Algebras of operators on Hilbert space, C^*-algebras, von Neumann algebras, non-commutative geometry

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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 ...
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1answer
351 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? ...
2
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1answer
413 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 ...
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233 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
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3answers
412 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
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1answer
184 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
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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 ...
3
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1answer
358 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 ...
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188 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 $II_1$ ...
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439 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
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1answer
167 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 ...
4
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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 ...
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186 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$. ...
7
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3answers
356 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
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1answer
153 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!
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2answers
388 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?
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2answers
346 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
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1answer
248 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
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1answer
197 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? ...
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Connected component of the identity in graded banach algebras

I search for a noncommutative idempotent less Banach algebra $A$ which is graded by a finite Abelian group $G$ such that a nontrivial homogenous element lies in the same connected component ...
4
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1answer
98 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 $K_0(A) \cong \mathbb{Q}^n$ ...
3
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1answer
84 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 = ...
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1answer
210 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. ...
5
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1answer
147 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 ...
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2answers
231 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?
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224 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 ...
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1answer
209 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 ...
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1answer
81 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? ...
7
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478 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 ...
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220 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 ...
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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 ...
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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
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1answer
172 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
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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 ...
2
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2answers
334 views

the spectrum of matrix with positive entries

It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true? Let A ...
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195 views

Essential unitary equivalence

Let us agree that heuristic meaning of the word "essential" is: up to compact operator. There is clear notion of unitary equivalent operators. What is the proper notion of two operators being ...
2
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2answers
184 views

Finding the commutant of a von Neumann algebra

Suppose you have a von Neumann algebra $A$ of operators on $H$ and would like to compute its commutant. You have constructed a collection $B\subset A'$ which you suspect generates it (i.e. you think ...
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1answer
263 views

Are all the $R$-$R$ bimodules completely reducible?

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R-R$ bimodule. Question : Is $X$ completely reducible (i.e. a direct integral of irreducible $R-R$ bimodules) ? Example : Let $N ...
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2answers
229 views

Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then : $$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$ Question : Is it singly generated (as von Neumann algebra)? how ? ...
3
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1answer
122 views

Are every finitely generated planar algebras, also singly generated?

Let $\mathcal{P}$ be a finitely generated planar algebra. Question : Is it also singly generated ? I ask this question, because, on one hand I've read on this paper of V. Jones and D. Bisch : ...
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Replacing commutative C*-algebras by simple ones

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$. I am particularly ...
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How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$. The following definition comes from this paper (p51-52) of ...
3
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1answer
136 views

Sets $E$ in $\mathbb{Z}$ such that any $l^2$ function with support on $E$ comes from Fourier of a continuous function

Are there infinite sets $E\subset\mathbb{Z}$ such that any $f\in l^2(\mathbb{Z})$ with support on $E$ comes from the Fourier transform of a continuous function on $\mathbb{T}$ ? If yes, is there a ...
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Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one

A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and ...
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1answer
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Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Representation of $ \mathscr{A} \rtimes_{\alpha} G $

Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and consider the twisted convolution $ * $-algebra $ ({L^{1}}(G,\mathscr{A}),\star,^{*}) $ defined by \begin{align*} \forall \phi,\psi ...
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Are the reduced group Von Neumann algebra/ Group $C^{\ast}$ algebra functorial in the case of LCH groups

Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann ...
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Lifting triangles in K-theory to KL-groups

Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
15
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1answer
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Operator Valued Kadison--Singer Problem

The Paving Conjecture, which is equivalent to the famous Kadison--Singer Problem, was spectacularly settled in the affirmative by Marcus--Spielman--Srivastava (arxiv:1306.3969). Let $E$ denote the ...
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Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by ...
8
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Observables and dimensional analysis

Here is a simple question about physical units that I hope has a simple satisfying answer. In mathematically sophisticated treatments of both quantum and classical physics one often speaks of an ...