# Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

1,342 questions

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### (Noncommutative) Tietze $C^*$ algebras

A unital $C^*$ algebra $A$ is said a Tietze algebra if it satisfies the following:
For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi}:...

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83 views

### examples of MF algebras [on hold]

Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra respectively? Thanks!

**5**

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**2**answers

166 views

### Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinuous on each compact set?

Let $H$ be a Hilbert space and $B(H)$ be its space of all (bounded) operators. A nuclear functional on $B(H)$ is a linear functional $f:B(H)\to{\mathbb C}$ that can be represented in the form
$$
f(A)=\...

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97 views

### On $s$-numbers in finite von Neumann algebra

$T$ is an operator in $M$, $M$ is finite von Neumann algebra. There is a notion of singular value function that is ($s$-numbers). My question is: what is $s$-number for tensor product of two operators ...

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53 views

### How to find the solution of the equation $b=1+P(ba)$?

We know the solution(commutative case of Spitzer's Identity) of the equation $b=1+\text{P}(ba)$ when the operator $\text{P}$ satisfies Rota-Baxter eqution
$\text{P}(x)\text{P}(y)=\text{P}(x\text{P}(...

**7**

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**1**answer

141 views

### Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra

In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^*_r(\Gamma)$-module $L^...

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**1**answer

237 views

+50

### Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras

In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.
Roughly, the group $K_0(A)$ is given by the ...

**2**

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**1**answer

138 views

### On diagonal part of tensor product of $C^*$-algebras

Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?

**3**

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**1**answer

105 views

### Ultraproduct of non-commuative $L^p$-spaces

Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...

**11**

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**2**answers

185 views

### Actions of locally compact groups on the hyperfinite $II_1$ factor

Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.
(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
(2) If so, how does one ...

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**0**answers

39 views

### When does there exist a faithful normal expectation onto von Neumann subalgebra (finite vNa)?

Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. What conditions must $S$ satisfy so that a faithful normal conditional expectation $\Phi : R \to S$ exists?
For ...

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**2**answers

86 views

### Polar decomposition of tensor product of operators in von Neumann algebra

If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-...

**2**

votes

**1**answer

81 views

### center of $C^*$-algebra and finite dimensional representation

The center of $K(H)$ is 0 and $K(H)$ has no nonzero finite dimensional representation. Can we conclude that if the center of a $C^*$-algebra $A$ is zero, then $A$ has no nonzero finite dimensional ...

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134 views

### Ultraweak topology in abelian von Neumann algebras

Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...

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56 views

### How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...

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45 views

### Adjoint for a non-densely defined unbounded operator on a Hilbert space

Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...

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174 views

### Noncommutative Fredholm operators

Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...

**6**

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136 views

### Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...

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**1**answer

90 views

### Computing multiplicity function for self adjoint operator with nonatomic spectral measure

Suppose $T$ is a self-adjoint operator in $B(H)$ with $\sigma(T)$ a spectrum of $T$. $\mu$ is a spectral measure. For the operators having a generally continuous spectrum how to calculate the ...

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55 views

### Sequence of unitaries in type III von Neumann algebra

Consider a type III von Neumann algebra $\mathcal{M}$ and an isometry $w$. How does one show that there exists a sequence of unitaries $u_n\in\mathcal{M}$ that converge strongly to $w$?
For instance,...

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**0**answers

65 views

### A cross product on $C^*_{red} G$

For every group $G$, the reduced group $C^*$-algebra $C^*_{red}G$ is equipped with the inner product $\langle a,b\rangle=tr(ab^*)$ where "$tr$" is the standard trace on group $C^*$-algebras.
For ...

**0**

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**1**answer

97 views

### Banach algebra $A$ without an approximate identity but $A^2=A$

Please help me with the following question.
What are some examples of Banach algebra $A$ satisfying the following two conditions?
$1$.$ A $ does not have an approximate identity.
$2$. $A^2=A$. ...

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**1**answer

122 views

### The compact open topology and the operator norm

Let $T:[-1,1]\rightarrow B(H)$ be a continuous family of bounded operators where $B(H)$ is endowed with the compact open topology for an infinite dimesional Hilbert space $H$. Is it true that if $T_0=...

