# Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].

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### Nuclearity of $C(\partial_F \mathbb{G})$

Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$....

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### Non-degenerate representation of a Banach algebra

Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...

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### Bochner theorem for (non-abelian) discrete groups

I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...

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### Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...

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### Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism

This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...

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### Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?

Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...

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### Comparison between the operator norm and the $L^1$ norm on group algebras

Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question:
The 2-norm given by $||\sum_{g \in G} c_gg||_2^...

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1
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### Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.
I also ...

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### Unitary representation of a group of automorphism on an abelian algebra

Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...

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### Borel functions in C*-algebras

Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that
$\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space.
There is a closure operation $A\...

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### Intersection of two intermediate subalgebras

Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...

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### On the Dunford-Pettis property and multiplier algebras

I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that:
Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...

3
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### Is a compact set of extreme points contained in a compact face?

I have run into the following question in convex analysis, which I haven't found answered in the literature:
Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...

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### A question regarding certain sequences in hyperfinite type $II_1$ factor

Let us consider the hyperfine type $II_1$ factor $\mathcal M$ arising from the inclusion $M_2\subseteq M_{2^2}\subseteq \dots M_{2^k}\subseteq\dots$ of matrix algebras with the normalised trace $\tau$....

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### Definite negative functions and length functions

$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups.
Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...

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### Doubts on convergence of series of operators

Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...

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### Two more topologies on unitary groups

Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...

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### Maps in the Künneth theorem for K-theory of C*-algebras

The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...

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### Irreducible representations of $\mathrm{UHF}_n$

I have a question about the irreducible representations of the $C^*$-algebra $\mathrm{UHF}_n = \bigotimes_{k=1}^\infty M_n$.
For every sequence of unit vectors $(\xi_k)$ in $\mathbb C^n$ there is a ...

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### Automorphisms of the injective envelope

Let $A$ be a separable $C^∗$-algebra and $(I(A),\kappa)$ be its injective envelope. WLOG assume that $I(A)$ is a monotone complete $C^*$-algebra, and $\kappa:A\to I(A)$ is the identity map.
Let $\...

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### Cocycle-conjugacy classes of flows on the C*-algebra of compact operators

A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$
such that the map
$$
t\in {\mathbb R}\mapsto \sigma _t(a)\in A
$$
is norm-...

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### Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems

In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...

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### Reference request for embedding of a tensor product $C^*$-algebra

I am studying Ruy Exel's paper "A new look at the crossed product of a $C^*$-algebra by a semigroup of endomorphisms." In the proof of Theorem 11.7 he writes:
Let $G$ be ameanable, thus $C^*(...

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1
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### Conditioning a $\mathrm{C}^*$-algebra state with infinite precision

This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success.
Let $\mathcal{A}$ be a unital $\...

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2
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### Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots?

Let $A$ be a finite-dimensional $*$-algebra over $\mathbb R$. We say that an element $x \in A$ is positive definite if $x$ admits an inverse and if $x = y y^*$ for some $y \in A$. Does every such $x$ ...

3
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### Property that follows from conditions involving slice maps on Hilbert module

Let $A,B$ be $C^*$-algebras and $E$ be a right $A$-Hilbert $C^*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^*$, there is a unique ...

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### Morphism of discrete quantum groups

In the paper Kazhdan's Property T for Discrete Quantum Groups
, we read the following fragment:
First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...

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### Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra

Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi_{\mathbb{G}}$. Consider the associated GNS-representation $\pi_{\mathbb{G}}: C(\...

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### Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...

4
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### The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras

I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper.
Let $A$ ...

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0
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### How do I show countable additivity of (proposed) spectral measure in the proof of the spectral theorem?

I'm currently writing a Bachelors thesis based on the following paper:
Douglas, R., & Pearcy, C. (1970). On the spectral theorem for normal operators. Mathematical Proceedings of the Cambridge ...

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0
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### Formula for the KK-theory groups $KK(A, C(S))$

I am studying $C^*$-algebras and their KK-theory. Let $A$ be a (unital if you wish) $C^*$-algebra and $S$ be a compact Hausdorff space. I am interested in computing the KK-theory groups $KK(A, C(S))$, ...

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1
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### A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property:
The algebra $A$ has trivial intersection with the set of commutator ...

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### Probabilistic interpretation of von Neumann's approach to quantum mechanics

One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...

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1
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### Convergence of the partial sum of a sequence strictly converging to zero

The following question comes from a statement in Lemma 16.4 in K-theory and $C^{\ast}$-Algebras written by N.E. Wegge-Olsen. Let $A$ be a non-unital $C^*$-algebra, $\{p_n\}_{n\in\mathbb{N}}$ be a ...

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### Relationship between the Penrose groupoid algebra and the group algebra of the symmetry group of a Penrose tiling

Given a Penrose tiling, there are two C*-algebras associated with it: the $\mathcal{O}_\text{Penrose}$ algebra (Penrose groupoid algebra) and the group algebra of the symmetry group of the tiling, ...

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1
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### Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections

I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
$\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...

3
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1
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### Closed prime ideal in $C[0, 1]$

I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
Is there any $\textbf{closed}$ prime ...

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### Property of pushouts in the category of unital $C^{\ast}$-algebras

Let $A$ be a unital $C^{\ast}$-algebra and $\{ f_i: A \rightarrow A_i \}_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod_i A_i$ is ...

3
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### Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short.
Consider the following fragments in Takesaki's second volume "Theory of operator ...

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### Higher theory o $C^{\ast}$-algebras and the $C^{\ast}$-algebra of a $\infty$-groupoid

Has someone already worked out what would be the infinity categorical analogue of the category of $C^{\ast}$-algebras? Given a groupoid $G$ we can associate a $C^{\ast}$-algebra $C^* (G)$, can we do ...

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1
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### Adjunction via Gelfand duality

$\DeclareMathOperator\Hom{Hom}$For which unital $C^{\ast}$-algebras $A$ does it hold that for all compact Hausdorff $S$ we have the bijection:
\begin{align*}
\Hom(A, C(S)) \cong \Hom(S, \Hom (A, \...

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1
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### Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity

Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...

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### A question on Stable rank 1

My apology in advance if my question is elementary
According to the initial definition of topological stable rank introduced by Marc Rieffel we have the following:
An algebra has tsr 1 if the space ...

2
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1
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### inclusion of von Neumann algebras implies reversing inequality of its modular operators

I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...

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2
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### Elements that commute with $1$ in the pushout of a $C^{\ast}$-algebra

Suppose $B$ and $C$ are commutative unital $C^{\ast}$-algebras with $B \subseteq C$ (unital). Let $c$ be an element of $C$ such that $c \ast 1 = 1 \ast c$ in the pushout (in the category of ...

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### Noncommutative condensed sets

Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories
\begin{align*} \mathrm{CHaus}^{\mathrm{...

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### Definition of condensed $C^{\ast}$-algebra

The classical definition of a $C^{\ast}$-algebra is a Banach algebra with an isometric antilinear involution map $a \mapsto a^\ast$. What would be a good definition for a condensed $C^{\ast}$-algebra? ...

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### Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$

The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...)
By ...

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### Why are projectionless $C^*$-algebras important (Kadison's conjecture)

It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...