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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

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11 views

Equivalent definition of decomposable map

Let $A\subset B(H)$ be an operator system and $B$ be a $C^*$-algebra. (1)$u:A\rightarrow B$ is called a decomposable map if $u$ is in the linear span of $CP(A, B)$, where $CP(A,B)$ is the set of all ...
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15 views

Construction of the free product of $C^*$-algebras

The content of the screenshot is from Pisier's book. I have two questions : $\mathcal{F}$ is an algebra, how to conclude that $\pi: \mathcal{F} \to B(H_{\pi})$ is a $*$-homomorphism? 2.Why is $j$ ...
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1answer
18 views

The rank of a spectral projection $E_A(\Delta)$ when $\Delta \bigcap \sigma_e(A) \neq \emptyset$

Let $H$ be a infinite dimensional Hilbert Space. $A \in B(H)$ and $A$ self-adjoint. Let $\sigma_e(A)$ be the essential spectrum of $A$. Since $A$ is self-adjoint, we can assume the convex hull of $\...
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19 views

incluson between kernels for two linearly dependent operators. [closed]

Let $X$ be a Banach space. Let $A$ and $B$ be two bounded operators in $B(X)$, assume that $Ax$ and $Bx$ are linearly dependent for each $x \in X$ but $A$ and $B$ are linearly independent. How we can ...
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1answer
19 views

Is there a proper ideal of $B(H)$ that contains a proper projection

Let $H$ be a infinite-dimensional separable Hilbert space and $\mathcal{I}$ be a proper closed two-sided ideal of $B(H)$. Can $\mathcal{I}$ contain a projection for a infinite dimensioal proper closed ...
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1answer
22 views

For every projection $p$ and normal $a$ in a C*-algebra $A$ (with $ap=pa$), there is a $*$-isomorphism $C(\sigma(a))\to C^{*}(a,p)$ such that …

Let $a$ be a normal element of a (non-unital) C*-algebra $A$. I am trying to prove that for every projection $p\in A$ that commutes with $a$ (i.e. $p=p^{2}=p^{*}$ and $ap=pa$), there is a $*$-...
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72 views

Let $\phi:A\rightarrow B$. Then if $b\in B_{sa}\cap\operatorname{ Inv}(B)$, there is $a\in A_{sa}\cap \operatorname{Inv}(A)$ s.t. $\phi(a)=b$

Let $\phi:A\rightarrow B$ be a surjective $*$-homomorphism between unitary C*-agebras (not necessarily commutative). The question is that if for any $b\in B$ self-adjoint and invertible, there has to ...
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1answer
33 views

If $B\subset B(H)$ is a C*-subalgebra and $T\colon B''\to B''$ is linear, bounded and weakly continuous, then $\|T\|=\|T|_{B}\|$

Let $H$ be a Hilbert space and let $B\subset B(H)$ be a C*-subalgebra. Suppose that $T\colon M\to M$ is linear, bounded and operator-weakly continuous, then I want to prove that $\|T\|=\|T|_{B}\|$. ...
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31 views

Is it true that all representations of $C_0(X,A)$ come from representations of $A$

Let $A$ be $C^*$- 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 $f^{\ast}(t)={f(t)}^{\ast}$ ...
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1answer
40 views

Inseparable $C^*$-algebras

In many theories about $C^*$-algebras, the $C^*$-algebras are always assumed to be separable. I have a question: Why few people discuss the inseparble $C^*$-algebras? Are they more difficult to handle?...
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1answer
100 views

kernel of Haagerup tensor product of maps

Haagerup tensor product $\otimes_{\rm h}$ is both injective and projective. Pisier, Gilles, Introduction to operator space theory, London Mathematical Society Lecture Note Series 294. Cambridge: ...
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1answer
32 views

Why shift operator is not homotopic to 1 ($K_1$-approach)?

Let us recall that via fourier transform it holds true that $C^*(S)\cong C(\mathbb{T})$, with map given by $S\mapsto e^{2\pi i x}$ (considering $\mathbb{T}=\mathbb{R}/\mathbb{Z}$). It is also true ...
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2answers
49 views

Existence of non- normal element element in a non commutative C$^*$-algebra

I'm not familiar with lots of examples of non commutative C$^*$-algebras, but there are $M_n(\mathbb{C})$, and $B(H)$. These have a non trivial non-normal element. My question is : what about the ...
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1answer
35 views

Ideals of $M_2(A)$ , $A$- non commutative non unital

Let $A$ be a non unital, non commutative $C^{\ast}-$ Algebra. Let $J$ be an ideal of $M_2(A)$. Assume $$J= \begin{bmatrix}P&Q\\R &S\end{bmatrix}$$ It is easy to prove that $Q$ is an ideal of $...
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1answer
37 views

Is the center $Z$ of a C*-algebra $A$ unital if and only if $A$ is unital?

