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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86 views

Epimorphisms of locally compact spaces

Let $LCH$ be the category of locally compact Hausdorff spaces with proper continuous maps. Question. What are the epimorphisms in $LCH$? I suspect them to be surjective, but I haven't been able to ...
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2answers
102 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$|\omega(A)|^2\leq\|\omega\|\cdot\omega(A^*A)$$
4
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1answer
43 views

A necessary and sufficient criterion for an element of a multiplier $ C^{*} $-algebra to be positive.

I am trying to find a reference for the following assertion: Let $ A $ be a $ C^{*} $-algebra, and let $ M(A) $ denote its multiplier algebra. Then $ m $ is a positive element of $ M(A) $ if and ...
1
vote
1answer
48 views

norms of row matrices

Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be $k\times k$ sqaure matrices and assume $\|x_j\|\leqslant\|y_j\|$ for all $j=1,\ldots,m$ (the norm in $B(\ell_2^k)$). Now define the block matrices $x,y\in ...
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5answers
942 views

Are commutative C*-algebras really dual to locally compact Hausdorff spaces?

Several online sources (e.g. Wikipedia, the nLab) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the ...
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1answer
28 views

continuous functional calculus for nonunital $c^*$-algebras

In lecture we had the continuous functional calculus for unital $c^*$-algebras: Let $A$ be an unital $c^*$-algebra, $a\in A$ normal and let $$alg(a,a^*)=\overline{ \{ ...
11
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2answers
357 views

When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
0
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1answer
86 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
0
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1answer
14 views

How a trace of an ideal act on an element of the whole algebra?

Let A be a C* algebra with an ideal I. Suppose $\tau$ is a trace on I. Let $x\in I$. Then how to understand $\tau(range(x))$? i.e. how $\tau$ acts on the range projection of x?
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votes
0answers
26 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
0
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1answer
23 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
2
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0answers
42 views

Noncommutative manifold: Spectral triples on noncommutative quotients

I'm interested in taking the noncommutative quotient of a manifold, and endowing it with a kind of noncommutative smooth structure. More formally I'm interested in the question: is there a canonical ...
4
votes
0answers
40 views

Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost ...
3
votes
0answers
50 views

Theorem about irreducible representation of $C^*$-algebra

I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've ...
0
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0answers
13 views

Reference request for various proofs of Bott periodicity for operator algebra K-theory

I am aware that there are multiple proofs of Bott periodicity for operator algebra K-theory and I would like to ask for references for them. The approach modeled after Atiyah's from topology can be ...
1
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1answer
46 views

What $\mathbb{C}I$ means?

I've come across this expression $$ \mathbb{C}I $$ while studying operators algebras. $C^*$-algebras and AF-algebras, concretely. In Kenneth R. Davidson's book $\boldsymbol{C^*}$**-algebras by ...
0
votes
0answers
27 views

“+” operator placed as index

What is the meaning of $(a-b)_+$? In other words, what is meant by the "+" operator when it is placed as an index. If I am comparing for example two variables $a$ and $b$. So what is the value of ...
2
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0answers
77 views

Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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0answers
17 views

Question about Noncommutative quotients

I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are ...
0
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0answers
15 views

How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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1answer
30 views

Projection matrix (C* algebra.. but linear algebra question) [closed]

The subject is $C^*$-algebra, but I think my question might be linear algebra related type. I have a question from the book Operator Algebras Theory of C*-Algebra by Blackadar. On page 351, in the ...
10
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3answers
237 views

If there is already enough room to add all projections, does passing to matrices change anything?

Throughout, $A$ denotes a $*$-algebra. We always assume $A$ is representable in the sense that $A$ can be embedded into $B(H)$ for some Hilbert space $H$. The particular embedding is not important, ...
2
votes
1answer
60 views

K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
3
votes
2answers
56 views

Existence of central cover for a representation of a C*-algebra

I've been trying to learn the basics about the representation theory of C*-algebras and came across the following in Pedersen's C*-algebras and their Automorphism Groups: With each ...
3
votes
1answer
45 views

Set-theoretic questions about the definitions of crossed-product $ C^{*} $-algebras and group $ C^{*} $-algebras.

