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Questions tagged [operator-algebras]

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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1answer
22 views

Inclusion of multiplier algebras

If $B$ is a $C^*$-subalgebra of $A$, i.e there exists an inclusion map $\phi\colon B \rightarrow A$, can we conclude that there exists a $*$-homomorphism beween the multiplier algebras $\bar{\phi} \...
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0answers
27 views

Tipe III Von Neumann algebras and spectra of the modular operators.

I´m studing the paper of Fredenhagen. There he said that he would to prove that the algebra of local observables under certain conditions is not of tipe III, by showing that all modular operators ...
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1answer
30 views

Essential ideals in sums of matrix algebras

Given the $C^*$-algebra $A=\prod_{n}M_n(\Bbb C)$, how many essential ideals has $A$? Is there a unique one ?
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2answers
43 views

Example of norm separable c-star algebras [on hold]

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.
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1answer
28 views

Norm closure of diagonalizable operator on Hilbert space

Problem: Prove the norm closure of diagonalizable operator in $\mathcal{B(H)}$ (bounded operators on Hilbert space which is not necessarily finite dimensional) is the set of normal operators. A ...
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1answer
26 views

examples of non-unital commutative $C^*$-algebras

I know that all the non-unital commutative $C^*$ algebras are isomorphic to $C_0(\Omega)$,where $\Omega$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I ...
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1answer
31 views

non isomorphic finite dimensional $C*$ algebras

How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
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1answer
64 views

Rigorous Computation of the Spectrum of a Certain C$^{*}$-algebra

Let $$ A=\left\{f\in C([0,1],M_{2}(\mathbb{C})):f(1)=\begin{pmatrix}\xi & 0 \\ 0 & \lambda\end{pmatrix} \text{ for some }\xi,\lambda \text{ in } \mathbb{C}\right\}. $$ In his Operator ...
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1answer
25 views

spectrum of $C^*$ algebras

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty $ direct sum with $c_0$ direct sum?
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1answer
26 views

$T$ is a matrix . How to find the desired $M$ such that for every $\alpha \in R^3$ $\|T(\alpha) \| \le M\| \alpha \|$ [closed]

Let $R^3$ denote the Euclidean space ,$\|.\|$ denote the usual norm. For a matrix $$T=\begin{bmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \\ \...
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1answer
31 views

arbitrary $n$-dimensional matrix algebras in II$_1$ factor

I'm struggling to show that in a type II$_1$ and any $n$ $\exists$ a subfactor $M$ such that $M \cong M_n$ . I suppose it should follow from the isomorphism between the equivalence classes of ...
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1answer
26 views

approximate identity element

Let $I\subset \prod_n B(H_{m_n})$ be a separable $C^*$ algebra,where $\prod B(H_{m_n})$ denotes the $\ell ^{\infty}$direct sum of $B(H_n)$ and $dim(H_{m_n})<\infty$. We suppose that there exits a ...
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1answer
15 views

equivalent projections in finite factors are unitarily equivalent

Why are two Murray von Neuman equivalent projections $p$ and $q$ in a finite factor unitarily equivalent?
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1answer
24 views

image of some ideal under the quotient map

! I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial. Is my understanding correct?
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1answer
13 views

Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?

Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...
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1answer
22 views

identify GNS construction as asubalgebra of $R^{\omega}$

I have two questions In lemma6.5.5. 1.why can we identify the GNS representation of $\prod M_{k(n)}(\Bbb C)/\bigoplus M_{k(n)}(\Bbb C)$ with respect to $\tau _{\omega}$ with a subalgebra of $R^{\...
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0answers
36 views

Trying to understand a proof from math overflow regarding direct limit

Direct limit of $\mathbb {C^*}$ behaves well with quotients. The following solution is from mathoverflow: Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a ...
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1answer
19 views

strictly positive element vs positive definite matrix

If $A=\prod_{n=1}^{\infty}M_{k(n)}(\mathbb{C})$,$x=(x_1,\cdots,x_n,\cdots)$ is strictly positive in $A$,does this mean that each $x_n\in M_{k(n)}\mathbb{C}$ is a positive definite matrix?
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1answer
22 views

tracial state of a orthogonal projection

Suppose $A\in M_n(\mathbb{C})$,$A$ has eigenvalues$\lambda_1,\cdots,\lambda_n$,$P$ is the orthogonal projection from $\mathbb{C}^n$ onto the span of eigenvectors associated with $\lambda_1,\cdots,\...
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1answer
24 views

Some clarifications required

Is the inductive limit of tensor product $L^{\infty}(X,\mu)^{\otimes \infty}$ is isomorphic to $L^{\infty}(X,\mu)$?
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2answers
41 views

k-rank numerical range of an operator.