Let $A$ be a C*-algebra. Its center is defined by $$Z:=\{z\in A:az=za \ \text{for all} \ a\in A\}.$$ It is easy to verify that $Z$ is a C*-subalgebra of $A$. Also, if $A$ is unital, then it is clear ...
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28 views

A questio about nilpotent contractions

Suppose $T:\mathbb C^3\to\mathbb C^3$ is a nilpotent contraction where $\mathbb C^3$ is equipped with the Euclidean norm. Then when can we say $I+T$ is a contraction on the range of $I-T$? What about $...
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1answer
22 views

Show $C^*(S)/K(l^2(\mathbb{N})) \cong C(T)$

Consider $l^2(\mathbb{N})$ and the shift operator $S: l^2(\mathbb{N}) \to l^2(\mathbb{N}): e_n \mapsto e_{n+1}$. It is easy to see that $C^*(S)$ contains the compact operators $K(l^2(\mathbb{N}))$ ...
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9 views

Deterministic matrix with prescribed limit distribution

I have a question about a common argument in free probability theory. Often times, when we want to prove a statement about free-ness or other properties of distributions of elements in a non-...
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1answer
24 views

Algebra of compact operators is CCR algebra

In Arveson's book "Invitation to $C^*$-algebra's", it is claimed that every algebra of compact operators is a CCR algebra. Concretely, let $\mathcal{A}$ be a $C^*$-subalgebra of some $B_0(\mathcal{H})...
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1answer
29 views

Is a representation $(H,\phi)$ of a simple C*-algebra $A$ always faithful?

Suppose that $A$ is a simple C*-algebra (i.e. there is no closed ideal $I\subset A$ such that $0\neq I\neq A$) and let $(H,\phi)$ be a representation. Can we conclude that $(H,\phi)$ is faithful, i.e. ...
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1answer
24 views

Prove the following are equivalent with irreducibility of representation on $C^*$-algebras

Let $\pi: A \to B(\mathcal{H})$ be a representation where $A$ is a $C^*$-algebra and $\mathcal{H}$ a Hilbert space. I'm trying to show the following four statements are equivalent: (1) $\pi$ is ...
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10 views

Set of representations of a $C^*$-algebra

I saw the following statement from a reference book. Let $S$ be the set of representations of a $C^*$-algebra $A$. Does it mean that $S$ is the set of unitary equivalence classes of the ...
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1answer
22 views

If $(H,\pi)$ is a finite dimensional irreducible representation of a C*-algebra $A$, then $\pi(A)=B(H)$

Let $A$ be a C*-algebra and $(H,\pi)$ an irreducible representation such that $\dim(H)<\infty$. Then how do I show that $\pi(A)=B(H)$? Here is what I tried: Since $(H,\pi)$ is irreducible, we ...
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1answer
30 views

If $S$ and $T$ are distinct self-adjoint operators on $H$ such that $S\leq T$, can we conclude that $\langle Sx,x\rangle<\langle Tx,x\rangle$?

If $S$ and $T$ are distinct self-adjoint operators a (separable) Hilbert space $H$ such that $S\leq T$, can we conclude that $\langle Sx,x\rangle<\langle Tx,x\rangle$ for all $x\in H$? Here $S\leq ...
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26 views

Spectrum of a Banach algebra VS spectrum of a $C^*$-algebra

The spectrum of a $C^*$-algebra $A$ is the set of unitary equivalence classes of irreducible $*$-representations of $A$. The spectrum of a unital commuative Banach algebra $B$ is the set of ...
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1answer
41 views

Number of nontrivial projection of a unital commutative C* algebra

Q.Can we construct a unital commutative C* algebra such that it admits exactly 5 non trivial projection ? I can't conclude that answer. I only know For some C* algebra , I and O is the only ...
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0answers
43 views

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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20 views

$*$-isomorphisms of cross product $C^*$-algebras

Let $G$ be a discrete countable group, $B$ is the CAR algebra $\otimes_{\Bbb N}M_2$. I feel confused about the statement marked green in the screenshot. How to construct the explicit bijections?
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1answer
14 views

Redundant assumption in proving the quotient of a Banach algebra over a modular ideal is a field

When reading Murphy's book $C^*$ algebras and operator theory, I found the Lemma 1.3.2 a bit strange. The original lemmas is as the following: If $I$ is a modular maximal ideal of a unital abelian ...
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1answer
37 views

Infinite tensor product

Suppose that $X$ is an infinite set and $A$ is a unital $C^*$-algebra. The tensor product $\bigotimes_X A$ is defined to be the closed linear span of $\bigotimes_{x\in X }a_x$, where $a_x\in A$ for ...
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1answer
22 views

Involution is not strongly continuous

Let $H$ be a Hilbert space and consider on $B(H)$ the strong topology, i.e. the topology induced by the seminorms $$x \mapsto \Vert x \xi \Vert, \quad \xi \in H$$ I.e. a net $(x_\alpha)_\alpha$ ...
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52 views

Univesal Coefficient Theorem for $C^*$-algebras

The UCT theorem is shown in the sreenshot. I have a question : What is the definition of $Ext_{\Bbb Z}^1(K_{*}(A),K_{*}(B))$? Does it have a relationship with Tor functor?
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1answer
52 views

$C^*$ algebras and differential geometry

If $(A, G, \alpha)$ is a $C^*$-dynamical system where $G$ is a Lie group and $\alpha: G \rightarrow \operatorname{Aut}(A)$ is a continuous homomorphism (where $\operatorname{Aut}(A)$ is equipped with ...
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32 views

Are bounded operators on a Banach space a $C^*$-algebra?

