**1**

vote

**0**answers

77 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 ...

**2**

votes

**0**answers

40 views

### Under what conditions is the primitive dual space of [SIN] group a Hausdorff space?

Recall 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 compact ...

**1**

vote

**1**answer

97 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 ...

**3**

votes

**0**answers

57 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 ...

**4**

votes

**0**answers

39 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 ...

**4**

votes

**1**answer

130 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 ...

**0**

votes

**1**answer

104 views

### adjoint of the operator rotation [closed]

I need to calculate the adjoint of the operator
$T_{a}=a i(x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x})$ with $i^{2}=-1, \quad a\in \mathbb{C}$ and domain
$D=\{\varphi\in L^{2}(R), ...

**6**

votes

**0**answers

139 views

### K theory as the fundamental group

There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...

**2**

votes

**1**answer

79 views

### About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...

**5**

votes

**0**answers

98 views

### Automorphisms of Cuntz algebra

Suppose, $ O_{\infty} $ is the cuntz algebra generated by the orthogonal isometries $ \{S_i\}_{i\in \mathbb{N}} $,i.e. $ S_i^*S_j=\delta_{ij}$ and $ O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}}) $.
Then ...

**4**

votes

**1**answer

65 views

### Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?

(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...

**3**

votes

**0**answers

59 views

### A continuous functional calculus on/positive elements in a Fréchet algebra?

I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all ...

**6**

votes

**1**answer

134 views

### Morita equivalence base equivalence relation for discrete groups

In the context of "discrete groups", is there an equivalence relation that implies the Morita equivalence of their reduced group C*-algebras?
We define $G \sim H$ for discrete groups $G$ and $H$, ...

**8**

votes

**0**answers

44 views

### Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?

Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...

**7**

votes

**3**answers

856 views

### Generalizations and relative applications of Fekete's subadditive lemma

Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...

**3**

votes

**1**answer

463 views

### Trace vs. state in von Neumann algebras

The question may be not appropriate with the title, since I do not know how to name it. I apologize.
Let $M$ be a finite $II_1$ factor, $\tau$ be the canonical trace. Let $p, q$ be two projections ...

**1**

vote

**1**answer

204 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

210 views

### Weak topology restricted to the unitary group of a von Neumann algebra

Consider the weak topology in a von Neumann algebra (weak in the sense of Banach spaces).
Does this topology coincide with the rest of the "weak" topologies when restricted to the unitary group?

**11**

votes

**0**answers

318 views

### Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...

**13**

votes

**1**answer

532 views

### A left inverse for the comultiplication on a Hopf von Neumann algebra

Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.
$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...

**3**

votes

**0**answers

138 views

### A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: ...

**1**

vote

**1**answer

112 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.

**4**

votes

**1**answer

130 views

### Most natural equivalence between $C^*$-algebras in NCG

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism.
Can someone explain this sentence or ...

**2**

votes

**1**answer

205 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):
...

**6**

votes

**1**answer

516 views

### Approximate units from strictly positive elements in $C^{*}$-algebras.

The existence of a countable approximate unit in a $C^{*}$-algebra $B$ is equivalent to the existence of a strictly positive element $h\in B$. There are several ways to construct an approximate unit ...

**0**

votes

**1**answer

118 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 $K(H)$ ...

**4**

votes

**2**answers

182 views

### On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces

I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy ...

**6**

votes

**1**answer

162 views

### von Neumann algebras as C*-algebras with multiplicative conditional expectation $A^{**}\to A$

Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions:
(1) $A$ is a von Neumann algebra.
(2) There is a multiplicative ...

**6**

votes

**1**answer

197 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 ...

**2**

votes

**0**answers

179 views

### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

**5**

votes

**2**answers

172 views

### Help in understanding result from publication on operator theory

in my research on dilations of contractions on Hilbert spaces and manifolds I have come across this nice publication concerning the classic Sz-Nagy theorem on the Arxiv by Levy and Shalit which states ...

**11**

votes

**1**answer

148 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 ...

**1**

vote

**0**answers

54 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: ...

**5**

votes

**1**answer

168 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

421 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

**4**

votes

**1**answer

260 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 ...

**6**

votes

**1**answer

117 views

### The quantum group SUq(n) as von Neumann algebra

i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...

**10**

votes

**3**answers

457 views

### Separable von Neumann algebra

What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...

**3**

votes

**0**answers

127 views

### isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...

**1**

vote

**0**answers

114 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 ...

**11**

votes

**2**answers

280 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
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**10**

votes

**1**answer

168 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 ...

**1**

vote

**0**answers

108 views

### Connecting homomorphism in generalized cohomology theory

I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence
$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ...

**7**

votes

**2**answers

653 views

### subfactor of finite rank but infinite index: is this possible?

A subfactor $N\subset M$ is essentially the same thing as an $N$-$M$-bimodule.
I'll recall the basic definitions in the language of bimodules, and I hope that subfactor people will excuse me.
...

**6**

votes

**1**answer

267 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 ...

**8**

votes

**1**answer

735 views

### The saturation of Murray von Neumann relation

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version:
Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff ...

**1**

vote

**1**answer

209 views

### Diagonalizable unitary operators [closed]

Let $u\colon H\to H$ be a unitary operator on a separable Hilbert space $H$ and let $(e_n)_n$ be a fixed orthonormal basis in $H$. Is it possible to decompose $u$ as $u=v^*dv$ where $v$ is a unitary ...

**8**

votes

**1**answer

101 views

### $\mathcal{O}_{\infty}$ and $\mathcal{Z}$ stable isomorphism as equivalence

If $O$ is a (strongly ?) self absorbing $C^*$-algebra, one has an equivalence relation on (separable) $C^*$-algebras: "being $O$-stably isomorphic" i.e. $A$ and $B$ are $O$-stably isomorphic if and ...

**4**

votes

**0**answers

65 views

### $p$-operator space structure on Banach algebras

There is an abstract characterization of operator algebras, which says that if $A$ is an operator space that is also an approximately unital Banach algebra, then the following are equivalent:
For ...

**2**

votes

**2**answers

180 views

### What is the dual of an semidefinitely representable (SDR) cone?

The Question
Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product.
Let $\mathcal K\subset V$ be a cone defined as
$$
\mathcal K=\Big\{x\in V\ ...