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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projective module over C*-algebra [closed]

Suppose we have $V$ a projective module over C*-algebra $A$. Suppose we define another projective module $W$ which is same as $V$ but the action is given by: $v.a := v.\alpha(a)$, where $\alpha$ is ...
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114 views

A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras. Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...
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106 views

A question about correlations between $ C^{*} $-algebras

I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras. Definition. Let $ A $ and $ B $ be ...
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1answer
176 views

Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes [closed]

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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102 views

Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$. Ultimately, I'm interested in finding a ...
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94 views

Determining the primitive ideal space of C-star algebras

Is there a general way of finding a primitive ideal space of $C^*$-algebra? For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how ...
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1answer
51 views

Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold

Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$. This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...
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1answer
99 views

A $C^{*}$ algebra associated to a graded $C^{*}$ algebra

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ ...
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36 views

Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$ By the definition of the quotient ...
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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 ...
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143 views

A $C^{*}$ algebra associated to a group [closed]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...
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106 views

Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is $$\text{Tr}(P(A)) = ...
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211 views

fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...
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80 views

A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to ...
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105 views

Conjugacy of circle actions on UHF C*-algebras

Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that ...
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101 views

Product of two foliations

1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...
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Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
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110 views

A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map ...
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54 views

Hilbert c*-module over approximately finite c*-algebra

Is there a construction of a Hilbert c*-module over an approximately finite c*-algebra using Hilbert c*-modules over finite algebras? How do we get a Hilbert c*-module over an inductive limit of such ...
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1answer
68 views

strong convergence in Hilbert c* module

For a monotone sequence of projections in Hilbert c* module, do we have similar conclusion as in Hilbert space (such as strong convergence)? If not, what could be said in that situation?
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178 views

Connes' correspondences of two $L^\infty$-algebras

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and ...
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1answer
199 views

Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality $$ |\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...
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1answer
223 views

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element? Let $n(A)$ be the infimum of such ...
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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 are path connected. Is the tensor product of two path connected algebra, a path connected algebra? What ...
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75 views

A possible characterization of finite dimensional $C^{*}$ algebras

Is there an infinite dimensional unital $C^{*}$ algebra $A$ with invertible group $U$ such that for every finite dimensional subvector space $Y\subset A$, the number of connected components of ...
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1answer
154 views

Totally non hereditary $C^{*}$-subalgebras

Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any ...
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186 views

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 ...
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1answer
210 views

Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that $$ \int d s \, f(s)\, \alpha_s(A) $$ is well defined as a ...
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2answers
209 views

The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...
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207 views

The kernel of $C^{*}(G)\to C_{r}^{*}(G)$

Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism. What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
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187 views

Can the full and reduced group $C^*$-algebras be “noncanonically” isomorphic?

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...
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1answer
240 views

Inner and extendible automorphisms of C*-algebras

If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$. Is the converse true: if an automorphism ...
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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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1answer
168 views

When do infinitesimals split in dimension groups?

Let $G$ be a dimension group (i.e. a directed, unperforated abelian group satisfying the Riesz interpolation property) with order unit $u\in G^{+}$. There is a canonical positive group homomorphism ...
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How the modular theory of von Neumann algebras, deal with generating C*-algebras?

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 ...
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1answer
192 views

Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

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 $A''=M$. Let $p,q \in M_{\infty}(A)$ be ...
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The letters of the word “ART”

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is ...
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1answer
221 views

A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$. For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...
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1answer
559 views

When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: ...
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1answer
264 views

applications of C$^*$-algebras in the field of PDEs

I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
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1answer
191 views

A question on K- theory of non commutative $C^\star$ algebra

Edit: According to the comment of Andre Henriques I revise the question: What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative ...
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1answer
116 views

Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra

As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...
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1answer
121 views

Simple $Z^{*}$ algebra

What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
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A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...
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1answer
154 views

Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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1answer
164 views

A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$. To what extent ...
2
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1answer
230 views

Non commutative topological manifolds

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the ...
2
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2answers
284 views

If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? Pietro Majer ...
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1answer
81 views

A Possible characterization of F.D or AF commutative $C^{*}$ algebras

By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra. Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...
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108 views

A topological space extracting from a group action

Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to ...