Algebras of operators on Hilbert space, C^*-algebras, von Neumann algebras, non-commutative geometry
2
votes
2answers
135 views
Lower bounds for norms of commutators
For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound ...
9
votes
0answers
235 views
Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?
(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...
2
votes
0answers
75 views
A section from subfactors to transitive groups
A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite ...
4
votes
1answer
308 views
Noncommutative geometry and category theory
The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...
13
votes
2answers
542 views
Groups which are only defined up to conjugation
I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...
4
votes
1answer
382 views
Goin' with the flow with Kummer and Pascal: Combinatorics and geometry underlying the logarithm of the derivative operator
In a MO-Q111165 and associated MSE-Q125343, I present a pair of raising / lowering (creation / annihilation) operators $R_x = log(D)$ and $L_x = -x·D$ with $D=d/dx$ (for a sequence of functions ...
0
votes
0answers
130 views
Positive definite matrix in Banach algebra
Let $V$ be a Banach algebra lattice, endowed with an ordering $\leq$. And $T:V\rightarrow V$ be a positive operator. For $v_1,v_2,...,v_n\in V$, I want to show that (following matrix is positive ...
1
vote
1answer
119 views
On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$
Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...
2
votes
2answers
84 views
Strong morita equivalence and morphims between $C^*-$algebras
Let $A$ and $B$ be two $C^*-$algebras, they are $strong$ $morita$ $equivalent$ if there exist a $(B,A)-$bimodule $E$, an $(A,B)-$bimodule $F$, such that $E\otimes_A F\cong B$, $F\otimes_B E\cong A$ ...
6
votes
0answers
179 views
C* algebras of free semicircular systems
It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, ...
5
votes
2answers
94 views
Extension of $C^*$ isomorphism to $W^*$ isomorphism
Let $\mathfrak{A}$ be $C^*$algebra, and $\pi$ its faithful representation on Hilbert space $\mathcal{H}$. Bicommutant $\mathfrak{B}=\pi(\mathfrak{A})''$ is the von Neumann algebra generated by ...
5
votes
1answer
188 views
Morita equivalence for operator algebras and tensor products, question about proof
This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...
6
votes
1answer
170 views
Realisation of noncommutative torus
One of the most basic examples in noncommutative geometry is the so called noncommutative torus to be denoted by $\mathbb{T}_{\theta}$. As far as I know, there are several equivalent constructions of ...
3
votes
1answer
289 views
Example of an infinite dimensional reflexive Banach algebra
If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
2
votes
1answer
160 views
Do all right orderable groups have the Haagerup property?
Do all right orderable groups have the Haagerup property?
Recall that a group is right orderable if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is ...
4
votes
1answer
109 views
Does noncommutative Lp-convergence respect orderings?
Let $M$ be a von Neumann algebra and $\tau$ a faithful (semi-finite?) normal trace on $M$; as is standard, the $L^p$-norm is defined as $||u||_p=\tau(|u|^p)^{1/p}$. Let $\{u_i\}_{i=1}^\infty$ be a ...
19
votes
0answers
381 views
When are two C*-algebras isomorphic as Banach spaces?
We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
1
vote
1answer
151 views
Computing noncommutative geometries
I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...
1
vote
1answer
311 views
Abelian subfactors, a relevant concept?
Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
3
votes
1answer
134 views
A generalized K- theory via generalized idempotents
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach algebra with $a^{n}=a$.
Can the 3 equivalent relations, Murray-Von Neumann, ...
3
votes
2answers
303 views
Jordan-Hölder theorem for subfactors?
All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
5
votes
1answer
169 views
Jordan-Hölder theorem for planar algebras?
First recall the Jordan-Hölder theorem for groups:
Theorem (Jordan-Hölder): Let $G$ be a group, and let $$ G=G_1 \supset G_2 \supset \dots \supset G_r = \{ e \} $$ be a normal tower such that ...
3
votes
0answers
120 views
A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity
In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...
2
votes
1answer
179 views
The category of subfactors extending the category of groups?
This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
0
votes
2answers
171 views
Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
11
votes
2answers
331 views
Can non-central projections still commute with all other projections?
Let $A$ be a C*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything ...