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87 views

### Existence of (generalized) Crossed Products

Compare the following construction with the crossed product construction for $C^*$-dynamical systems:
Let $W\curvearrowright X$ be a group action of a discrete group on a compact Hausdorff space $X$ ...

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**1**answer

143 views

### Bicommutant theorem for commutative operator algebras

Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says:
Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...

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67 views

### Morita equivalence for graded von Neumann algebras

I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...

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votes

**1**answer

80 views

### On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...

**5**

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**1**answer

180 views

### Property $\Gamma$ in terms of Correspondences

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...

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**1**answer

122 views

### Need a reference of a fact given in B. Blackadar's Operator Algebras

I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that
Maximal Tensor products commute with arbitrary limits.
In the same book the proof of this fact is not given....

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**1**answer

75 views

### Strictly increasing approximation of the identiy

Is there always a strictly increasing approximation of the identity in a separable $C^*$-algebra?

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77 views

### $T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions

The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field
theories (CFT) by an operator that is quadratic in the stress-...

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86 views

### Pimsner-Popa basis dealing with higher relative commutants

Let $(N \subseteq M)$ be a finite index inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection.
A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-Popa ...

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**1**answer

226 views

### Why is the definition of von Neumann trace independent of the choice of the Hilbert space?

A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ ...

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**1**answer

74 views

### Real rank 0 implies stable rank 1 on $C^\ast$-algebras?

A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...

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49 views

### Inclusion preserving map between ideal spaces is continuous

Let $A$ be a $C^*$-algebra with ideal space $\mathcal{I} (A)$ and equip $\mathcal{I} (A)$ with the Fell topology, i.e. the topology generated by the subbase $U_{I}:=\left\{ J\in{\cal I}\left(A\right)\...

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76 views

### Is the set of points in the irreducible decompositions of this C$^{*}$ -algebra's representations closed?

Suppose $X$ and $Y$ are compact Hausdorff spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily ...

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394 views

### Why is Planar algebras I (by Vaughan Jones) not published?

On Saturday 4 September 1999, Vaughan Jones put on arXiv a paper entitled Planar algebras, I.
Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the ...

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**1**answer

144 views

### construct a concrete $C^*$ algebra

Does there exist a concrete $C^*$ algebra $A$ such that that the following conditions hold:
(1) $A$ is unital and $A$ has no tracial state.
(2)there exists a closed ideal $I$ of $A$ such that $I$ ...

**3**

votes

**1**answer

137 views

### What is the story behind this Hilbert space in the definition of Hilbert Modules

Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...

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**0**answers

108 views

### A single vertex operator - bare bones explanation?

There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. ...

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**1**answer

128 views

### Projections in the tensor product of von Neumann algebras

This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...

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**1**answer

211 views

### Stone–von Neumann theorem?

The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...

**2**

votes

**3**answers

229 views

### center of a $C^*$-algebra

Does there exist a $C^*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?
I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no ...

**10**

votes

**0**answers

197 views

### What are these matrices called?

A paper I'm writing heavily uses block diagonal matrices with the property that each block is upper triangular and constant along its diagonal. Like this:
$$\begin{bmatrix}
A_1 \\
& \ddots \\
&...

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votes

**2**answers

124 views

### Spectrum equals eigenvalues for unbounded operator

Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...

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**1**answer

66 views

### strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...

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53 views

### equivalent definition of 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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125 views

### A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product.
Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...

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**1**answer

130 views

### irreducible representation of a $C^*$ algebra

Suppose we have a $C^*$ algebra $A=\{(x_n)\in \prod M_n(\Bbb C),lim_n tr_n(x_n^*x_n)=0\}$.
If $B$ is any nonzero $C^*$-sub algebra of $A$,does there exist a finite dimensional irreducible ...

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votes

**1**answer

115 views

### A generalization of invariant and coinvariant subspaces

Let $\mathcal{A}$ be a subalgebra of $M_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for ...