Questions tagged [oa.operator-algebras]
Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
1,694
questions
7
votes
0answers
53 views
Completely positive maps on Coxeter groups - the general case
In the reference below Bozejko and Speicher showed the following (for the full statement see the remark on page 9):
Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a Hilbert space and denote ...
0
votes
0answers
62 views
Normal states in the factor of type III $_1$
Let $M$ be a factor of type III$_1$. If $w_1$ and $w_2$ are two normal states on $M$, by Connes-Stormer transitivity theorem, for any $\delta>0$, there exists a unitary $u\in M$ such that $\|w_1–...
2
votes
1answer
97 views
Direct integral decomposition relative to a given measure space
It is well known that a separable Hilbert space $H$ decomposes as a direct integral in the presence of an abelian von Neumann algebra $\mathscr A\subseteq B(H)$.
More precisely, and quoting from ...
7
votes
1answer
173 views
Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?
Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
1
vote
2answers
85 views
Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...
2
votes
1answer
126 views
Opposite $C^*$ algebras
$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
17
votes
3answers
664 views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
2
votes
1answer
74 views
Covariant representations and crossed products of von Neumann algebras
Let $(M,G,\alpha)$ be a $W^\ast$-dynamical system with $G$ locally compact abelian (I am mostly interested in the case $G=\mathbb{R})$. A covariant representation of $(M,G,\alpha)$ is a pair $(\pi,u)$ ...
0
votes
0answers
73 views
Questions on unbounded derivations of C* algebra
In Sakai note, on the fourth part differentiation. Sakai stated the following:
It is an open question whether the result can be extended to $n=2,3,...$
What $n$ he is referring to? Also Sakai stated ...
10
votes
1answer
228 views
Factor states on C*-algebras
Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
0
votes
0answers
87 views
relationship between two modular automorphism groups associated n.s.f weights
Let $w$ be a normal semifinite faithful weight on a von Neumann algebra $M$. If $w_n$ is a sequence of weights which converge to $w$ in norm topology.
Does there exist relationship between $\{\sigma_t^...
0
votes
0answers
125 views
Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
7
votes
1answer
233 views
Induction and restriction of unitary representations
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$,
let $\Rep(G)$ and $\Rep(H)$ denote their ...
3
votes
2answers
176 views
Elements of the minimal tensor product of a finite dimensional operator system and a $C^*$-algebra
I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is the same question on MSE.
Let $E\subset A$ be a finite dimensional operator ...
7
votes
1answer
416 views
Motivation for $C^*$-algebras
I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
0
votes
0answers
84 views
Confusion: Normal homomorphism vs. weak*-continuous vs $\sigma$-weakly continuous
$\newcommand\M{\mathcal M}
\newcommand\N{\mathcal N}
\newcommand\A{\mathcal A}
\newcommand\B{\mathcal B}$
In Takesaki [1], I find the following theorem:
Proposition 5.13. Let $\M_1,\M_2,\N_1,\N_2$ be ...
4
votes
1answer
131 views
Polar decomposition in abstract von Neumann algebra
Probably an easy question, but here goes:
In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \...
2
votes
1answer
120 views
Trying to understand Haagerup tensor product $B(H)\otimes_{\rm h}B(K)$
I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded ...
4
votes
1answer
207 views
Non-unital Russo-Dye Theorem
Let $A$ be a C$^*$-algebra and let $\phi$ be a positive linear map from $A$ to $B(H)$ (bounded linear operators on Hilbert's
space). If $A$ is unital, then the Russo-Dye Theorem implies that $\|\phi\...
8
votes
1answer
150 views
Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?
Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
3
votes
1answer
162 views
English translation of von Neumann's Algebra der Funktionaloperationen (1930)
Does anyone know if there exists an English translation of von Neumann's early work in operator theory, in particular the paper Zur Algebra der Funktionaloperationen und Theorie der normalen ...
1
vote
0answers
106 views
Haagerup tensor product
Let $H$ and $K$ be Hilbert spaces. Recall that a closed subspace $V\subset B(H,K)$ is called a ternary ring of operators(TRO) if $xy^*z \in V$ for all $x,y,z \in V$. Obviously a TRO is also a ...
6
votes
1answer
158 views
Embedding of $C(X)$ into $B(H)$ where $H$ is separable
I would like to ask a question which may look strange at the first sight nevertheless I find it interesting. Let $H$ be a separable Hilbert space: for any separable $C^*$-algebra $A$ one can embed $A$ ...
3
votes
0answers
85 views
When can a state on a C*-algebra descend to a quotient?
Has anything been written on the following question:
Let $(\mathfrak{A}, \phi)$ consist of a C*-algebra $\mathfrak{A}$ equipped with a tracial state $\phi$. Let $\pi : \mathfrak{A} \twoheadrightarrow ...
4
votes
0answers
204 views
Understanding vector-valued analytic functions vs holomorphic functional calculus
Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\...
0
votes
0answers
110 views
von Neumann algebras with certain properties
Let $\mathcal M$ be a von Neumann algebra which is equipped with a normal faithful tracial state. Further assume that there exist projection $e\in\mathcal{Z}(\mathcal M)$, i.e. the center of $\mathcal ...
7
votes
1answer
94 views
Is the bitranspose continuous for the $\sigma$-strong topology?