It is well-know that if $H$ is a complex Hilbert space then the set of all bounded operators $\mathcal{B}(H)$ is a $C^*$-algebra. If we change $H$ with a complex Banach space $E$, do we have the same ...
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1answer
70 views

Gauss divergence theorem applied to operator valued functions

I'm studing quantum field theory. Especifically the procedure called second quantization for the complex scalar field. I noticed that I can derive the Klein Gordon equation from the Heisemberg ...
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27 views

Bott-Periodicity for unitary matrices with entries in general $C^{*}$-algebra

When working through the proof of Bott-Periodicity (the original proof by Bott), I noticed that compactness of $U(n)$ is important as it gives us that every geodesic looks like an exponential map (...
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3answers
25 views

Every *-isometric isomorphism of $B(\mathcal{H})$ keep compact operators?

Let $\mathcal{H}$ be a Hilbert space, $B(\mathcal{H})$ denotes the $\mathcal{C}^*$-algebra consisting of bounded linear transformation on $\mathcal{H}$ ($*$ is the adjoint). Now consider a isometric *-...
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1answer
33 views

Show that $C_{0}(X,A)$ is a sub space of $B(H)$ for some Hilbert space $H$

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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1answer
35 views

Spectral Radius Formula

An element of a Banach algebra $\frak{U}$ has spectral radius $r(A)$ given by the formula $$r(A) = \lim_{n \to \infty} ||A^n||^{1/n}$$. In particular, $\{||A^n||^{1/n}\}$ has a limit. Here is the ...
2
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1answer
23 views

Strong operator topology on a $C^*$-algebra?

As far as I know, the Strong Operator Topology (SOT) is defined for the space of operators $B(H)$ for any Hilbert space H. The paper I am reading implicitly make references to the 'fact' that we can ...
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1answer
49 views

The unbounded antipode for Woronowicz's quantum group $\operatorname{SU}_q(2)$

For non-zero $q\in [-1,1]$, Woronowicz's quantum group $\operatorname{SU}_q(2)$ is given as the universal unital $\mathrm{C}^*$-algebra generated by elements $a,c\in C(\operatorname{SU}_q(2))$ subject ...
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0answers
21 views

Derivative of the residue logarithm of a formal pseudo-differential series

In my lecture notes in the proof of Adler's theorem they use $$ D_t(res \log A)= res (D_t(A) \circ A^{-1}). $$ Where $D_t$ is derivation of the differential field $F$ and $A$ is a formal pseudo-...
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1answer
13 views

Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
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1answer
36 views

Trace norm of rank one operator $x\otimes y$ for $x,y\in H$

Let $H$ be a Hilbert space. The trace norm on $B(H)$ is defined as $$\|u\|_{1}:=\operatorname{tr}(|u|):=\sum_{e\in E}\langle|u|(e),e\rangle,$$ where $|u|:=(u^{*}u)^{1/2}$ and $E$ is (any) orthonormal ...
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2answers
49 views

How to find vectors u,v such Su=Tv where S and T are linearly independent?

Let X be a Banach space, and S, T two surjective bounded operator linearly independent. How we can find two vectors u,v in X, linearly independent such that Su=Tv? X has infinite dimension.
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1answer
44 views

Generators for $K_1(A\otimes \mathbb{K})$

I've been working computing generator for several $C^*$-algebras involved in my Master's thesis, however I've got stucked with the generators of $K_1(C(\mathbb{T})\otimes\mathbb{K})$, which is in my ...
2
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1answer
20 views

Understanding the 'dilation' of Stinespring theorem

Stinespring's theorem states the following: Let $\mathfrak{A}$ be a unital $C^*$-algebra and $\Phi: \mathfrak{A} \rightarrow B(\mathcal{H})$ a completely positive map. Then there exists $\mathcal{K} ...
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2answers
26 views

Simple $C^*$-algebras are not commutative

Is it true that for simple $C^*$-algebras, meaning that they don't have non-trivial two-sided ideals, it holds that they are necessarily non-commutative? And why?
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16 views

$C^*$- systems with positive maps

Which is the main reason of defining a $C^*$- system as a pair of a $C^*$- algebra and a (unital) positive map $\Phi : \mathfrak{A} \rightarrow \mathfrak{A}$ ? What is the main reason behind choosing ...
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1answer
22 views

Algebra product on the commutative $C^*$-algebra $\mathbb{C}^N$

If we think of $\mathbb{C}^N$ as a commutative $C^*$-algebra, then it will be the algebra of continuous functions on $X=\{1,2, \dots, N\}$. So the only functions that are continuous are constants, (do ...

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