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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$C^*$-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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188 views

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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24 views

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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39 views

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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38 views

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 ...
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Self-adjoint and positive operator minimal polynomial on complex inner product spaces

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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Positive operator minimal polynomial [duplicate]

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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An strange trace class operator

Assume $H$ is a non separable Hilbert space. 1- Let $\{a_n\}$ be a sequence of bounded linear operators on $H$. Any operator $a_n$ induces the bounded linear functional $x\to tr(xa_n)$ on $L^1(H)$, ...
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Showing that there exists a functional on $l^\infty$ whose value lies between the infimum and supremum of a sequence-is my solution correct?

I'm trying to show that there exists a functional in $l^\infty$ whose value lies between the infimum and supremum of a sequence. That is, there exists a functional $\phi:l^\infty\longrightarrow ...
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C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...
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Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want ...
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Does category theory help in operator algebras?

I'm currently studying the basics of Banach and $C^*$-algebras. Almost all the proofs i've seen so far are very simple but some of them are extremely tricky (in my opinion). This tricky interplay ...
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Operator realizations of algebras

I want to realize the algebra $A_q(\tilde{S}^{n-1})$ as introduced in the acticle of Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf) as bounded operators on a Hilbert space $H$. Can ...
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Norm of a Positive Operator

For a positive operator $A\in B(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, I want to prove that $\|A\|=\sup\{\lambda: \lambda \in \sigma(A)\}$, where $\sigma(A)$ is the spectrum of $A$. ...
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1answer
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$\mathcal{A}+K$ is norm-closed where $\mathcal{A}$ is a $C^*$-algebra and $K$ is the compact operators.

Let $\mathcal{A}\subset B(H)$ be a unital $C^*$-algebra and let $K$ be the closed ideal of compact operators. I need to show that $\mathcal{A}+K$ is also a $C^*$-subalgebra of $B(H)$. I am stuck at ...
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Maximal ideal space and quotient space in abelian Banach algebra

I have a short question regarding operator algebras. Given an abelian Banach algebra $\mathcal{A}$. Assume that $\phi \in \big\{ \phi : \phi \text{ is a non-zero linear multiplicative functional} ...
3
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1answer
51 views

Gelfand-Naimark for $C^*$-categories

What is a reference for the following Theorem? If $A$ is a small $C^*$-category, then there is a faithful $C^*$-functor $A \to \mathsf{Hilb}$. $C^*$-categories with exactly one object are just ...
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Unit of a purely infinite, simple C*-algebra

Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$? Well, there are two ...
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A question about corners of $C^\ast$-algebras

Let $\mathcal{A}$ be a $C^\ast$-algebra, $p\in M_n(\mathcal{A})$ a projection, is there a $k\in\mathbb{N}$ such that $pM_n(\mathcal{A})p\cong M_k(\mathcal{A})$ ? Thanks a lot!
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States on a non-unital $C^*$-algebra

Let $\mathcal{A}$ be a unital $C^*$-subalgebra of $B(H)$. Then the definition $\phi(a):=\langle ah,h \rangle$ for a fixed $h\in H, \|h\|=1$ and for all $a\in\mathcal{A}$ defines a state on ...
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When can $*$-algebras be turned into $C^*$-algebas?

Let $A$ be a (not necessarily unital) complex $*$-algebra, i.e. an algebra over $\mathbb{C}$ together with an involution $*: A \to A$. There exists at most one norm on $A$ turning $A$ into a ...
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Matrices over the Cuntz algebra

Consider the Cuntz algebra $O_2$. Is it true that $M_2(O_2)$ is isomorphic to $O_2$? I was trying to show that is impossible but now I am not sure.
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star-modules and star-algebras

Is the following concept already known, perhaps with a different name? Let $R$ be a commutative ring and $* : R \to R$ be an involutive homomorphism (the most typical case is $R=\mathbb{C}$ and $* = ...
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Subordinate projections: Transitivity

Let $A$ be a $C^*$-algebra. If necessary, let us assume that $A$ is a von Neumann algebra. For projections $p,q \in A$ one writes $p \prec q$ if $p$ is Murray-von Neumann-equivalent to a subprojection ...
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a point on trace class operators

Assume $H$ is separable Hilbert space and fix an orthonormal basis $\{e_n\}_1^{\infty}$. Let us denote $p_n$ by the projection onto the subspace generated by $\{e_1\cdots,e_n\}$. Let $a$ be a ...
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Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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Characterization of normal operators on Hilbert space as function of a self-adjoint operator

My question : Suppose T is a normal operator on a Hilbert space H. Show that there exists a self-adjoint operator S on H such that T=f(S), where f is continuous function from spectrum of S into S. My ...
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Normal element is sum of four commuting positive elements

I am stuck with the following problem : Every normal element in a $C^*$ algebra can be written as a linear combination of four commuting positive elements. I had tried along the following lines. ...
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Introductory books for ‎ ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
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1answer
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How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space ...