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Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

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358 views

Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
9
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131 views

Isomorphic matrix algebras with non-isomorphic C*-algebras

Let $A$ and $B$ be two $C^{\ast}$-algebras which their matrix algebras, $M_2(A)$ and $M_2(B)$, are $\ast$-isomorphic $C^\ast$-algebras. Question 1: Are $A$ and $B$ isomorphic $C^\ast$-algebras? In a ...
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148 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
7
votes
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152 views

Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
7
votes
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188 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
6
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74 views

Cuntz algebra and Schur-Weyl duality

Let $\mathcal{O}_n$ be the Cuntz algebra with generators $a_1,...,a_n$. We can define an action of $U(n,\mathbb{C})$ (the group of $n\times n$ unitary operators) on $\mathcal{O}_n$ in a very natural ...
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129 views

Understand representations of c*-algebras from a categorical point of view

In my lecture on von Neumann algebras we have defined a representation of a c*-algebra $\mathcal{A}$ as a *-homomorphism $\pi$ into $\mathcal{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$. ...
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100 views

Projections in C*-algebras

Let $A$ be a C*-algebra and $A_0$ a dense *-subalgebra. Is there a condition which ensures that all projections in $A$ already lie in $A_0$ ? In particular, I think of $A$ being an inductive limit of ...
5
votes
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140 views

A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
5
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62 views

Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
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73 views

Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$ (but not necessarily WOT-closed). What one has to assume about $A$ and $B$ to ensure that if $M\subset A$ is a maximal ...
4
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102 views

Closeness of points in the irreducible decomposition of a C$^{*}$-algebra representation

Suppose $X$ and $Y$ are compact metric 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 ...
4
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51 views

A separable $C^*$ algebra which contains all separable $C^*$ algebras.

Is there a unital separable $C^*$ algebra which unitaly contains all unital separable $C^*$ algebras? The motivation is that the answer is positive in the commutative case since every compact ...
4
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43 views

An iff condition for the existence of a $\Gamma$- invariant sub-algebra of $C(X)$

There is an action of $\Gamma$ on a compact Hausdorff space $X$. The question is to find an iff condition for the existence of a $\Gamma$ invariant sub-algebra of $C(X)$. I started out with the ...
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35 views

Elliott-Natsume-Nest proof of Bott periodicity for $K$-theory of $C^\ast$-algebras

The following is an exercise in the book "$K$-theory and $C^\ast$-algebras" by Wegge-Olsen: Let $S$ denote the suspension functor. Suppose that we have natural transformations $\Phi^0$ from the ...
4
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103 views

Evaluating Gibbs state in the second quantized formalism

First, let us fix our notation. If $A:\mathcal H \to \mathcal H$ is a linear operator on a single-particle Hilbert space $\mathcal H$, we can lift $A$ on the Fock space $\mathfrak F_{s/a}(\mathcal H)$ ...
4
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37 views

tensorial product of C*-algebras and adjointness

Given $M,N$, $R$-modules, it is standard verification that $-\otimes Z$ is left adjoint to $Hom(Z,-)$. If we have more structure, particularly if $\mathcal{A}$ and $\mathcal{B}$ are C*-algebras, ...
4
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306 views

Are the continuous functions dense in the set of bounded measurable functions?

Let $X$ be compact and Hausdorff, and let $\mathcal{B}(X)$ be the set of bounded Borel-measurable functions $X \to \mathbb{C}$. Also let $\mathcal{C}(X)$ be the set of continuous functions $X \to \...
4
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49 views

Associated C*-Categories and the reduced crossed product

Let $G$ be a discrete group and $\mathcal{G}$ a groupoid, that is a small category in which every arrow is an isomorphism. Wolfgang Lück explains how we can construct a $C^*$-category from $\mathcal{G}...
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96 views

State space of $C([0,1])$

Consider the $C^*$-algebra $A := C(X)$, where $X$ is a compact Hausdorff space. Denote by $S(A)$ the state space of $A$. In the weak$^*$ topology this is a convex compact set such that $S(A)$ is the ...
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105 views

Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
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votes
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116 views

Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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95 views

$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as C*-...
4
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229 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
4
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0answers
189 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
115 views

Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
4
votes
0answers
177 views

Is exponential function in a C*-algebra injective on self-adjoint elements?

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?
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Representations of a C*-algebra of bounded Borel functions

Let $X$ be a compact Hausdorff space. Let $B(X)$ be the C*-algebra of bounded Borel measureable functions on $X$ (under the supremum norm). I am curious whether the (say unital) $*$-representations of ...
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27 views

Regarding equality of norms in Hilbert bimodules

I was reading the book Elements of noncommutative geometry and in page 160 lemma 4.21 the authors state that in a Hilbert B-A bimodule $E$ the two norms induced by the two inner products coincide. ...
3
votes
0answers
41 views

Real content of Chebyshev sum inequality

There's well-known Chebyshev's sum inequality. I'm looking for possible generalization of this statement; problem is that I'm not sure what is the right direction, but here's one which I'd prefer. ...
3
votes
0answers
81 views

In the construction of the free product of C*-algebras, is the seminorm from which we quotient actually a norm?

