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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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 ...
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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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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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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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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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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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32 views

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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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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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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47 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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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 ...
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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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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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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 ...
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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 ...
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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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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? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes ...
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$\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 ...
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C*-Algebra generated by self-adjoint Elements of commutative unital C*-Algebra

I want to proof the following theorem: Let A be a commutative unital C*-Algebra and $a_1, ..., a_n\in A_{sa}$. Then $C^*(1_A, a_1, ..., a_n)\subseteq A$ is *-isomorphic to $C(\Omega )$ for a compact ...
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An strange operator in B(H)

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis for $H$. Let $E_0$ be a countable subset of $E$ and $\{\delta_i\}_1^{\infty}$ be a bounded sequence of $(0,\infty)$. For ...
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What is the definition of $\ell^2(G)$ where $G$ is a group?

First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have ...
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Is a crossed product of a separable $C^\ast$-algebra by a finite group separable?

If $A$ is a separable $C^\ast$-algebra, $\alpha$ is an action on $A$ by a finite group $G$, then is the crossed product $A\rtimes_\alpha G$ separable?
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$C^*$-seminorm smaller than C*-norm?

I am currently reading Ruy Exel's Partial Dynamical Systems, Fell Bundles and Applications where he mentions that for every $C^*$-seminorm $p$ on a C* algebra $A$ one has $$p(a)\leq ||a||$$ for all $a ...
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Construct a projection satisfying a certain property

Let $\cal G$ be a group of finite order $n$. For every prime divisor $p$ of $n$, construct a projection $P\in \cal N(G)$ such that $\operatorname{tr}_{\cal N(G)}(P)=1/p$. Here $\cal N(G)$ denotes ...
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Differential Operators and Coefficients

First question on Math StackExchange here. I have been staring at this for a bit, but wasn't quite able to get the hang of it. Here it goes. We are given \begin{align} \frac{\partial}{\partial x} = ...
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semifinite projections

Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$. ( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
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Maximal Ideals in $L^1(\mathbb{R})$

Define $I_\xi = \{ f \in L^1(\mathbb{R}) : \hat{f}(\xi)=0 \}$. I have to prove that $I_\xi$ is a maximal ideal in $L^1(\mathbb{R})$. The following are my attempts at solution : Attempt 1 : Consider ...
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finite and properly infinite projections

Let $M$ be a von Neumann algebras and $p$ be a projection in $M$. $Q1:$I want to prove that there is a central projection $z \in M$ such that $pz$ is finite and $P(1-z)$ is properly infinite. $Q2:$ ...
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prove that all pure states in a commutative C* algebra are multiplicative linear functionals

I am trying to prove this , but can not see it clearly. it was given as some sort of converse of the fact that all multiplicative linear functionals are pure states
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idempotents in a subalgebra of $B(H)$.

Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is closed under weak operator topology. Suppose that there exist idempotents ...
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$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
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Functoriality in $K$-theory for $C^*$-algebras or Banach algebras

I'm trying to clear up some confusion I'm having over how one establishes functoriality in $K$-theory for $C^*$-algebras or Banach algebras. Let me stick to $K_0$. Given a *-homomorphism (or bounded ...
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Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...
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Relative weak-star topology on pure states

Let $A$ be a (unital) C*-algebra and consider $PS(A)$, the set of all pure states on $A$ with the relative weak-star topology. I would like to check (a weaker form of) Uryshon's lemma on $PS(A)$ in ...
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Inverse Tensor map

Let $$\phi: M_n (\mathbb{C})\otimes M_m (\mathbb{C})\to M_{nm} (\mathbb{C})$$ $$ \phi({A}\otimes{B}) = \begin{bmatrix} a_{11} {B} & \cdots & a_{1n}{B} \\ \vdots & \ddots & \vdots \\ ...
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Holomorphic Functional Calculus for the Square Root

I'm working on a problem set, so I'm not looking for a solution, but just maybe a pointer on where I'm going wrong. I want to use the holomorphic functional calculus to determine the square root of ...
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A formula for representation

Let $A$ be a C*-algebra. Do you confirm the following discussion? Let us consider a representation $\pi:A\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$. If we denote ...
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Approximation by elements in intersection of two Banach subalgebras

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there ...
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Minimal projections II

Let $M_1$ and $M_2$ be two W*-algebras. Let $A$ be a C*-algebra and $\pi_j:A\to M_j$ be two faithful representations with $M_j=\overline{\pi_j(A)}^{w^*}$. Assume that $$\textrm{The unit of}~ ...
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Minimal projections

Assume $M$ is a W*-algebra such that the set of minimal projections is not empty. Let $z(M)$ be the supremum of all minimal projections in $M$. It is well-known that $z(M)$ is a central projection. ...
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Projections in *-homomorphism

Let us consider the commutative C*-algebra $C_0(\Omega)$ and a representation $\pi:C_0(\Omega)\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(C_0(\Omega))$. It is ...
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Operator system of minimal dimension with one dimensional projections

Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in ...
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Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
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Norm on reduced crossed product - $C^*$ version v.s. $L^p$ version

Let $(G,A,\alpha)$ be a $C^*$-dynamical system where $G$ is a countable discrete group. When defining the reduced crossed product, one can proceed as follows: Let $\pi$ be a faithful representation ...
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projection in a factor von Neumann algebra.

We know that center of a factor von Neumann algebra $\mathcal{A} $ is trivial. Let $P_1$ be a projection in $\mathcal{A} $ such that $P_1\neq I,0$ . undoubtedly there exist another projection like ...
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$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras.

Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$. I read in Murphy 2.1.11 that this is true if $B$ is ...