Let $\varphi\colon A\to B$ be a bounded, linear map between C*-algebras. Is the bitranspose $\varphi^{**}\colon A^{**}\to B^{**}$ continuous when the von Neumann algebras $A^{**}$ and $B^{**}$ are ...
6
votes
1answer
280 views
Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
3
votes
1answer
97 views
Is restriction to the center an open map?
Given a type one $C^*$-algebra $A$, its center $Z$ acts by scalars on each irreducible representation space. Mapping a representation to its central character yields a continuous map from the ...
2
votes
0answers
56 views
Are quasitrace extensions unique?
I'm trying to understand the basics of quasitraces on $C^*$-algebras. Using the terminology of Haagerup, given $n \geq 2$, an $n$-quasitrace $\tau$ on a $C^*$-algebra $A$ is a 1-quasitrace on $A$ ...
5
votes
1answer
136 views
Relating different constructions of the universal compact quantum group
Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...
2
votes
1answer
164 views
Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
10
votes
1answer
756 views
Making sense of “every non-commutative algebra has its own internal time evolution (aka a one-parameter group)”?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
1
vote
1answer
101 views
Calculation of the norm of linear combinitation of two states on a $C^*$-algebra
Let $A$ be a unital $C^*$-algebra. Suppose $\rho_1$ and $\rho_2$ are two states on $A$. If $\rho_1=\rho_2$, we have $\|\rho_1+i\rho_2\|=\sqrt{2}$.
If we have $\|\rho_1+i\rho_2\|=\sqrt{2}$, can we ...
8
votes
1answer
154 views
Reference for “Every compact quasinilpotent operator is the limit of nilpotent ones”
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
3
votes
0answers
93 views
Cuntz semigroups of basic C*-algebras
I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103):
"[...] $A_i$ is ...
1
vote
0answers
63 views
Showing the existence of a right-inverse in a von Neumann probability space
Disclaimer: This is my first post here on Overflow as opposed to the "normal" forum, so if this question is too elementary for this forum, I'd appreciate y'all letting me know. I posted it ...
2
votes
1answer
95 views
Uniqueness of the direct sum of $C^*$ algebras as quotient of free products
Suppose that you have $A, B$ two unital $C^*$ algebras and let $A \ast B$ the reduced free product (I think that it is the reduced amalgamated product over the common $*$-subalgebra $\mathbb{C} 1$) ...
5
votes
1answer
238 views
Almost commuting matrices, one a projection, is there a nearby projection that commutes?
Suppose that $P, A, Q \in \mathbb{M}^{n \times n}(\mathbb{R})$ (I'm still interested if it must be done over $\mathbb{C}$), (EDIT:) suppose that $A$ is given, $P$ is an orthogonal projection, and $\...
1
vote
0answers
62 views
Extension of states satisfying a certain inequality
Let $A$, $B$ be (unital) $C^\ast$-algebras with $B \subseteq A$, let $\phi$ be a state on $B$ and let $b \in B$ be an element with $b b^\ast, b^\ast b \geq C$ for some $C>0$ and $\phi(b^\ast ab) \...
6
votes
1answer
164 views
Image of $L^2M$ inside $L^1M$, for $M$ a von Neumann algebra
Let $M$ be a factor (von Neumann algebra with trivial center), and let $L^1M:=M_*$ be its predual.
Let $\omega:M\to\mathbb C$ be a faithful normal state.
The Hilbert space $L^2M:=L^2(M,\omega)$ admits ...
0
votes
0answers
126 views
Morphism of non-commutative algebras
Disclaimer: this question is a "big picture" one that comes from my personal thoughts in physics. If it doesn't fit this site, please tell me.
While having a walk, I thought a bit about what ...
2
votes
0answers
99 views
Is there a finite depth irreducible subfactor of prime index and not group-subgroup?
Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
4
votes
2answers
165 views
Positive maps on finite group algebras and group homomorphisms
Let $G$, $H$ be finite groups. Consider the group algebra $\mathbb{C}G$ acting on $L^2(G)$, making $\mathbb{C}G$ into a C* algebra, and the resulting positive elements, say $P_G\subset \mathbb{C}G$. ...
2
votes
0answers
65 views
Construction of non-commutative torus using ergodic action of $\mathbb{T}^{n}$
It is well known that non-commutative torus can be constructed using universal C* algebra, by n unitary elements and twisted relations. It can also be constructed using ergodic action of torus group $\...
2
votes
0answers
101 views
What are all the possible indices for the finite depth subfactors?
Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
6
votes
1answer
184 views
Amenable groupoid C*-algebras satisfy the UCT in English?
As is by now well known, Tu proved in 1998 that the C*-algebras coming from amenable groupoids satisfy the so-called UCT (universal coefficient theorem). Unfortunately, I don't speak french and I've ...
3
votes
1answer
150 views
Coincidence of two topology on a bounded subset of a finite von Neumann algebra
Let $M\subset B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$. There is a norm $\|.\|_\tau$ on $M$ given by $\sqrt{\tau(xx^*)}$. How to show the $\|.\|_\tau$-topology ...
2
votes
1answer
102 views
About nuclear-by-exact extensions
I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...
17
votes
1answer
367 views
For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...
