# Tagged Questions

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: [tag:banach-algebras], [tag:von-neumann-algebras], [tag:operator-algebras], [tag:spectral-theory].

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### Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
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### Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group. General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal coefficient theorem (UCT)? I am mainly ...
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### 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 ...
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### $C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
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### Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
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### Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
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### Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
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### 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 ...
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### The approximation property of group C*-algebras

Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...
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### Quantum Braid Group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
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### Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
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### Verifying Woronowicz’s proof that all ${\text{SU}_{q}}(2)$’s are isomorphic as $C^{*}$-algebras, where $-1 < q < 1$

This question is related to one that I asked some time ago. Definition 1. Let $q \in (-1,1)$. Let $A$ be a unital $C^{*}$-algebra and $(x,y)$ a pair of elements of $A$. Then define the ...
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### Unique maximal ideal in group C*-algebras

Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$ ...
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### Non Commutative Hyperspaces

Let $X$ be a compact metric space. Recall that the hyperspace $2^{X}$ is the set of all non empty compact subsets of $X$ with the Hausdorff metric. Assume that $\mathcal{C}$ is the category of all ...
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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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### Is the crossed product $\mathcal{K} \rtimes G$ a groupoid algebra?

Suppose G, a discrete group acting on the compact operators $\mathcal{K}$ by automorphism of C*-algebra $\mathcal{K}$. Can we view the crossed product as a groupoid C*-algebra of some groupoid? This ...
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### 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 ...
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### Extending Akemann's Non-Commutative Urysohn Lemma

Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections. Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$? Note if $||pq||=1$ this is immediate, ...
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### Closed containment of open projections in C*-algebras

For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements. $\overline{p}\leq q$ $p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$ $p\leq q$ ...
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### Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra

Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?
Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...