# Tagged Questions

**6**

votes

**2**answers

176 views

### Literature on “real” $C^*$-algebras

I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper
D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98,
which cites
G.G. ...

**5**

votes

**2**answers

321 views

### Metrics on the space of $C^{*}$ algebras

I think that there is a metric on the huge space of all $C^{*}$ algebras. What is the explicit
definition of this metric?may you introduce me a reference?
Moreover is the restriction of this ...

**7**

votes

**1**answer

190 views

### Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?

So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following.
Theorem The universal $C^*$-algebra generated by one element ...

**2**

votes

**1**answer

104 views

### Reference request for a type III action of a group on a manifold

Let an action of a group $\Gamma$ on a manifold $M$ such that $L^{∞}(M)⋊Γ$ is a type $III$ factor.
André Henriques posted here the following comment :
I don't know the literature, so I can't ...

**4**

votes

**1**answer

123 views

### Passing automorphism group through a representation

This is a very general question-- I'm really after any references to anything similar...
Let $A$ be a $C^*$-algebra equipped with a continuous one-parameter group of automorphisms ...

**6**

votes

**1**answer

281 views

### Tannaka duality for C*-algebras?

Tannaka-Krein
duality shows
how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional
complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to
...

**1**

vote

**0**answers

297 views

### Hans Saar's thesis

I would love to have a look on some results which are claimed by some people to be in Saar's thesis:
H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen ...

**1**

vote

**1**answer

799 views

### PhD in operator algebras and non-commutative geometry [closed]

I do not know whether it is a good place to ask this question or not.
I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...

**3**

votes

**0**answers

103 views

### reference for direct finiteness of the ring of affiliated operators

Let $\Gamma$ be a group, $N(\Gamma)$ its group von Neumann algebra,
$\newcommand{\cUG}{{\mathcal U}(\Gamma)}$
and $\cUG$ the ring of all densely-defined, closed operators ...

**3**

votes

**1**answer

151 views

### When is a $*$-homomorphism between multiplier algebras strictly continuous?

(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad ...

**14**

votes

**0**answers

386 views

+200

### 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) ...

**5**

votes

**1**answer

286 views

### Fuglede-Kadison determinants in $L(\mathbb{F}_2)$

Given $M$ a finite von Neumann algebra with trace $\tau$, $T\in M$ invertible.
The Fuglede-Kadison determinant is defined as
$\Delta(T)=e^{\tau(log|T|)}$,
where $|T|=(T^*T)^{\frac{1}{2}}$, and ...

**0**

votes

**0**answers

197 views

### Series of linear maps: on a paper by Evans and Hanche-Olsen

I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...

**6**

votes

**2**answers

1k views

### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...

**7**

votes

**1**answer

213 views

### Is there an upper bound on the dimension for irreducible representations of a continuous trace $C^{*} $-algebra?

The following are questions of Don Hadwin:
If $A$ is a unital continuous trace C*-algebra, is there an upper bound on the dimension of all the irreducible representations?
It is known that all ...

**10**

votes

**2**answers

369 views

### C*-algebras with no nontrivial endomorphisms

Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial ...

**0**

votes

**1**answer

129 views

### Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras

Suppose $A\otimes B$ is the minimal tensor product of two unital $C^*$ algebras $A$ and $B$.
We know that the set of states, $\{\phi\otimes\psi|\phi\in S(A),\psi\in S(B) \}$ on $A\otimes B$ ...

**9**

votes

**1**answer

207 views

### Is there a Dedekind-Frobenius group determinant for infinite groups?

If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a ...

**3**

votes

**1**answer

275 views

### When is a $\ast$-algebra a $C^{\ast}$-algebra?

The purpose of this question is to collect sufficient conditions on a unital $\ast$-subalgebra $\mathcal{A}$ of the algebra of bounded linear operators $B(\mathcal{H})$ on a separable Hilbert space ...

**3**

votes

**1**answer

290 views

### Injective von Neumann algebra

Let $G$ be a non-amenable countable discrete group. How can I show that the group von Neumann algebra $L(G)$ has no injective direct summand?

**6**

votes

**3**answers

451 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 ...

**32**

votes

**8**answers

1k views

### How to quantify noncommutativity?

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...

**11**

votes

**1**answer

499 views

### Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in ...

**10**

votes

**1**answer

487 views

### Strong Atiyah conjecture

Who introduced the Strong Atiyah Conjecture?
Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator ...

**8**

votes

**2**answers

319 views

### What is the physical difference between states and unital completely positive maps?

Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into ...

