**0**

votes

**0**answers

45 views

### Ideal structure of group $C^*$-agebras [on hold]

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ?
If G to be the group of integers $Z$ , then $C^*$($Z$)= C($T$) so because ideal structure of $ ...

**3**

votes

**1**answer

64 views

### Is the module action $M\times M^*\to M^*$ 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 ...

**9**

votes

**0**answers

81 views

### How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...

**5**

votes

**2**answers

182 views

### Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?

Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...

**4**

votes

**0**answers

104 views

### References for a lemma about compact operators on a Hilbert module

I am looking for a reference for the following result:
If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta ...

**0**

votes

**1**answer

75 views

### Equivalent projections in von Neumann algebras

Let $M$ be a von Neumann algebra in $B(H)$. Let $p$ and $q$
be projections in $M$. Assume that they are equivalent in $B(H)$, i.e there is a partial isometry $u$ in $B(H)$ with $p=uu^*$ and ...

**2**

votes

**0**answers

87 views

### Sequence of Projections in C*-Algebra

Claim: Let $A$ be a unital C*-Algebra and $q\in A$ a projection. Given a sequence $\left(p_n\right)_{n\in\mathbb{N}}\subseteq Ae$ of projections with $p_n\rightarrow q$ for $n\rightarrow \infty$ there ...

**3**

votes

**0**answers

151 views

### Uniqueness of the reduced free product of unital completely positive maps

For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = ...

**0**

votes

**0**answers

64 views

### Representation of a C*-algebra

Let $A$ be a C*-algebra. It seems that for given an *-representation $\pi$ of $A$, there is unique central projection $z_{\pi}$ in $A^{**}$ such that $\pi$ is just (unitary equivalent to) $\rho_z$ ...

**0**

votes

**1**answer

166 views

### The total variation of a complex measure

Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C_0(\Omega)^*$ , dual of the commutative C*-algebra $C_0(\Omega)$, is just the space of ...

**5**

votes

**1**answer

199 views

### $C^{*}$-correspondences viewed as generalized endomorphisms

I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...

**9**

votes

**1**answer

117 views

### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...

**0**

votes

**0**answers

76 views

### semifinite projection

Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...

**6**

votes

**0**answers

197 views

### Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...

**4**

votes

**0**answers

136 views

### Connectivity of the group of invertible elements of $C(S^{2})\otimes A$

For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group?
All finite dimensional $A$ satisfy this property.
Is it true to say ...

**3**

votes

**0**answers

94 views

### Extending Akemann's Non-Commutative Urysohn Lemma

Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections.
Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$?
Note if $||pq||=1$ this is immediate, ...

**10**

votes

**0**answers

225 views

### Almost idempotent approximate units in C*-algebras

As in Blackadar's "Operator Algebras" Definition II.4.1.1., call an approximate unit $(a_\lambda)$ in the positive unit ball of a C*-algebra almost idempotent if $a_\lambda a_\gamma=a_\lambda$ ...

**3**

votes

**0**answers

113 views

### Closed containment of open projections in C*-algebras

For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements.
$\overline{p}\leq q$
$p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$
$p\leq q$ ...

**7**

votes

**2**answers

474 views

### $H^{*}$ algebras as a generalization of $C^{*}$ algebras

Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties:
$\forall \lambda ...

**3**

votes

**1**answer

184 views

### simple and non nuclear $C^*$-algebra

Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?

**2**

votes

**1**answer

87 views

### Why is the ker-hull-topology on $Irr(A)$ is the discrete topology?

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that ...

**1**

vote

**1**answer

150 views

### Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...

**3**

votes

**0**answers

222 views

### Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...

**5**

votes

**1**answer

190 views

### Power's Theorem for irreducible representations

Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...

**9**

votes

**2**answers

260 views

### Is this a functor on the category of $C^{*}$ algebras?

The category of $C^{*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...

**7**

votes

**1**answer

254 views

### What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?

Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...

**7**

votes

**0**answers

188 views

### Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $

This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the ...

**7**

votes

**1**answer

159 views

### Hopf Galois extensions and conditional expectations for C* algebras

Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map ...

**5**

votes

**2**answers

115 views

### Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...

**5**

votes

**0**answers

224 views

### C$^*$-algebras isomorphic after tensoring

From the negative answer to this question we know that C$^*$-algebras that are isomorphic after tensoring with $M_n$ for all $n\geq 2$ need not be isomorphic. So what happens when we strengthen this?
...

**15**

votes

**1**answer

353 views

### The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...

**2**

votes

**1**answer

63 views

### short question about biduals of $C^\ast$-algebras

Let $A$ be a $C^\ast$-algebra. Consider the canonical embedding $A\to A^{**},\; a\mapsto i(a)$, such that $i(a)(a^*)=a^*(a)$ for all $a\in A$. Here is $A^{**}$ considered as a Banach space. It's well ...

**16**

votes

**2**answers

443 views

### C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$

In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes ...

**3**

votes

**2**answers

209 views

### map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$?

Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows ...

**1**

vote

**1**answer

90 views

### examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set ...

**6**

votes

**1**answer

199 views

### Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states:
Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on ...

**20**

votes

**2**answers

943 views

### Separating pure states on the $2\times 2$ matrix algebra

I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of ...

**6**

votes

**1**answer

274 views

### Does the following $ C^{*} $-algebraic result have a purely algebraic proof?

While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * ...

**4**

votes

**1**answer

290 views

### A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $

Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the ...

**3**

votes

**0**answers

66 views

### An estimate for the maximal C* norm in the group algebra of a free group

Let F--->G be an epimorphism of groups, F being finitely generated and free. Let H be its kernel. Consider a lifting i: G--->F of the epimorphism. Every element of C[G] is of the form
a=sum a(g) i(g) ...

**2**

votes

**2**answers

251 views

### unitization-process of unital- and non-unital $C^*$-algebras

I have a small question about unitization of (unital) $C^*$-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link ...

**3**

votes

**1**answer

139 views

### questions about the proof of the theorem of completely positive order zero maps

I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.
I want to understand the proof of the theorem (which you can find in the ...

**1**

vote

**1**answer

182 views

### Almost complex structure and nontrivial idempotents

Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...

**6**

votes

**0**answers

184 views

### A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...

**4**

votes

**1**answer

140 views

### A question about correlations between $ C^{*} $-algebras

I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras.
Definition. Let $ A $ and $ B $ be ...

**1**

vote

**1**answer

259 views

### Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]

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 ...

**1**

vote

**1**answer

153 views

### Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$.
Ultimately, I'm interested in finding a ...

**1**

vote

**0**answers

130 views

### Determining the primitive ideal space of C-star algebras

Is there a general way of finding a primitive ideal space of $C^*$-algebra?
For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how ...

**1**

vote

**1**answer

55 views

### Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold

Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...

**1**

vote

**1**answer

117 views

### A $C^{*}$ algebra associated to a graded $C^{*}$ algebra

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...