Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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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 ...
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100 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
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101 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, ...
4
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1answer
227 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 ...
12
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1answer
260 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 $a_\lambda a_\gamma=a_\lambda$ ...
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176 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 ...
3
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117 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 $r\vee q=1$ $p\leq q$ ...
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2answers
508 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 ...
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1answer
205 views

simple and non nuclear $C^*$-algebra

Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?
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61 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 ...
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229 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 ...
11
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1answer
261 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 $x^*x=a$ ...
4
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1answer
88 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 $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?
6
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1answer
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 ...
5
votes
1answer
193 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 ...
8
votes
1answer
189 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 ...
12
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1answer
213 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 ...
3
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1answer
94 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 ...
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138 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 $U_q(sl_n)$ ...
10
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2answers
277 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, ...
11
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1answer
239 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 ...
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45 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 ...
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337 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 ...
15
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551 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 ...
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2answers
597 views

Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the classical Borsuk Ulam theorem for the Euclidean n-sphere, $n>2$ that is based only on the theory of Banach algebras. I checked on MR but had no ...
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197 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 ...
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187 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
1answer
102 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 ...
3
votes
1answer
140 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 $2$ ...
7
votes
1answer
163 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 ...
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2answers
291 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 ...
5
votes
1answer
170 views

Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...
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200 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 ...
4
votes
1answer
232 views

Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$. Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...
6
votes
1answer
507 views

An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
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233 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? ...
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600 views

Schur's Lemma for Hilbert spaces

Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...
7
votes
1answer
353 views

Fredholm operators in $K$-theory?

Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...
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475 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 ...
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Rank–nullity theorem for finite von Neumann algebras

The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$ Let $M \subset B(H) $ be a finite von Neumann ...
4
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2answers
404 views

$C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism. I ...
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1answer
157 views

Faithul map and (minimal) tensor product of $C^*$-algebras

Let $f$ be a faithful state on a $C^*$-algebra $A$, i.e. $f(a^*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^*$-algebras faithful if $T(a^*a)=0$ implies $a=0$. How to prove ...
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94 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 ...
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1answer
165 views

A relative property gamma and $L(\mathbb F_2)$

Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?
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An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$. The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$. A ...
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968 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 ...
7
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1answer
578 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 ...
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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 ...
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95 views

Cuntz comparison of strictly positive elements in finite C*-algebras

Let $A$ be a finite, non-unital C*-algebra, $s\in A$ a strictly positive element and $a\in A$ a positive element that is Cuntz-equivalent to $s$, i.e. there exist sequences $\{x_n\},\{y_n\}\subset A$ ...
4
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1answer
243 views

reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...