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
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1answer
13 views

central projections

Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
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1answer
17 views

strictly positive elements under a nonzeo $*$ homomorphism

Suppose $A$ is a separable $C^*$ algebra,x is a strictly positive element in $A$,$\phi:A\rightarrow B$ is a nonzero $*$ homomorphism,is $\phi(x)$ also strictly positive in$B$?
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1answer
17 views

the spectrum of self-adjoint element

If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $\sigma_A(x)\subset \mathbb{R}$,my question is :Is the following form possible for $\sigma_A(x)$?1.$\sigma_A(x)$ be unions of ...
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1answer
35 views

functional calculus

Suppose $A$ is a non-unital $C^*$ algebra,$B$ is another $C^*$ algebra.Suppose $\phi:A\rightarrow B$ is a non-zero $*$ homomorphism and $x_0$ is a normal elememt in $A$,by continuous functional ...
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1answer
11 views

Uniqueness of faithful (!) tracial states on separable $C^*$-algebra

Let $A$ be a separable $C^*$-algebra and $S\subseteq A$ a norm-dense, countable set in $A$. Assume that there are two faithful (meaning that the corresponding GNS-construction gives a faithful ...
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1answer
35 views

strictly positive element

If $A$ is a non-unital separable $C^*$ algebra,does there exist a strictly positive idempotent element in $A$ ?
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1answer
36 views

Do all *-isomorphisms between von Neumann algebras preserve strong operator topology?

Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology? Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other ...
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0answers
16 views

On support of a spectral measure

Suppose $A$ is the self-adjoint unbounded operator in Hilbert space $\mathcal{H}$, if spectral measure $E$ of $A$ is supported on $[0,a]$, then prove that the resolution of identity $E_{a}=1_{(-\infty,...
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1answer
20 views

composition of finite rank projection and bounded operator

If $P\in B(H)$ is a finite rank projection,we assume the rank is $n$,I know the fact $PB(H)P\cong M_n(\mathbb{C})$,but how to construct the isomorphism?
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0answers
26 views

How much there exists operators of $a\in End(\mathbb{F}_p^3)$

How much there exists operators of $a\in End(\mathbb{F}_p^3)$ such that a((2, -1, 3)) = (1, 1, -1), a((1, 2, 3)) = (1, 0, 1), a((3, 1, -1)) = (2, 1, 0). I know how to solve such tasks for $a\in(\...
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1answer
26 views

$\mathcal{B}(H)^+$ is closed and generated by $id$

Let $H$ be a Hilbert space. Define its positive cone by $$\mathcal{B}(H) = \{A \in \mathcal{B}(H) : \langle v, A(v)\rangle \geq 0, \forall v \in H\}.$$ Show that $\mathcal{B}(H)^+$ is closed with ...
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2answers
36 views

Extending a $*$-homomorphism between $C^*$-algebras to $*$-homomorphism between generated von Neumann algebras

Let $A \subseteq \cal B(H)$, $B \subseteq \cal B(H')$ be $C^*$-algebras, where $\cal H$, $\cal H'$ are Hilbert spaces and let $\psi: A \rightarrow B$ be a $*$-homomorphism. My question: When does ...
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1answer
25 views

minimal projection in a $C^*$ algebra

If $(H,\pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $E\in A$ such that $\pi(E)\neq 0$?
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1answer
20 views

a question on partial isometry

This is a statement from wikipedia.I don't understand why can we deduce that $C$ is a partial isometry if $A^*A=B^*B$.
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1answer
35 views