**10**

votes

**1**answer

606 views

### Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?

Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...

**8**

votes

**1**answer

291 views

### The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$

Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as ...

**5**

votes

**1**answer

291 views

### Reference for embedding an infinite direct product of matrix algebras into the hyperfinite $II_1$ factor

In some calculations I am writing up,
$\newcommand{\cR}{{\mathcal R}}$
I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be ...

**15**

votes

**3**answers

869 views

### Reference request for translating from Top to C*-alg

Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality-- namely, that the categories of ...

**6**

votes

**0**answers

348 views

### References for “folklore” on strong amenability of (group) C*-algebras?

[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.]
$\newcommand{\ptp}{\widehat{\otimes}}
\newcommand{\co}{\operatorname{co}}
...

**7**

votes

**1**answer

340 views

### When do tensor products of C*-algebras commute with colimits?

Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...

**20**

votes

**2**answers

628 views

### Nonseparable disintegration theory: references

I mean a theorem of the following kind. Let $A$ be a C*-algebra, and let $\pi: A\to B(H)$ be its representation. Then there exist a set $P$ with a positive measure $\mu$, a field of Hilbert spaces ...

**5**

votes

**5**answers

1k views

### Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...

**13**

votes

**1**answer

683 views

### P-adic C* algebras

I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of ...

**2**

votes

**1**answer

153 views

### Uniqueness of free complements

Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ ...

**4**

votes

**1**answer

498 views

### equality in noncommutative Hölder inequality

Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S_p$ denote the Schatten space. For any $x\in S_p$ and any $y\in S_q$ we have
$$
||xy||_{S_r} \leq ...

**17**

votes

**2**answers

700 views

### Commutators in the reduced C*-algebra of the free group

Is it known whether any element of trace 0 in the reduced $C^*$-algebra of a non-abelian free group, is a limit of sums of (additive) commutators?

**2**

votes

**1**answer

216 views

### Nuclearity of certain semigroup crossed product C*-algebras

This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...

**2**

votes

**1**answer

334 views

### About properties of groupoid C*-algebras

I'm interested in the following kind of questions about groupoid $C^*$-algebras.
1) If $G_1 \times_{H} \ G_2$ is a fibre product of (nice) groupoids do we have something like $$C^\star(G_1 \times_{H} ...

**2**

votes

**0**answers

204 views

### Strict complete contractions

Let $A$ be a $C^*$-algebra. The Stinespring construction shows that a completely positive contraction $T:A\rightarrow B(H)$ has the form $T(x) = U^* \pi(x) U$ where $U:H\rightarrow K$ is a ...

**9**

votes

**4**answers

851 views

### What kind of completion is this?

Let $X$ be a compact Hausdorff space, and $C(X)$ the unital commutative C*-algebra of continuous functions on it. The double Banach dual $C(X)^{**}$ is a commutative von Neumann algebra and hence has ...

**1**

vote

**1**answer

620 views

### Strict positivity in dense subalgebras of $C^{*}$-algebras

Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let
...

**5**

votes

**1**answer

373 views

### Is there a trivial construction of the trace on the Jones basic construction?

Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably ...

**8**

votes

**0**answers

467 views

### Faa di Bruno and Free Probability?

It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. For many examples, see David Vella’s paper. The ...

**9**

votes

**1**answer

734 views

### Connes's unpublished manuscript on correspondences, anyone?

There exist unpublished notes on correspondences of von Neumann algebras due to Connes. This is often cited, but I've never seen a copy. It would be nice to have this, say, to maybe look further into ...

**10**

votes

**1**answer

239 views

### Ideals in smooth subalgebras of C*-algebras

Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a ...

**7**

votes

**0**answers

267 views

### Canonical Time Evolution for Type $II_{1}$-Factors?

This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor ...

**7**

votes

**2**answers

839 views

### Completely positive maps as “positive operators”

Let $A$ be a unital $C^{*}$-algebra and $\phi:A \rightarrow A$ be a completely positive map, i.e. $\phi^{(n)}:M_{n}(A) \rightarrow M_{n}(A)$ preserves positivity for any natural number $n$, where ...

**0**

votes

**1**answer

185 views

### How coarse is the coarse correspondence?

Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) ...

**3**

votes

**2**answers

464 views

### Positive extension of functionals on a subset of the state space of a $C^*$ algebra

Let $A$ be a finite dimensional $C^*$ algebra and $S(A)$ the state space. Let $K\subset A$ be an intersection of $S(A)$ with a vector subspace $J\subset A$ and let $f$ be a positive affine functional ...