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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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 $I_{\alpha}$...
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52 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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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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20 views

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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26 views

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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48 views

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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23 views

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 ‎‎$‎‎K(H)‎\subseteq ‎\varphi‎(A)$ ‎(‎‎$‎‎K(...
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35 views

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: $$|f|(1)=||f||...
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28 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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37 views

Sequences of operators in $B(H)$ which converge in the weak operator topology

Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has ‎$$‎‎‎‎\...
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46 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 C*-...
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65 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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34 views

*-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 $A^+=A+\mathbb{C}I\subset B(L^p(X,\mu))$, 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 ...
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105 views

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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32 views

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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44 views

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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21 views

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} \\ \frac{...
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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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2answers
34 views

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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38 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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47 views

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 diagonalizing):...
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37 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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38 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} \int_0^...
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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
43 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)'= ⊕(Aα'...
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61 views

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: $$e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X} \ ?$$
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positive elements of $C^*$-algebra

If $A$ is a abelian $C^*$-algebra and $a,b$ are elements in $A$ such that $0‎\leq ‎a‎\leq ‎1,0‎\leq ‎b‎\leq ‎1‎‎$ then $0‎\leq ‎a‎b\leq ‎1‎$. My problem is:" Is it true if $A$ is not abelian?"
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States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
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37 views

Why is a von Neumann algebra is closed with respect to weak * topology?

I was trying to prove that the identity map between a von Neumann algebra $(A,\mbox{ultra weak topology})$ with respect to ultra weak topology and the von Neumann algebra $A$ with respect to weak* ...
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49 views

Ultra weakly continuous linear functional on von neuman algebras

I started to learn von Neumann algebras and I wonder about the proof of the following. Can anyone outline it? this is theorem 4.2.10 in Murphy's book: A linear functional $\tau$ on von Neuman ...
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1answer
29 views

inner product and hermitian matrices

One of my professors mentioned that since a matrix A is positive semi definite and B is hermitian, hence the inner product $<A,B>$ is real. Is this an if and only if condition? So if we know ...
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78 views

If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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1answer
50 views

Unitary elements in C*-algebras

Let $\mathscr A$ be an unital $C^*$-algebra and let $u \in \mathscr A$ be an unitary element (I.e., $u^*u=uu^*=1$). Is it true that $$u=e^{ia}$$ for some hermitian element $a\in \mathscr A$? Im not ...
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24 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let $\{\zeta_n\}...
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1answer
55 views

Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
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40 views

A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
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20 views

Operators problem

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having $$L=\frac{1}{2}\left(\frac{\partial}{\...
3
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136 views

Measure theory , Functional calculus, Self Adoint

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
3
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1answer
81 views

Hilbert Space multiplication Operator, shift operator

I have this problem and am not sure how to even approach it.. Hilbert space $l^2(\mathbb{Z})$ with orthonormal basis$ $$(e_n)$ and Hamiltonian operator $He_n=i(e_{n+1}-e_{n-1})$ a)I need to use ...
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1answer
30 views

adjoint operator of the partial trace map

Could someone explain to me, what is the adjoint map of the partial trace map the (tensored with the identity map), or why does the following equality hold? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes ...
4
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76 views

$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as C*-...