# Tagged Questions

**5**

votes

**1**answer

57 views

### Real rank zero of group $C^*$-algebras

The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...

**2**

votes

**0**answers

46 views

### isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...

**1**

vote

**0**answers

74 views

### Ideal structure of group $C^*$-agebras [closed]

Let $G$ be a locally compact groups and $C_r^*(G)$ be a reduce group $C^*$-algebra.
$\ Question:$What is the ideal structure of reduce group $C_r^*(G)$?

**3**

votes

**1**answer

174 views

### $K$-Theory of finite dimensional Banach algebras

Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
...

**3**

votes

**0**answers

65 views

### Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...

**8**

votes

**0**answers

90 views

### Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...

**2**

votes

**0**answers

64 views

### The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment.
Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...

**1**

vote

**1**answer

78 views

### When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [on hold]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...

**20**

votes

**3**answers

1k views

### What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...

**7**

votes

**1**answer

343 views

### Is the fundamental group of $II_{1}$ factors invariant under a relation?

In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and ...

**3**

votes

**2**answers

56 views

### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann ...

**1**

vote

**0**answers

17 views

### Difference between Schmidt decomposition and singular value decomposition [migrated]

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbart space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...

**3**

votes

**2**answers

550 views

+100

### Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the classical Borsuk Ulam theorem
for the Euclidean n-sphere, $n>2$ that is based only on the theory
of Banach algebras. I checked on MR but had no ...

**0**

votes

**0**answers

69 views

### Circle actions on graph C*-algebras [closed]

Do all graph C*-algebras admit actions of the circle?
Suppose we have a graph C*-algebra which we know is the quotient of a graph algebra by a circle action. Is it possible to read off the original ...

**28**

votes

**3**answers

2k views

### What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...

**3**

votes

**1**answer

90 views

### Is the module action $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...

**7**

votes

**1**answer

273 views

### Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...

**2**

votes

**0**answers

177 views

### Number of connected components of a $C^{*}$ algebra

Inspired by the concept in the following post
What are these compact sets called?
We introduce the following concept:
Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent ...

**3**

votes

**0**answers

154 views

### Uniqueness of the reduced free product of unital completely positive maps

For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = ...

**3**

votes

**0**answers

54 views

### Unitizations of Banach algebras and matrix norms

Consider the short exact sequence of Banach algebras $0\rightarrow A\rightarrow A^+\rightarrow\mathbb{C}\rightarrow 0$ where $A$ is a Banach algebra without unit and $A^+$ denotes the unitization of ...

**1**

vote

**2**answers

84 views

### Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?

**1**

vote

**0**answers

47 views

### Can a semigroup be defined on a Banach algebra? [closed]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...

**2**

votes

**1**answer

199 views

### Has a subfactor with lattice $B_3$, a singly generated identity biprojection?

Let $(N \subset M)$ be an irreducible finite index subfactor.
If its lattice of intermediate subfactors is equivalent to $B_3$ (the lattice of divisors of $n=p_1p_2p_3$ square free):
...

**11**

votes

**0**answers

320 views

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

**9**

votes

**0**answers

86 views

### How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...

**1**

vote

**1**answer

58 views

### A relation among projections of a von Neumann algebra

This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice.
For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = ...

**8**

votes

**1**answer

240 views

### Correspondences as generalized morphism between $C^*$-algebras

While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...

**5**

votes

**2**answers

186 views

### Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?

Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is ...

**-1**

votes

**1**answer

111 views

### How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? [closed]

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...

**3**

votes

**0**answers

274 views

### Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
A subalgebra $I$ of $H$ is called a left coideal subalgebra if $\Delta(I) \subset H \otimes I$.
$H$ is ...

**1**

vote

**0**answers

41 views

### Support vectors and relative modular operator

I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...

**4**

votes

**0**answers

104 views

### References for a lemma about compact operators on a Hilbert module

I am looking for a reference for the following result:
If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta ...

**6**

votes

**1**answer

346 views

### Who gave the generalized Stone-Weierstrass Theorem?

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on ...

**6**

votes

**1**answer

121 views

### Spectral mapping theorem for polynomials in $z,\overline{z}$ and direct construction of the function calculus for a normal operator

Suppose that $A$ is an element in Banach algebra and $p$ is a polynomial. Then we have an equality $p(\sigma(A))=\sigma(p(A))$ where $p(A)$ has an elementary meaning. This theorem (the spectral ...

**6**

votes

**1**answer

483 views

### An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**7**

votes

**2**answers

377 views

### States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...

**3**

votes

**2**answers

137 views

### Weak convergence implies norm convergence for trace class operators?

It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here.
Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...

**3**

votes

**0**answers

223 views

### Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...

**6**

votes

**1**answer

126 views

### The positive cone of the standard representation of a Von Neumann algebra

Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). ...

**4**

votes

**0**answers

123 views

### Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra

While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove:
Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...

**5**

votes

**1**answer

201 views

### $C^{*}$-correspondences viewed as generalized endomorphisms

I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...

**2**

votes

**1**answer

217 views

### Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the “other” Schur multipliers of a group?

The name for the the following 2 mathematical objects:
$$H_2(G,\mathbb{Z})$$ and
$$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times ...

**3**

votes

**0**answers

97 views

### The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...

**4**

votes

**1**answer

220 views

### Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$.
Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...

**2**

votes

**1**answer

165 views

### Polar decomposition

Let $x$ be a trace class operator on a Hilbert space $H$. Then $x$ induces unique normal functional on $B(H)$, which we denote it by $f_x$.
Let us consider the polar decomposition $x=u|x|$ and ...

**20**

votes

**8**answers

4k views

### Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra ...

**4**

votes

**0**answers

119 views

### Noncommutative Poincaré inequalities

This is a question on how (or if) people in the community think about the Poincaré inequality in noncommutative geometry. In geometry, the Poincaré inequality (when it exists) gives a bound on a ...

**14**

votes

**0**answers

419 views

### Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
Is the tensor product of two path connected algebra, a path ...

**0**

votes

**1**answer

447 views

### Why does an infinite tensor product depend on some vectors for operator algebras?

I have read that, in the definition of the infinite tensor product of operator algebras such as ${\rm C}^\ast$-algebras and ${\rm W}^\ast$-algebras, every factor in the product is associated with a ...

**2**

votes

**1**answer

79 views

### Commutator representation of certain smoothing operators

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...