Questions tagged [oa.operator-algebras]

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

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Is the ring of $p$-adic integers extremally disconnected?

We call a topological space $X$ extremally disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremally disconnected spaces are totally disconnected in the sense that ...
Rick Sternbach's user avatar
3 votes
0 answers
217 views

Inverse Limit in the category of $C^{\ast}$-algebras or operator spaces

Does the inverse limits (Projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces? I tried to search but could not find a proper reference. Any reference or comments about ...
Math Lover's user avatar
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Subalgebras of $II_{1}$ factor

Let $M$ be a type $II_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II_{1}$ ?
user136400's user avatar
2 votes
1 answer
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Analogue of spectral values of automorphisms in vN algebra

Is there any analog of studying spectral properties of automorphisms of von Neumann algebra? Does it make sense, if anybody knows please give a reference.
user136400's user avatar
1 vote
2 answers
322 views

Regarding Haagerup $L^{P}$ spaces

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. ...
user136400's user avatar
3 votes
0 answers
276 views

Tensor product of compact operators on Banach modules

Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...
Benedikt Hunger's user avatar
6 votes
2 answers
536 views

Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)?

Both the ultraweak and ultrastrong topologies are intrinsic topologies in the sense that the image of a continuous (unital) $*$-homomorphism between von Neumann algebras (in either topology) is a von ...
condexp's user avatar
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1 answer
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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}:...
Ali Taghavi's user avatar
5 votes
2 answers
267 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)=\...
Sergei Akbarov's user avatar
0 votes
0 answers
143 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 ...
user136400's user avatar
8 votes
1 answer
336 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^...
Alexander Alldridge's user avatar
8 votes
1 answer
530 views

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 ...
Dave Shulman's user avatar
2 votes
1 answer
341 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?
user136400's user avatar
3 votes
1 answer
198 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 $\tau_i,$ $i\in I.$ ...
A beginner mathmatician's user avatar
11 votes
2 answers
444 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 ...
André Henriques's user avatar
1 vote
0 answers
92 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 ...
condexp's user avatar
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3 votes
2 answers
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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-...
user136400's user avatar
2 votes
1 answer
151 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 ...
math112358's user avatar
3 votes
2 answers
246 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 ...
condexp's user avatar
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7 votes
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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 ...
Miguel Moreira's user avatar
1 vote
0 answers
238 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 ...
Max Schattman's user avatar
8 votes
1 answer
270 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 ...
ernest's user avatar
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6 votes
0 answers
179 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 ...
Eric Schlarmann's user avatar
0 votes
2 answers
267 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 ...
user136400's user avatar
1 vote
0 answers
83 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,...
Nima Lashkari's user avatar
1 vote
0 answers
85 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 ...
Ali Taghavi's user avatar
0 votes
1 answer
227 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. $2$. $A^2=A$. ...
Fermat's user avatar
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1 vote
1 answer
339 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=...
ernest's user avatar
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4 votes
1 answer
446 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 ...
Jochen Glueck's user avatar
9 votes
0 answers
108 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 ...
Anton Kapustin's user avatar
-1 votes
1 answer
166 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 ...
mathlover's user avatar
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6 votes
1 answer
458 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 $...
Jon Bannon's user avatar
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2 votes
1 answer
179 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....
Math Lover's user avatar
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-1 votes
1 answer
89 views

Strictly increasing approximation of the identiy

Is there always a strictly increasing approximation of the identity in a separable $C^*$-algebra?
User's user avatar
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6 votes
1 answer
437 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-...
wonderich's user avatar
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5 votes
0 answers
118 views

Pimsner-Popa basis dealing with higher relative commutants

Let $(N \subseteq M)$ be a finite index unital 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-...
Sebastien Palcoux's user avatar
7 votes
1 answer
322 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 $V$ ...
Meisam Soleimani Malekan's user avatar
6 votes
1 answer
359 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 ...
Miguel Moreira's user avatar
3 votes
0 answers
96 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 ...
ervx's user avatar
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13 votes
1 answer
1k 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 ...
Sebastien Palcoux's user avatar
4 votes
1 answer
180 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 $I$ ...
math112358's user avatar
3 votes
1 answer
230 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 ...
Meisam Soleimani Malekan's user avatar
7 votes
0 answers
180 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. ...
მამუკა ჯიბლაძე's user avatar
6 votes
1 answer
426 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 ...
Masayoshi Kaneda's user avatar
0 votes
1 answer
369 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) $$ U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t $$ ...
SerkanSüner's user avatar
2 votes
3 answers
615 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 ...
math112358's user avatar
10 votes
0 answers
249 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 \\ &...
Nik Weaver's user avatar
0 votes
2 answers
429 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 ...
Bas Winkelman's user avatar
0 votes
1 answer
205 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 ...
math112358's user avatar
4 votes
0 answers
145 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 $A''$...
Ali Taghavi's user avatar

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