**3**

votes

**1**answer

93 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:...

**0**

votes

**1**answer

38 views

### A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result:
For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...

**4**

votes

**0**answers

62 views

### Equivalence of questions regarding restrictions of pure states

In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...

**2**

votes

**0**answers

60 views

### Approximate unit in C*-algebra with additional properties

In the book about K-theory (a friendly approach) by Wegge and Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(...

**9**

votes

**2**answers

299 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 \...

**10**

votes

**0**answers

91 views

### On spatial tensor products of von Neumann algebras

Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras.
We write $\odot$ for the algebraic tensor product,
and $\bar\otimes$ for the spatial tensor ...

**4**

votes

**1**answer

134 views

### isomorphism of noncommutative tori

I have a kind of vague question.
Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't ...

**2**

votes

**2**answers

743 views

### Does a conditional expectation from a von Neumann algebra to its center exist?

In a finite von Neumann algebra, the unique tracial state serves as one, then for a general von Neumann algebra, does it exist?

**11**

votes

**1**answer

170 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 ...

**5**

votes

**0**answers

104 views

### Compactum of Banach algebra

What is an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties?
There exists a bounded approximate identity in $I$ for $I$ i.e., a net $\{e_\alpha\}...

**1**

vote

**0**answers

107 views

### On self-dual group-subgroup subfactors

Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction.
The subfactor $(N \subset M)$ is called self-dual if it is isomorphic to ...

**5**

votes

**1**answer

153 views

### Convergence of functionals on compact projections on a separable Hilbert space

Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...

**5**

votes

**0**answers

80 views

### Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...

**2**

votes

**0**answers

70 views

### On finite index infinite depth subfactors and reduction to depth 2

Let $(N \subset M)$ be a finite index irreducible subfactor (with $N$ and $M$, ${\rm II}_1$ factors).
Let $N \subset M \subset M_1 \subset M_2 \subset \cdots$ be the tower of basic constructions.
...

**8**

votes

**3**answers

330 views

### What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...

**3**

votes

**1**answer

71 views

### Is there no extra intermediate subfactor for the basic construction?

Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, the basic construction is $N \subset M \subset M_1 = \langle M , e^M_N \rangle$.
Question: For any intermediate subfactor $N \subset P \...

**2**

votes

**2**answers

104 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

756 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 ...

**2**

votes

**0**answers

127 views

### Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices

A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...

**4**

votes

**1**answer

182 views

### Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...

**1**

vote

**0**answers

107 views

### Is there a non-integer in the dimension spectrum for the Heisenberg group?

Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group.
Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...

**2**

votes

**1**answer

71 views

### Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...

**6**

votes

**2**answers

426 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 ...

**3**

votes

**3**answers

471 views

### Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...

**2**

votes

**0**answers

96 views

### Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...

**5**

votes

**2**answers

229 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 ...

**6**

votes

**1**answer

284 views

### Decomposability of positive maps

By results of Størmer and Woronowicz, every positive map $\Phi \colon \mathcal{M}_{d \times d} \rightarrow \mathcal{M}_{d' \times d'}$ for $dd' \leq 6$ can be decomposed as a convex combination
$$\...

**7**

votes

**1**answer

172 views

### K theory for pre $C^*$-algebras

In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...

**1**

vote

**0**answers

53 views

### Tor functor in the case of algebra of smooth functions

Let $A=C^{\infty}(\mathbb{S}^{1})$, let $B$ be the sub-algebra $C^{\infty}(0,1)$. Here we identify $\mathbb{S}^{1}$ by $\mathbb{R}/2\pi \mathbb{Z}$. I want to ask if there is a way I can decompose $A$ ...

**1**

vote

**0**answers

66 views

### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...

**4**

votes

**1**answer

79 views

### Terminology: jointly completely bounded?

This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those ...

**1**

vote

**1**answer

151 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 ...

**3**

votes

**0**answers

64 views

### Algebras involved to define spectral triple

A spectral triple consist from $(A,H,D)$ where $A$ is some unital $*$-subalgebra of $B(H)$ and $D$ is unbounded operator with compact resolvent such that for all $a\in A$ the commutatot $[D,a]$ is ...

**3**

votes

**0**answers

51 views

### Isometry from a representation to the representation tensored with itself

Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $.
(The group $ S(2^{\infty}) $ is the direct limit of the following ...

**5**

votes

**1**answer

114 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

119 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 ...

**3**

votes

**2**answers

123 views

### The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on:
If $ \mathbb{H} $ is an RKHS and we denote the ...

**1**

vote

**0**answers

58 views

### weak convergence in operator space structure

Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...

**2**

votes

**0**answers

60 views

### (Topological) K-theory for commutative $C^*$-algebras: operator and standard approaches

Let $A$ be a commutative unital $C^*$ algebra. Then $A=C(X)$ for some compact Hausdorff space $X$. Topological $K$-theory group (namely $K_0$) is defined in terms of vector bundles as a Grothendieck ...

**4**

votes

**3**answers

375 views

### What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$?
I think it would be the permutations of the integers. Is this right?

**7**

votes

**2**answers

297 views

### “Identity tensor transpose” as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Equipping $M_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...

**7**

votes

**1**answer

135 views

### Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...

**2**

votes

**1**answer

220 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):
...

**16**

votes

**1**answer

625 views

### Explicit path in the unitary group of a $C^*$-algebra

For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...

**13**

votes

**2**answers

453 views

### A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**6**

votes

**1**answer

117 views

### Two notions of bundle of C* algebras

One can find in the literature two notions of $C^*$-algebra over a topological space $X$.
The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a $C^*$-...

**6**

votes

**1**answer

167 views

### Time-Energy Uncertainty Relation in relativistic Quantum Mechanics

There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation.
Unfortunately, in QM time is not an operator like space, ...

**2**

votes

**0**answers

87 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

135 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 ...

**1**

vote

**1**answer

121 views

### Does a $W^*$ envelope exist?

I know that an operator algebra $A$ has a "minimal" $C^*$-algebra $C$ containing $A$, which is known as the $C^*$ envelope of $A$. The existence of such a minimal $C^*$-algebra generated by $A$ (a ...