**4**

votes

**2**answers

336 views

### Operator Theoretical Models for $K(\mathbb{Z}, 3)$

I am looking for a reference concerning operator theoretical Models of $K(\mathbb{Z},3)$. Stolz-Teichner briefly say in "what is an elliptic object" that a certain hyperfinite Type ...

**4**

votes

**2**answers

246 views

### unitary reduction of $q$-normal matrices

The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...

**0**

votes

**0**answers

48 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 ...

**8**

votes

**1**answer

3k views

### Double Orthogonal Complement

Let $V$ be a complex inner product space. If $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in ...

**3**

votes

**1**answer

158 views

### An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group.
(i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$).
Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ?
Here, $\beta_{2}^1(G)$ stands ...

**8**

votes

**2**answers

123 views

### States and left ideals

Given a nontrivial left ideal $J$ of a unital $C^*$ algebra $A$, is there a state on $A$ which vanishes on all elements of $J$? (Left or right doesn't matter, just not 2-sided.)
The problem came ...

**10**

votes

**0**answers

207 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

**1**answer

133 views

### What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ ...

**2**

votes

**1**answer

60 views

### Approximation of the central support

Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.
a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ ...

**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

151 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 ...

**1**

vote

**0**answers

41 views

### Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues [migrated]

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...

**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 ...

**4**

votes

**0**answers

133 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 ...

**7**

votes

**1**answer

553 views

### A problem in functional calculus

This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...

**2**

votes

**1**answer

187 views

### Has a subfactor with lattice $B_3$, a singly generated identity biprojection?

Let $(N \subset M)$ be an irreducible finite index subfactor.
If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):
...

**1**

vote

**1**answer

98 views

### Comparison between spectra

Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$.
...

**3**

votes

**0**answers

82 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

453 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 ...

**4**

votes

**1**answer

208 views

### A question on complex line bundle over $S^{2}$

Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...

**8**

votes

**2**answers

279 views

### Complete regularity in C*-algebras

It is clear that commutative C*-algebras correspond to locally compact Hausdorff spaces. And locally compact Hausdorff spaces are completely regular. Now, does the complete regularity statement have ...

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

**4**

votes

**0**answers

130 views

### Discrete groups G whose full C*-algebra C*(G) is not MF?

This is a cheap rip-off of this question, but I am genuinely interested in an answer.
Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF?
Let us ...

**11**

votes

**1**answer

247 views

### Are free positive operators equivalent to almost-commuting operators?

Set $A:=C_0((0,1]) * C_0((0,1])$ (the free product C*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^*x=a$ ...

**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

164 views

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

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

**0**

votes

**0**answers

50 views

### On the tensor product of irreducible finite index subfactors

This post extends this question on maximal subfactors, admitting this answer of Feng Xu.
Statement: A tensor product of irreducible finite index subfactors has a non-obvious intermediate if and only ...

**7**

votes

**2**answers

186 views

### An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...

**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

187 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 ...

**1**

vote

**0**answers

43 views

### Is 6 the smallest index for an irreducible subfactor to have a principal graph with a multiplicity >1 edge?

The irreducible subfactor $(R^{S_3} \subset R)$, of index $6$, admits a principal graph with a multiplicity $2$ edge because the group $S_3$ admits an irreducible complex representation of dimension ...

**6**

votes

**1**answer

187 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**9**

votes

**2**answers

251 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, ...

**4**

votes

**1**answer

70 views

### second dual of minimal tensor products of $C^*$-algebras

Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?

**10**

votes

**1**answer

166 views

### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
...

**10**

votes

**0**answers

276 views

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

**7**

votes

**1**answer

166 views

### Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...

**11**

votes

**1**answer

185 views

### q-Virasoro and q-Heisenberg algebras

The literature has definitions (seemingly plural, though they might be linked) of a $q$-deformed Virasoro algebra. But is there any link of these to a $q$-deformed Heisenberg algebra? (Classically ...

**2**

votes

**1**answer

81 views

### Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...

**3**

votes

**0**answers

136 views

### Deformation and Representations

Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...

**3**

votes

**0**answers

39 views

### Isometric domain of a unital completely positive map with respect to $L_p$-norms

This question can be formulated for general ($\sigma$-finite) von Neumann algebras, but for me it is enough to consider matrix algebras.
So let $M$ be a matrix algebra and $\rho$ a faithful state ...

**5**

votes

**0**answers

109 views

### Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra?
In other words, of the following form ...

**0**

votes

**1**answer

194 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**12**

votes

**2**answers

498 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**2**

votes

**0**answers

161 views

### Infinite number of non-isomorphic von Neumann algebras with property Gamma?

A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element ...

**3**

votes

**1**answer

96 views

### Point-ultraweak limit of *-homomorphisms/cpc order zero maps

Suppose we have the following:
A C*-algebra $A$ and a von Neumann algebra $M$ (we can assume that $M$ is $\mathcal B(H)$).
A sequence of *-homomorphisms $\phi_i\colon A\to M$
an ultrafilter ...

**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 ...