4
votes
1answer
127 views
Examples of special isometries
Are there examples of (distinct) Hilbert spaces $H_1$=$(H,\langle\cdot,\cdot\rangle_1)$, $H_2 $=$(H,\langle\cdot,\cdot\rangle_2)$ and a linear operator $V: H_1\to H_2$ such that $V^n: H_1\to H_2$ is ...
2
votes
0answers
78 views
semigroup-analogue of an inequality
I want to prove the following result. Almost similar result has already been proved for positive functionals. I want a semigroup-analogue.
Let $V$ be a Banach lattice algebra (endowed with ordering ...
4
votes
1answer
86 views
Does the group of compact perturbations of the identity act transitively on the compact operators?
Let $H$ be an infinite dimensional (separable if necessary) complex Hilbert space, and denote by $K(H)$ the ideal (in $B(H)$) of compact operators on $H$. Let $G_c=\{I+K\in B(H): I+K \text{ is ...
1
vote
0answers
172 views
Fusion categories with permutation “associativity matrices”
Let $\mathcal{C}$ be a fusion category and let $(H_1,...,H_r)$ be its simple objects.
$\mathcal{C}$ is non-pointed if at least one of its simple object has Perron-Frobenius dimension $ \neq 1$.
...
6
votes
2answers
173 views
Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?
My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
4
votes
1answer
138 views
K-Theory of Algebra of Zeroth Order Pseudo differential operators
Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?
Thanx!
17
votes
2answers
341 views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
5
votes
2answers
312 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
234 views
characterization of commutative Banach algebras
Let $A$ be a Banach algebra with the following property:
For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.
...
2
votes
1answer
168 views
Functors with an epi-mono factorization property
This is a simple question about terminology and a request for any related references. Specifically, what would you call a functor $F:\mathbf{D}\rightarrow\mathbf{C}$ with the following property?
...
4
votes
1answer
90 views
Automorphisms of “rational” Kirchberg algebras
Let $M_{\mathbb{Q}}$ be the universal UHF-algebra and let $\mathcal{O}_{\infty}$ be the infinite Cuntz algebra. Let $A$ be a Kirchberg algebra that satisfies the UCT with $K_0(A) \cong \mathbb{Q}^n$ ...
2
votes
1answer
68 views
Crossed Products by Permutation Groups
What can be said about the following crossed product $C^*$-algebra?
Let $A$ be a Kirchberg algebra with $K_0(A) = \mathbb{Q}$ and $K_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = ...
0
votes
0answers
136 views
$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$
Let $F_{2}$ be the free group with two generators.
Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way.
...
4
votes
1answer
133 views
Free C^*-algebra
Let $A_0$ be a set of all polynomials with complex coefficients of infinitely many noncommuting (free) variables, denoted by $X_1,X_2,...,X_1^*,X_2^*,...$. We equip $A_0$ with the operation $*:A_0 \to ...
3
votes
2answers
169 views
Type III factor representation
Does there exist any theorem which permits, under suitable hypotheses, to represent a particular complete orthomodular lattice as the projection lattice of a Type III von Neumann factor?
6
votes
1answer
183 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 ...
1
vote
1answer
140 views
A norm one projection
Let $\mathcal{H}$ be Hilbert space and $\mathfrak{B(}\mathcal{H}\mathcal{)}$ of all bounded linear operators on $\mathcal{H}$. Let $\mathcal{A}$ be a maximal commutative sub-algebra of ...
1
vote
1answer
76 views
modul projection map
Let H be a Hilbert space and B(H)be space bounded linear operators on H. Let A be a commutative maximal sub-algebra of B(H). Is there a conditional expectation ( a norm one projection) from B(H) to A?
...
6
votes
3answers
275 views
Universal C^*-algebra with generators and relations
We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations ...
10
votes
0answers
182 views
Groups with reduced C*-algebras of stable rank 1
Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in ...
5
votes
5answers
294 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
0answers
116 views
Planar algebraic translation of a subfactor property
Let $N \subset M$ be an irreducible finite depth and finite index subfactor.
$M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows :
...
5
votes
1answer
137 views
A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
4
votes
0answers
168 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 ...
