ be a faithful normal state on $M$. Suppose that $T \colon M \to M$ is a normal unital $*$-homomorphism preserving $\varphi$, i.e. $\varphi \circ T ...
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union of group von neumann

If we have an increasing chain of group von Neumann algebras such as $L(G_1)\subseteq L(G_2)\subseteq\ldots$ what can we say about the weak closure of their union? Is it a group von Neumann algebra? ...
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If$ p \in B(H)$ is a projection, then $r \in A'$ if and only if the closed vector subspace $p(H)$ of $H$ is invariant for $A$.

In the proof of the theorem $4.1.12$ on the page $120$ in Murphy, he uses a central remark that: If $p$ is a projection in $B(H)$ , then $p$ belong to $A'$ if and only if the closed vector subspace ...
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Prove: the density operator of a pure state has exactly one non-zero eigenvalue equal to unity

What is the proper way of proving : the density operator $\hat{\rho}$ of a pure state has exactly one non-zero eigenvalue and it is unity, i.e, the density matrix takes the form (after ...
2
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1answer
32 views

nilradical of a finite-dimensional algebra

I'm trying to understand/solve an exercise problem in "Algebras of linear transformations" by Farenick. The following is the problem: [exercises 4.6-5] Let $\mathfrak{A}$ be a finite-dimensional ...
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28 views

Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} ...
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31 views

Can anyone prove this equation? (Eq. with operators)

I am trying to understand the last equation from page 2 of this pdf http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap7.pdf but I am not being able to develop as here it says. Could ...
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1answer
34 views

direct sum of a collection of Von-Neuman algebras is still Von-Neuman

If {Aα} be a collection of some Von Neuman algebras then their direct sums is still Von Neuman ? I can prove that if Aα are unital then (⊕ Aα)= (⊕ Aα)" that is because of the fact that (⊕ Aα)'= ...
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2answers
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Prove $e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X}$ if $\hat{X}^{2}=I$.

If we have an operator $\hat{X}$ such that $\hat{X}^{2}=I$ (the identity), how do we prove that: $$e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X} \ ?$$