**5**

votes

**0**answers

36 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

42 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 ...

**-1**

votes

**0**answers

46 views

### finite and properly infinite projections [closed]

If $M$ is a Von Neumann algebra and $p$ is projection in $M$ then there is a central projection $z \in M$ such that $ pz $ is finite and $p(1-z)$ is properly infinite.

**10**

votes

**0**answers

200 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 $a_\lambda a_\gamma=a_\lambda$ ...

**1**

vote

**1**answer

89 views

### Quick question about conjugate equivalence bimodules and inner products

Let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, ...

**6**

votes

**0**answers

141 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

133 views

### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...

**10**

votes

**0**answers

325 views

### Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
Is the tensor product of two path connected algebra, a path ...

**3**

votes

**0**answers

128 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

81 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, ...

**7**

votes

**2**answers

452 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

**0**answers

107 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 $r\vee q=1$
$p\leq q$ ...

**1**

vote

**1**answer

124 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 ...

**15**

votes

**0**answers

265 views

### Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group.
General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal
coefficient theorem (UCT)?
I am mainly ...

**3**

votes

**1**answer

163 views

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

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

**2**

votes

**1**answer

82 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

**0**answers

150 views

### Connected component of the identity in graded Banach algebras

I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as ...

**3**

votes

**0**answers

129 views

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

Edit: According to the comment of Qiaochu Yuan I realize that $\mathbb{C}^{2}$ is a counter example. So I add the assumption "simplicity" to this edited version
Note: In this post, the cyclic ...

**5**

votes

**1**answer

186 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

250 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, ...

**0**

votes

**0**answers

335 views

### Amenability of an “almost Hamiltonian” group

here is another interesting question that I can't answer on my own.
Let G be a countable, discrete group such that for any subgroup H of G and any element s of G we have: [H : sHt] is finite and [H ...

**7**

votes

**1**answer

239 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

169 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

143 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 ...

**4**

votes

**0**answers

218 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?
...

**5**

votes

**2**answers

108 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 ...

**15**

votes

**1**answer

343 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 ...

**20**

votes

**2**answers

929 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 ...

**15**

votes

**2**answers

414 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 ...

**2**

votes

**1**answer

58 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 ...

**3**

votes

**2**answers

198 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

85 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 ...

**5**

votes

**1**answer

190 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 ...

**12**

votes

**1**answer

938 views

### Matrices with entries in a $C^*$-algebra

Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution ...

**6**

votes

**1**answer

265 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

283 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 ...

**9**

votes

**2**answers

562 views

### General recipe for building C*-algebras out of combinatorial object

I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...

**2**

votes

**2**answers

236 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 ...

**7**

votes

**1**answer

209 views

### allowing `discontinuous functions' into a C* algebra

There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any ...

**3**

votes

**0**answers

64 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) ...

**1**

vote

**1**answer

177 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 ...

**3**

votes

**1**answer

134 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 ...

**9**

votes

**3**answers

903 views

### Relative K-theory and split exact sequences of C* algebras

Let $A$ be a C* algebra, $J$ an ideal, $\pi: A \to A/J$ the quotient map. Recall that the relative K theory group $K_0(A, A/J)$ consists of equivalence classes of triples $(p,q,x)$ where $p$ and $q$ ...

**6**

votes

**0**answers

182 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 $A\subset B$ ...

**6**

votes

**1**answer

156 views

### Relation between graphs and groupoid $C^*$-algebras

In the paper "Graphs, groupoids and Cuntz-Krieger algebras" by Kumijan, Pask, Raeburn, Renault it was shown (if I understand it correctly) that whenever $G$ is a row-finite directed graph
with no ...

**4**

votes

**1**answer

138 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 ...

**6**

votes

**2**answers

1k views

### Space of compact operators

I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...

**0**

votes

**1**answer

238 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 ...

**5**

votes

**2**answers

245 views

### fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...

**1**

vote

**1**answer

144 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 ...