On proof of weak operator closed ness of kernel of the representation

Let $\pi:L^{\infty}(X,\mu)\mapsto B(\mathcal{H})$ be a representation of an abelian von Neumann algebra. Where $\mu$ is a probability measure and $\mathcal{H}$ is a seperable Hilbert space. Prove that ...
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0answers
52 views

Projections correspond to double duals of $C(X)$-algebras fibers

Let $A$ be a $C(X)$-algebra ($X$ compact). For $x\neq y$ in $X$, we have the Glimm ideals in $A$: $I=C_0(X\setminus \{x\})A$ and $J=C_0(X\setminus \{y\})A$. The fibers are denoted by $A_x=A/I$ and $...
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1answer
45 views

Why is the Rational Rotation Algebra not a Matrix Algebra?

Let $A_{\theta}$ be the rotation C$^{*}$-algebra with rotation $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $vu=e^{2\pi i \theta}uv$. Suppose that $\theta=p/q$, where $p$ and $q$ are non-zero ...
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1answer
27 views

Gelfand-Naimark Theorem of non-unital case

Let $X$ be a non-compact, locally compact Hausdorff space. Then $C_0 (X)$, the space of complex valued continuous functions on X vanishing at infinity, is a non-unital $C^*$-algebra and $\Omega (C_0 (...
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1answer
34 views

nuclear $C^*$ algebra

If $(A_i)$ is a sequence of nuclear $C^*$ algebras,Is $\oplus_{c_0}A_i$ ($c_0$ direct sum)and $\prod A_i$($\ell ^\infty $ direct sum) also nuclear?
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1answer
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functional calculus under the $*$ homomorphism

If $A,B$ are two $C^*$ algebras,$\psi:A \rightarrow B$ is a non $*$ homomorphism.Suppose $b$ is a nonzero normal element in $B$,we have a $*$ isometric isomorphism $\phi:C(\sigma_B(b))\to C^*(b,b^*),\;...
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1answer
17 views

functional calculus for non-unital $C^*$ algebras

If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(\sigma_{A}(a))$ to $C^*(a)$. I have a question:what is the set of $C_0(\sigma_{A}(a)...
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0answers
111 views

Extension of a von Neumann algebra by a von Neumann algebra

Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras: $$0\to A\to C\to B\to 0$$ Assume that $A,B$ are generated by their ...
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1answer
29 views

Centralizer of projections

Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$, i.e. $$p^2=p=p^* \space \text{ and } \space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$. Is it true that ...
3
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1answer
43 views

trace and operator norm of $\exp^A$

If $A$ is an n by n complex matrix, 1.How to compute $tr(\exp^A)$ .Can we use the Taylor expansion as following: $\exp^A=\sum_{k=0}^{\infty}\frac{A^k}{k!},$then $tr(\exp^A)=\sum_{k=0}^{\infty}tr(\...
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1answer
24 views

elements in $C^*$ algebra

If $x$ is an element of a $C^*$ algebra $A$,Is $exp^x$ an element of $A$? My thought: compute the Taylor expansion of $exp^x$,but it is a sum of countable terms.Is sum closed in $A$?
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1answer
46 views

infinite dimensional positive matrix

Suppose $A$ is a infinite dimensional positive complex matrix,what is the operator norm of $A$? In the finite dimensional case,we can use the spectral theorm,$\|A\|=sup|\lambda_i|$,where $\lambda_i$ ...
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1answer
39 views

What Are the Irreducible Representations of the Rational Rotation C$^{*}$-algebra?

Let $m$ and $n$ be integers, with $n>0$ and $\gcd(m,n)=1$. Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra generated by two unitaries $u$ and $v$, satisfying the ...
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0answers
24 views

For every normal operator, is there a unitary operator with the same spectral measure?

Given a (bounded) normal operator $N$ (satisfying $N^*N=NN^*$) in a von Neumann algebra $A$, does $A$ always contain a unitary operator $U$ having the same spectral (projection-valued) measure as $N$? ...
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0answers
15 views

construct an element in $\prod M_{k(n)} (\mathbb{C})$

Suppose $A$ is a $C^*$ algebra,$\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is a essential ideal of $A$ and there is an element $(x_n) \in A$ such that $(x_n) \in \prod M_{k(n)} (\mathbb{C})$ and $tr(x_n) \...