# Questions tagged [oa.operator-algebras]

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

1,543
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### G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...

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49 views

### General construction of enveloping C*-algebra, left/right-regular representation, etc

In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...

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35 views

### Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?

Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$.
Is it true that $F$ is completely contractive? If it is, I would be very grateful ...

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45 views

### Lower bounds in the space of compact operators

Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the "unit sphere" $S:=\{T\in K(H)|T\geq 0\text{ and }||T||=1\}$. Is it true that, given any pair ...

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62 views

### Observable nearly commuting with a “complete” set of commuting observables

Consider the Hilbert space $H = E^{\otimes n}$ where $E=\mathbb{C}^2$.
On $E$ we have an observable $O$ (i.e. a Hermitian matrix) that is diagonalizable in the standard basis with eigenvalues $1$ and ...

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98 views

### Need reference for ideals and representations of $C_0(X,A)$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...

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133 views

### Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?

My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \...

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90 views

### Characterizing discrete quantum groups

Let $M$ be a von Neumann algebra, and let $\Delta$ be a unital normal $*$-homomorphism $M \rightarrow M \mathbin{\bar\otimes} M$ that satisfies the coassociativity condition $(\Delta \mathbin{\bar\...

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54 views

### Spectral decomposition of a particular operator [closed]

Let $\mu$ be a lebesgue measure on $\mathbb{R}$, Consider the operator on $L^{2}(\mathbb{R},\mu)$ given by multiplication by $\sin(x)$. What is its spectral multiplicity, measure, and decomposition?

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85 views

### Is the reduced group $C^*$-algebra quasidiagonal

Let $G$ be an amenable group. I wonder whether it is true that the reduced group $C^*$-algebra $C_r^*(G)$ is quasidiagonal.

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127 views

### Continuous functions on a compact $T_1$ space

Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, ...

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261 views

### A question about comparison of positive self-adjoint operators

I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...

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168 views

### Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page)
I find that towards the end of the ...

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126 views

### For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$

Is there a terminology for the following property of $C^*$ algebra $A$:
For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...

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52 views

### Jordan isomorphisms of type I von Neumann algebras

Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras. A linear map $J:\mathcal M\to\mathcal N$ is said to be a Jordan isomorphism if $J$ is bijective, $*$-preserving and $J(xy+yx)=J(x)J(y)+J(...

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94 views

### closed ideals in L(L_1)

Denote $L_1=L_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L_1)$ includes the chain
$$
\{0\}\subsetneq\mathcal{K}(L_1)\subsetneq\mathcal{FS}(L_1)
\subsetneq\mathcal{J}_{\ell_1}(L_1)\subsetneq\...

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64 views

### Dixmier trace, Wodzicki residue and topological index

There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...

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126 views

### What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?

I've been working computing several K-groups associated to some $C^*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K_1(C(\mathbb{T})\otimes\...

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107 views

### Showing a product on a character space is continuous

Quoting from Timmermann's An invitation to quantum groups and duality:
Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact
quantum group. Then there exists a compact group $G$ ...

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116 views

### Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]

Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...

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137 views

### A question on quantum tori

Let $\mathbb T_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?

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76 views

### Wave equation for smooth Schwartz kernels

Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...

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138 views

### Approximating a projection by a sum of elementary tensors with a certain property

Let $A$ and $B$ be two C$^{*}$-algebras and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.)
Does there ...

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207 views

### Embedding of Cuntz algebras $O_2\subseteq O_3$?

The Cuntz algebra $O_n$ is the (universal) C*-algebra generated by n-isometries $s_1,...,s_n$ such that
$$\sum_{i=1}^n s_is_i^\ast =\mathbf{1}, \ \hbox{and}\ s_i^\ast s_j=\delta_{ij} \mathbf{1}\ (\...

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69 views

### A quantitative characterization of bounded approximation property

Recall that a Banach space $X$ has the approximation property (AP for short ) if for every compact subset $K$ of $X$ and every $\varepsilon > 0$, there exists a finite rank operator $S$ on $X$ such ...

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93 views

### When is rank-1 perturbation to a positive operator still positive?

Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...

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116 views

### Separability of an algebra is equivalent to separability of its spectrum

Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...

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129 views

### Representation of algebras as bounded nilpotent operators

Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})...

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117 views

### A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm

Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.

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125 views

### Representations of Banach algebras

If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)...

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109 views

### When a finite codimensional subalgebra contains a finite codimension ideal?

What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ ...

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119 views

### Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...

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106 views

### Commutator of translation invariant operators on $L^2(\mathbb{R})$

I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$.
Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $...

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85 views

### Left and right topological K-theory of Banach algebras

Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...

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155 views

### Non-commutative analogue of a certain fact in differential geometry

In the literature, is there a non-commutative analogue of the fact that every Riemannian manifold whose isometry group has sharp dimension must be a constant curvature manifold?

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83 views

### Reduced compact quantum group and left and right multiplication

Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product
$$
\...

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145 views

### $K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...

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### Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...

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### Support of commuting elements in a von Neumann algebra

Let $\mathcal M$ be a semifinite von Neumann algebra and $a$ be an unbounded positive self-adjoint operator affiliated to $\mathcal M$. Suppose $x\in\mathcal M$ is such that $x$ commutes with all ...

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### Limit of spectral projection of increasing sequence of positive operators

Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...

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79 views

### Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...

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### Corepresentations on Hilbert modules

In the seminal paper "Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres" by Baaj and Skandalis, we find the following very general definition of what a corepresentation ...

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81 views

### Numerical range of tensor product of two matrix

Let $T\in M_n$. Is the following true
$$\bigcap\limits_{B\in M_2\\\text{tr}(B)=0}\left\{X\in M_2: W(X)\subseteq W(T)\text{ and } W(B\otimes X)\subseteq W(B\otimes T)\right\}\subseteq\bigcap\limits_{...

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55 views

### Characterizing spectral radius using invertible elements in unital C* algebra [closed]

Consider A a unital C* algebra, I want to show that the spectral radius of a satisfies the following: $𝑟(𝑎)= $ inf$_{𝑏∈𝐼𝑛𝑣(𝐴)}||𝑏𝑎𝑏^{−1}||$, where Inv(A) is the set of invertible elements in ...

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107 views

### Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces.
Given two $\Cst$-...

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36 views

### Factors of polar decomposition of an operator A bounded below are contained in C*(A)

I have a question about a problem I come across in An Invitation to C''-Algebra by Arveson. The problem is as follows. Suppose we have a bounded operator A on a Hilbert space that is bounded below, ...

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143 views

### Completely positive, unital maps acting on unitary operators [solved]

Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...

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71 views

### Non-separable non-postliminal $C^*$-algebra with injective $\pi\mapsto\ker\pi$

It is known that for a postliminal/GCR $C^*$-algebra the map $\pi\mapsto\ker\pi$ from (equivalence classes of) irreducible representations to their kernels is injective. If the algebra is separable, ...

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66 views

### induced maps between group $C^*$-algebras

Suppose $G,H$ are two locally compact groups, if there is a injective homomorphism $\phi:G\to H$, can $\phi$ induce the $*$-homomorphism between group $C^*$-algebras $C^*(G)$ and $C^*(H)$? If it ...

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88 views

### Topology on function spaces for pointwise convergence

Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...