In his book Crossed Products of $ C^{*} $-Algebras, Dana P. Williams defines the crossed product of a $ C^{*} $-algebra $ A $ by a locally compact group $ G $ as the completion of $ {C_{c}}(G,A) $ ...
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votes
1answer
28 views

question about a proof on Murphy's book about $C^*$-algebras

I'm reading Murphys book "$C^*-$algebras and operator theory" and I have a question about a proof in chapter 3. The statement is (Theorem 3.1.8): Let I be a closed ideal in a $C^*$-algebra A. Then ...
3
votes
1answer
42 views

Projections on a Hilbert space

Suppose $P$ and $Q$ are self-adjoint projections on a Hilbert space such that $P+Q+\lambda I$ is a self-adjoint projection for some $\lambda \in \mathbb{R}$. Does it follow that $P$ and $Q$ commute?
5
votes
1answer
54 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
2
votes
1answer
43 views

A Lemma about the operator space

The following lemma comes from the book "C*-algebras Finite-Dimensional Approximations" by N.P. Brown and N. Ozawa P379 Lemma 13.2.3 Let $X_{i}\in B(H_{i})$ (i=1,2) be unital operator subspaces ...
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0answers
26 views

Primitive ideal space of C*(Z2*Z2)

Find the primitive ideal space, the center, a continuous field of $C^*(Z_2*Z_2)$. Here, $C^*(Z_2*Z_2)$ is the full group $C^*$-algebra. I know the definitions of all of them, but I'm having hard ...
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2answers
51 views

Universal properties of certain crossed products

So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. ...
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1answer
25 views

Is an ultrapower of the hyperfinite factor still hyperfinite?

Let $\mathcal{R}$ be the hyperfinite type $II_{1}$ factor and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Is it true that $\mathcal{R}^{\mathcal{U}}$ is never hyperfinite ? How can I see ...
2
votes
0answers
13 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
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0answers
21 views

Lifting invertible elements in a $C^*$-algebra connected to the identity

Let $A$ and $B$ be unital $C^*$-algebras and suppose that there is a surjective *-homomorphism $f:A\rightarrow B$. Then any invertible element in $B$ that is connected to $1_B$ can be lifted to an ...
2
votes
0answers
36 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$ [closed]

Let $a,b$ be elements of a unital C*-algebra $A$ with $0\leq a,b\leq 1$ (e.g., $a,b$ are projections). Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$? If $ab$ is normal ...
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3answers
49 views

Non-closed ideals in $C^*$-algebras

What is an example of an ideal in a commutative $C^*$-algebra that is not closed? If by chance every ideal in a commutative $C^*$-algebra is closed, how about in non-commutative $C^*$-algebras? ...
2
votes
1answer
46 views

A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
0
votes
0answers
32 views

Gelfand representation of the algebra $C^1([0,1])$

In Murphy's book about C* algebras, exercise question 1.10 asks the reader to show that the Gelfand representation of the algebra $C^1([0,1])$ is not surjective. Just before the reader is asked to ...
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votes
0answers
19 views

C* Algebra, f(x,z)

Let $A$ be a $C^*$ algebra, $x\in A$ and $||x|| < 1$. Let $f(x,z) = (1-x x^*)^{-\frac{1}{2}}(1+zx)$, $|z|=1, z\in \mathbb{C}$, $\mathbb{C}$ is the complex field. How to prove: $$ f(x,z)^* f(x,z) ...
1
vote
1answer
42 views

A question about $n$-dimensional operator space

Let $F_{n-1}$ be the free group of rank $n-1$ and $C^{*}(F_{n-1})$ be the universal group C*-algebra of $F_{n-1}$. And if $E_{n}$ is the $n$-dimensional operator space in $C^{*}(F_{n-1})$ spanned by ...
0
votes
0answers
8 views

Determining spectral bounds variationally.

I'm learning C0-semigroup theory (mainly from Arendt et al. (vector-valued Laplace-transforms and Cauchy problems), Engel & Nagel (One par. semigroups for linear evolution eq.),Evans (partial ...
3
votes
1answer
29 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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0answers
32 views

Why does one only consider one-parameter groups in Borchers-Arveson theorem?

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ ...
7
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2answers
481 views

A question about pure state

For every unit vector $x$ in a Hilbert space $H$,let $F_x$ be the linear functional on $\mathcal B(H)$ (bounded linear operators) defined by $F_x(T)=(Tx,x)$. Prove that each $F_x$ is pure state and ...
2
votes
0answers
14 views

amenable groups versus amenable graphs

In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ...
4
votes
1answer
120 views

Where does the double commutant theorem fails for $AW^*$-algebras?

Commutative $AW^*$-algebra are characterized as those $C^*$-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra). Von Neumann ...
2
votes
1answer
79 views

Morita equivalence and KK-theory

Let $A,B,C$ be $C^\ast$-algebras. Suppose $B$ and $C$ to be strongly morita equivalent. Then $KK(A,B)\cong KK(A,C)$. Could someone provide a reference or proof of this fact? I guess the ...
0
votes
1answer
58 views

Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
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16 views

States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
1
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1answer
35 views

spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...