7
votes
2answers
236 views

Arveson's extension for normal completely positive maps

My question deals with a version of Arveson's extension theorem (for the standard version, see, e.g., Paulsen's book Completely Bounded Maps and Operator Algebras). Let $\mathcal A$ be a von Neumann ...
7
votes
2answers
217 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
2answers
342 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
1answer
216 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
1answer
105 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
1answer
128 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
1answer
303 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
0answers
303 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
1answer
839 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
0answers
108 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
1answer
154 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 ...
15
votes
1answer
492 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) ...
5
votes
1answer
313 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
0answers
198 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
2answers
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
1answer
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
2answers
372 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
1answer
130 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
1answer
213 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
1answer
280 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
1answer
302 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
3answers
484 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
8answers
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
1answer
515 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
1answer
510 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
2answers
337 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
1answer
623 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
1answer
312 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
1answer
294 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
3answers
886 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
0answers
372 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
1answer
349 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
2answers
635 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 ...
6
votes
5answers
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
1answer
696 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
1answer
155 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
1answer
508 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
2answers
720 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
1answer
224 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
1answer
348 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
0answers
211 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
4answers
870 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
1answer
621 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
1answer
375 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 ...
7
votes
0answers
474 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
1answer
747 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 ...
12
votes
1answer
254 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
0answers
270 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
2answers
858 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
1answer
191 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) ...