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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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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91 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 ...
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37 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 ...
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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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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 ...
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137 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 ...
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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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72 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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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 nuclearity?...
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62 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?
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70 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} ...
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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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28 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) ...
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27 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 ...
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2answers
56 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 ...
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1answer
53 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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72 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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48 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 $t \rightarrow\alpha_t$ ...
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2answers
169 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 ...
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45 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 ...
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1answer
42 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 ...
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131 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 ...
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Commutant of algebra of multiplication operators

Let $L^2(X)$ be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space $X$. Define $\mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}$, where $M_f$ is the the ...
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46 views

almost unital Banach algebra's

Let $A$ be an "almost unital Banach algebra", in the sense that it satisfies all the usual axioms but not necessarily that $\|1\|=1$. From the product inequality $\forall x,y \in A$ \begin{equation} \|...
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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 (non-degenerate)...
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37 views

Existence of certain subalgebras of $C(X)$

Suppose $A$ is a commutative Banach algebra.Knowing that the Gelfand transform is not surjective but injective does it imply that $A$ is not isomorphic to $C(X)$ ? by $M$ we mean the maximal ideal ...
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3answers
124 views

Characterisation of a Commutative C* Algebra which is an Integral Domain

Let $X$ be a compact hausdorff topological space with more than one element.Then prove that the ring $C(X)$ of complex valued continuous functions on $X$ is not an integral domain. Thanks for any help....
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35 views

Interpreting the lingo of a definition

The Terms I grew up with: A bounded linear operator $U$ on a Hilbert space $H$ is a partial isometry if there exists a subspace $M$ of $H$ such that $\|Ux\| = \|x\|$ for all $x\in M$, and $Ux = 0$ ...
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48 views

Involution on the Disc Algebra

We know that the disc algebra $A(D)$={$f$ in $C(DUBd(D))$: f is analytic on $D$} wher $D$ is the open unit disc in the complex plane is a Banach algebra under the usual sup norm.Is it possible to ...
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69 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
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69 views

Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and $\omega_{\phi}:\mathbb{C}[\Gamma]\rightarrow\mathbb{C}:\sum_{t\in\Gamma}...
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Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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Order zero maps in matrix algebra

Let $a$ and $b$ are two elements in a $C^*$algebra $A$. We say $a\perp b$ if $ab=ba=a^*b=ab^*=0$. We say a completely positive map $\phi: A \rightarrow B$ is of order zero if for any positive elements ...
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282 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 ...
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Existence of idempotents versus existence of projections in a C*-algebra

Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections? Clearly, when $x$ is ...
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78 views

What's the difference between a Banach Algebra and a C*-algebra?

I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at C*-algebras based on my interests. ...
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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? (...
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1answer
75 views

Norms on unitization of nonunital Banach algebra

Let $A$ be a nonunital Banach algebra and denote by $A^+$ the unitization of $A$. One commonly used Banach algebra norm on $A^+$ is given by $||(a,\lambda)||=||a||+|\lambda|$ (where $a\in A,\lambda\in\...
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$C^*$ algebra generated by a $C^*$ algebra and a group

In this article, "Spectral measures in C∗-algebras of singular integral operators with shifts", in chapter 3.1. They have a $C^*$ algebra $U$, and an unitary representation $\pi$ of a discrete group ...
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1answer
76 views

when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, $G=H\...
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104 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
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1answer
34 views

Are ideals generated by separable subspaces separable?

Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space? It looks to me ...
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1answer
49 views

C* Algebra Positivity

STATEMENT: This is a proof from one of Qiaochu's notes on $C^*$ algebras. Proof: Let A be a $C^*$ algebra.We now want to show that for any $c\in A$ we have $c^*c\geq 0$. Suppose otherwise.We know ...
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80 views

Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra

I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra. Thinking about it for 10 minutes, I didn't ...
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Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into $\mathscr{L}_\infty$...
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51 views

A question about induced $C^\ast$-algebra

Recently, I read the book Crossed Products of C*-algebras, and meat a question. The question is how to prove $\mathrm{Ind}_c(A,\alpha)$ is dence in $\mathrm{Ind}(A,\alpha)$. On the page 102, the ...
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38 views

Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions $...
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80 views

Partial Isometries: Characterization

Given a C*-algebra $\mathcal{A}$ Consider an element: $$J\in\mathcal{A}:\quad P:=J^*J$$ Then the equivalence holds: $$JJ^*J=J\iff P^2=P=P^*$$ How can I prove this?
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Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to \mathcal{B}(\...
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A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...