**4**

votes

**0**answers

76 views

### C$^*$-algebras in which the spectral radius is comparable to the norm

For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \...

**3**

votes

**1**answer

94 views

### Linear independency and compactness of the set of pure states of a $C^*$-algebra

Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...

**11**

votes

**2**answers

293 views

### Is a C*-algebra with an isomorphic predual a von Neumann algebra?

It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...

**7**

votes

**1**answer

250 views

### Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?

Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...

**6**

votes

**1**answer

349 views

### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective?
The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...

**16**

votes

**5**answers

3k views

### Reference: Learning noncommutative geometry and C^* algebras

I am starting to study noncommutative geometry and C^* algebras so my question is
Does anyone know a good reference on this subject?
I would like a basic book with intuitions for definitions and ...

**3**

votes

**1**answer

136 views

### The C*-envelope of the algebra of continuous functions on a compact topological space is commutative

In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:...

**3**

votes

**1**answer

75 views

### commutativity of a diagram in cohomology of $C^*$-algebras

The setting is the same as in my last question commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras) :
Let $A$ be in the bootstrap category (=N in the other ...

**11**

votes

**1**answer

173 views

### What does it tell us, if we know a unital C*-algebra has approximately inner (half-)flip?

This is a somewhat vague question, but I think it is not too open-ended and should admit well-circumscribed answers by specialists in operator algebras.$\newcommand{\Cst}{{\rm C}^*}$ It arises from ...

**4**

votes

**1**answer

99 views

### commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras)

I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the ...

**3**

votes

**0**answers

42 views

### bootstrap class ($C^*$-algebras): comparison of two definitions

I want to clarify the relationship between two (at first sight) different definitions of the bootstrap class for $C^*$-algebras, in order to understand which $C^*$-algebras satisfy the universal ...

**0**

votes

**0**answers

362 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 ...

**2**

votes

**2**answers

120 views

### Multiplier algebra of $A \otimes \mathcal{K}$

If $A$ is unital C$^*$-algebra, is it true that the multiplier algebra of $A \otimes \mathcal{K} $ is $ A \otimes \mathcal{B}(\mathcal{H})$? Where $\mathcal{K}$ is C$^*$-algebra of compact operators ...

**10**

votes

**5**answers

764 views

### If two projections are close, then they are unitarily equivalent

Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...

**3**

votes

**1**answer

97 views

### busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras".
Let $0\to B\to E\to A\to 0$ be a short exact ...

**6**

votes

**2**answers

449 views

### Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...

**5**

votes

**2**answers

234 views

### $id:A\to A^{op}$ is completely positive iff $A$ is abelian

Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive.
The claim is: $id$ is completely positive iff $A$ is abelian.
I need ...

**1**

vote

**1**answer

155 views

### Uniform Roe algebras and exact groups

Let $\Gamma$ be a discrete group.
Q: If $l^\infty(\Gamma)\rtimes \Gamma=l^\infty(\Gamma)\rtimes_r \Gamma$ canonically, can we conclude that $\Gamma$ is an exact group?
The converse implication is ...

**0**

votes

**0**answers

61 views

### A point on the absolute value of a bounded linear functional.

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...

**3**

votes

**0**answers

96 views

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

Let $F\twoheadrightarrow G$ be an epimorphism of groups, $F$ being finitely generated and free. Let $H$ be its kernel. Consider a lifting
$i:G\hookrightarrow F$ of the epimorphism.
Every element of $...

**5**

votes

**1**answer

119 views

### is the maximal tensor product of compact operators an essential ideal?

I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...

**3**

votes

**1**answer

125 views

### is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?

First of all, sorry for my bad english. I tried to find out whether the following statement is true or not:
Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...

**4**

votes

**4**answers

425 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 http://math....

**2**

votes

**0**answers

92 views

### Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...

