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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Universal C*-algebraic presentations of $C_0(X)$ for locally compact $X$

There are quite a number of well-known commutative $\mathrm{C}^*$-algebras that have presentations as universal $\mathrm{C}^*$-algebras (for example see here). All the examples I have seen are unital,...
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Primitive ideals of inductive limits of $C^*$-algebras

I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras. Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be a primitive (modular) ideal of ...
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132 views

Two isomorphic reduced group $C^*$-algebras

Suppose that $C^*_r(G)\cong C^*_r(H)$, can we conclude that $G\cong H$?
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Monotone series of projections converging to 1 in von Neumann algebra

The following statement is being used a lot in the literature, and I wonder how to prove it. Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an ...
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Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
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G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state. Consider the covariant GNS ...
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General construction of enveloping C*-algebra, left/right-regular representation, etc

In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...
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109 views

Need reference for ideals and representations of $C_0(X,A)$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
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Proof of universality of Toeplitz algebra

It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^*...
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101 views

Is the reduced group $C^*$-algebra quasidiagonal

Let $G$ be an amenable group. I wonder whether it is true that the reduced group $C^*$-algebra $C_r^*(G)$ is quasidiagonal.
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Question on Cuntz' proof of Bott periodicity

I am reading the presentation of Cuntz' proof of Bott periodicity for $C^*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^*$-algebras $$0 \longrightarrow \...
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Continuous functions on a compact $T_1$ space

Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, ...
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For what kind of $C^*$ algebra $A$ every normal element $y\in A$ has a normal lift for every given surjective $C^*$ morphism $\phi:B\to A$

Is there a terminology for the following property of $C^*$ algebra $A$: For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal ...
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Jordan isomorphisms of type I von Neumann algebras

Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras. A linear map $J:\mathcal M\to\mathcal N$ is said to be a Jordan isomorphism if $J$ is bijective, $*$-preserving and $J(xy+yx)=J(x)J(y)+J(...
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What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$?

I've been working computing several K-groups associated to some $C^*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K_1(C(\mathbb{T})\otimes\...
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Showing a product on a character space is continuous

Quoting from Timmermann's An invitation to quantum groups and duality: Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact quantum group. Then there exists a compact group $G$ ...
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Relationship between canonical commutation relations and projective representations?

$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\...
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Approximating a projection by a sum of elementary tensors with a certain property

Let $A$ and $B$ be two C$^{*}$-algebras and suppose we have a non-zero projection $p\in A\otimes B$. (We can assume $A$ is nuclear, so that there is only one possible tensor product.) Does there ...
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Embedding of Cuntz algebras $O_2\subseteq O_3$?

The Cuntz algebra $O_n$ is the (universal) C*-algebra generated by n-isometries $s_1,...,s_n$ such that $$\sum_{i=1}^n s_is_i^\ast =\mathbf{1}, \ \hbox{and}\ s_i^\ast s_j=\delta_{ij} \mathbf{1}\ (\...
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1answer
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Involution in a commutative unital real C* algebra

It follows immediately from Gelfand duality that the involution in a commutative unital real C* algebra is the identity. Is there a direct proof from the axioms of C* algebras?
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Separability of an algebra is equivalent to separability of its spectrum

Let $A$ be a commutative C*-algebra. I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable. Notes ...
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A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm

Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide? A somewhat similar question is discussed here.
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When a finite codimensional subalgebra contains a finite codimension ideal?

What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property: Every finite codimensional subalgebra $B$ of $A$ ...
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Left and right topological K-theory of Banach algebras

Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette. The definition is based on ...
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Reduced compact quantum group and left and right multiplication

Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product $$ \...
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$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
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113 views

Can C*-algebras be characterized among Banach *-algebras by the spectral radius?

Let $(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an ...
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Is the switch automorphism inner for continuous-trace $C^*$-algebras?

If $R$ is a commutative ring, and $A$ is an Azumaya algebra over $R$, then the switch (or flip, or exchange, etc.) automorphism of $A\otimes_R A$, given by $a\otimes b\mapsto b\otimes a$, is inner: it ...
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107 views

Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces. Given two $\Cst$-...
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Non-separable non-postliminal $C^*$-algebra with injective $\pi\mapsto\ker\pi$

It is known that for a postliminal/GCR $C^*$-algebra the map $\pi\mapsto\ker\pi$ from (equivalence classes of) irreducible representations to their kernels is injective. If the algebra is separable, ...
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561 views

How are number theory and C*-algebras connected?

I came across this research profile where under Research Overview, it states that These days C*-algebra theory is a very active area of mathematical research in its own right, and enjoys deep ...
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Real Rank of $M_n(A)$

The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest ...
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1answer
156 views

When do completely positive maps have a closed image?

Let $\mathcal{A}, \mathcal{B}$ be C*-algebras. A map $\phi \colon \mathcal{A} \rightarrow \mathcal{B}$ is completely positive (cp) if it's linear, * preserving and all of its' coordinatewise ...
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Connes Embedding Conjecture is false [closed]

This preprint from yesterday claims to prove that Connes Embedding Conjecture fails. Since the paper is from outside Operator Algebras (Computer Science/Quantum Computing) and they actually work on ...
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Need help in proving an inclusion between some subspaces of operators

The following question was first posted on Math.Stackexchange.com but unfortunately I didn’t get any answer. This might be obvious for many researchers but I can’t see how this is so, thus I am asking ...
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Why C*-algebras is not as popular as other areas of pure mathematics? [closed]

I am applying for graduate school in pure mathematics and I recently got very interested in C*-algebra. I am definitely wrong but I get the feeling that C*-algebras is not as popular as other areas ...
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Morita-invertible C*-algebras

I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of C*-...
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Projection (or idempotent) graph of a $C^*$ algebra(or a ring)

In the literature, are there some research around a directed graph associated to a $C^*$ algebra or a ring $A$ whose vertices are projections or idempotents of $A$ and $e$ is connected to $f$ iff $ef=...
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Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$

Revision: According to comment of Wojowu we give a complete revise for this post. A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
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1answer
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Automorphism of algebras with certain initial conditions on given idempotents

The First question Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \...
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)

The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
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Bisector Projection

Let $p,q$ be two projections of a $C^*$ algebra. A projection $l$ is called a bisector projection to $(p,q)$ if $$|pl-l|=|ql-l|$$ The motivation comes from the geometric intuition of "...
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Reference request for representation theory of TRO

Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
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275 views

How can one define a kind of “Determinant” on reduced group $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra which is equipped with a faithfull trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...
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1answer
301 views

On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra

Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$. Obviously the singularities of this systems are just the idempotents of the ...
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a closed projection on a C*-algebra is compact iff it is closed on the multiplier algebra

I'm trying to understand the proof for the equivalence of (i) and (v) in the following picture. I don't quite understand what the highlighted sentence means. I want to know why there is a surjection ...
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109 views

On crossed product of L^{P} spaces

Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true \begin{align*} L^{p}(M\rtimes G, \varphi^{M\rtimes G})\...
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2answers
383 views

Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
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Existence of surjective map for TRO

Recall that a Ternary ring of operator(TRO) $V$ is a closed sub space of a $C^{\ast}$- algebra $A$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$ that’s every TRO $V$ comes with an embedding of $V$...
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Algebra structure on Haagerup tensor product of operator spaces

Let $A$ and $B$ be operator spaces. Is there any algebra structure on Haagerup tensor product of operator spaces such that the Haagerup tensor product becomes Banach Algebra? Any references or ideas?

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