Context Let $(A_\lambda)_{\lambda \in \Lambda}$ a family of unital C*-algebras. I am trying to see that they have a coproduct. In order to construct it, I understand that we take the coproduct of $(...
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votes
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120 views

A precise definition of contractible Banach algebras.

I am sorry if this question is elementary: What is a precise definition of a contractible Banach algebra? What is my mistake to think that there is a possible (conflict) contradictory ...
3
votes
0answers
74 views

Is the bidual of a C*-algebra isomorphic to the universal enveloping von Nemann algebra as a Banach algebra?

Let $A$ be a C*-algebra and $(\pi, H)$ a universal representation of $A$. We know that there is a linear isomorphism $\tilde{\pi}$ between the bidual $A^{**}$ of $A$ and the universal enveloping von ...
3
votes
0answers
79 views

Spectral decomposition of hilbert modules

Let $A$ be a $C^*$-algebra, $E$ a Hilbert $A$-module and $T:E\rightarrow E$ a bounded self-adjoint regular operator. Is there something like a spectral theorem for Hilbert modules that gives a ...
3
votes
0answers
105 views

What pre-requisites does non commutative geometry has?

I'm a masters student currently deciding in which area should I focus on. So far my primary interest has been C* algebras and operator algebras (already have some knowledge on K-theory for C* algebras ...
3
votes
0answers
60 views

Any reference concerning factorization of partial isometries in $B(H)$

Let $H$ be a Hilbert space. An operator $x\in B(H)$ is called a partial isometry if it is an isometry on Ker$x^{\perp}$. Two well-behaved classes of partial isometries are maximal partial isometries (...
3
votes
0answers
95 views

Lifting of matrix units

My question is originally about the analysis of extensions of $M_n$ by $K(H)$,"$K$-theory for operator algebras"- Blackadar, Exmaple $15.4.1.(b).$ We examine extensions of $M_n$ by $K(H)$, so that we ...
3
votes
0answers
90 views

Positive Projections on Matrix Subspaces

Let $\mathcal{M}_d$ be the space of complex valued $d\times d$ matrices, and $\mathcal{S}\subset \mathcal{M}_d$ a subspace. Let $P_{\mathcal{S}}$ be a projection onto $\mathcal{S}$. My questions is ...
3
votes
0answers
81 views

Approximate Unit: Boundedness

Given a Banach algebra $A$. Consider a onesided approximate unit: $$e:\Lambda\to A:\quad ae_\lambda\stackrel{\lambda}\to a\quad(\forall a\in A)$$ Does it follow that it is eventually bounded: $$\...
3
votes
0answers
51 views

If $H\subseteq G$ is a subgroup then $C^*(H)\subseteq C^*(G)$

This is a proposition $2.5.8$ in Brown&Ozawa: Let $H$ be a subgroup of $G$. There is a canonical inclusion $C^*(H)\subseteq C^*(G)$. By universality we have a canonical $*$-homomorphism $\pi: C^*...
3
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0answers
61 views

A problem on exact $C^\ast$-algebra with a finite-dimensional operator subsystems

I am stuck in problem 3.9.5 in the book of Brown and Ozawa (C$^*$-algebras and Finite-dimensional Approximations). The problem is: Let $A\subset B(H)$ be exact and let $E\subset A$ be a finite ...
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votes
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82 views

Orthogonal $*$-homomorphisms satisfy $(\phi+\psi)_*=\phi_*+\psi_*$?

Let $A,B$ be $C^*$ algebras and $\phi, \psi : A\to B$ be *-homomorphisms. We say that they are mutually orthogonal if for all $x,y\in A$ we have $\phi(x) \psi(y)=0$. It can be shown then that $\phi +...
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votes
0answers
48 views

Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for $M_\...
3
votes
0answers
68 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
3
votes
0answers
74 views

Special case of Green's imprimitivity theorem and related question

Consider a locally compact group $G$ and a closed subgroup $H$ of $G$, and let $G$ act on $G/H$ by left translation. Green's imprimitivity theorem implies that the crossed product $C_0(G/H)\rtimes G$ ...
3
votes
0answers
63 views

Is the bilinear map $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
3
votes
0answers
96 views

Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} \...
3
votes
0answers
38 views

Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
3
votes
0answers
34 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle f.a,F\...
3
votes
0answers
413 views

$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...