**2**

votes

**0**answers

139 views

### The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...

**7**

votes

**2**answers

1k views

### The functoriality of group C* algebra structure

Let $G$ and $H$ be discrete groups and $f:G \rightarrow H$ be any homomorphism of these groups. I have three questions about it:
1) How to prove the functoriality of the construction of universal $C^*...

**1**

vote

**0**answers

81 views

### Lower bound for the $C^*$-unitisation norm?

Let $A$ be a $C^*$-algebra without unit. Its unitisation $A^+$ carries the $C^*$-norm
$$\|(a,\lambda)\|_{A^+} = \sup \{\|ax+\lambda x\|_A \;\big|\; \|x\|_A = 1 \},$$
which is the operator norm of $...

**1**

vote

**1**answer

219 views

### On the second dual of $C[0,1]$

I have two questions on the second dual of $C[0,1]$:
R. D. Mauldin ([1]) proved that: For a given bounded linear functional
$T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on ...

**1**

vote

**1**answer

127 views

### Representations of Calkin algebra

Let $H$ be a separable Hilbert space and consider the Calkin algebra $C(H)=\frac{B(H)}{K(H)}$.
Q) True or false: Any representation of $C(H)$ is a direct sum of irreducible representations.

**0**

votes

**1**answer

122 views

### A point-wise separation Hahn-Banach theorem in C*-algebras

Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$.
Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...

**7**

votes

**1**answer

209 views

### When does a $C^*$-algebra have no nonzero projection?

Let $A$ be a $C^*$-algebra and $\hat{A}$ its spectrum of $A$,the set of classes of non-zero irreducible representation of $A$ endowed with hull-kernel topology. suppose $\hat{A}$ is a non-compact ...

**1**

vote

**0**answers

60 views

### Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra

Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: \...

**7**

votes

**1**answer

196 views

### Is this a characterization of commutative $C^{*}$ algebras?

Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$
Is $A$ necessarily a commutative ...

**5**

votes

**1**answer

263 views

### A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity

In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...

**4**

votes

**1**answer

122 views

### $C(X)$ as finitely generated $C^*$-algebra

Let $X$ be a Hausdorff space. Suppose that $C(X)$ (or $C_0(X)$) is a finitely generated $C^*$-algebra. What we can say about $X$ ? For example can we characterize its inductive dimension, axioms of ...

**10**

votes

**1**answer

189 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 ...

**6**

votes

**1**answer

281 views

### Weakly amenability and exactness for discrete groups

A countable discrete group $\Gamma$ is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions $(\phi_n)$ on $\Gamma$ such that $\phi_n\...

**1**

vote

**0**answers

167 views

### A functor on the category of rings, algebras or compact Hausdorff topological space

Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...

**4**

votes

**2**answers

271 views

### The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...

**6**

votes

**2**answers

253 views

### Generators of the $ K_{0} $-group of the non-commutative torus $ A_{\theta} $ with $ \theta \in \mathbb{Q} $ (i.e. rational rotation algebra)

I am studying the non-commutative torus $ A_{\theta} $.
When $ \theta $ is irrational, $ {K_{0}}(A_{\theta}) $ is generated by $ [1] $ and $ [p_{\theta}] $.
(Note: $ p_{\theta} $ is a projection in $...

**12**

votes

**1**answer

269 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 $a_\lambda a_\gamma=a_\lambda$ ...

**8**

votes

**1**answer

172 views

### Real rank zero of group $C^*$-algebras

The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...

**2**

votes

**1**answer

390 views

### Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be (...

**1**

vote

**0**answers

96 views

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

Let $G$ be a locally compact groups and $C_r^*(G)$ be a reduce group $C^*$-algebra.
$\ Question:$What is the ideal structure of reduce group $C_r^*(G)$?

**4**

votes

**0**answers

148 views

### Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...

**8**

votes

**0**answers

145 views

### Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...

**3**

votes

**0**answers

118 views

### The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment.
Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...

**1**

vote

**1**answer

128 views

### When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...

**3**

votes

**2**answers

71 views

### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann ...

**3**

votes

**0**answers

61 views

### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...